Directional transience implies ballisticity

Proof. Uniform ellipticity implies the strict ellipticity assumed by the companion’s directional zero–one law. It therefore upgrades P_0(A_\ell)>0 to P_0(A_\ell)=1. Since A_\ell=A_{\ell/\|\ell\|}, Theorem 1.1 gives the stated velocity and positive projection.

Fix this vector v and a direction u\in S^{d-1}. If v\cdot u>0, the vector law of large numbers gives X_n\cdot u/n\to v\cdot u>0 almost surely, hence P_0(A_u)=1. Conversely, if P_0(A_u)>0, the preceding argument applied to u gives a deterministic vector w with X_n/n\to w almost surely and w\cdot u>0. The two probability-one convergence events intersect, and uniqueness of a vector limit gives w=v. Thus v\cdot u>0. These two implications prove the equality of direction sets and the assertion on the closed opposite hemisphere.□

Proof. The global height minimum is attained almost surely. Thus for some deterministic n,x the event that X_n=x and that all later heights are at least h(x) has positive probability. The quenched Markov property at n gives

Since the first factor is at most one, translation invariance gives p_h>0.

At a first strict record, every earlier departure row has smaller height. The fresh-row rule therefore gives conditional raw probability p_h that the record is true. Test the first positive-time strict record. If an actual drop later detects its failure, test the next strict record above the whole past observed by that failure time. All candidates and failure detections are path stopping rules. Transience supplies another candidate after every finite failure, and each candidate has success probability p_h. Thus a true record is eventually found almost surely.

For the iid assertion, enumerate the possible first words. A word from 0 to z has positive length, every preterminal height in [0,h(z)), and every intermediate strict record invalidated by a drop before its end. Together with a suffix staying at or above h(z), these conditions say exactly that this is the first word under D_h. Indeed a later suffix above h(z) cannot invalidate any earlier lower record. The prefix weight and the suffix factor p_h use disjoint departure rows. Dividing by the initial conditioning probability p_h leaves the raw prefix weight as the word probability and an independent translated suffix of the original conditioned law. Countable enumeration and iteration give the iid words and infinitely many true boundaries. For the first raw true record the same argument applies, without the restriction that its prefix have nonnegative height. This argument does not apply an annealed Markov property at a future-dependent cut.

Under D_h, a fixed positive ordinary record index i is true with probability equal to the raw probability of reaching that record before dropping below zero: the fresh suffix contributes p_h, which cancels the conditioning. This probability is at least p_h, since all record indices are reached on D_h. The true indices are a renewal process with iid increments S_*. Its positive-index count through m, divided by m, converges almost surely and in expectation to 1/\mathbb E S_*. This remains true with limit zero when the mean is infinite, by the strong law on truncations; the ratios are bounded by one. The preceding lower bound on each indicator therefore yields \mathbb E S_*\le p_h^{-1}. Each ordinary record increases height by at most one, so L\le S_*. This argument uses record indices, not a lattice structure for the values of h.□

Proof. Suppose instead that \mathbb E R=\infty, and put I(s)=\mathbb E\min(R,s). This function is increasing, concave, unbounded, and satisfies I(s)=o(s), since R is finite almost surely. Write H_k=\sum_{j=1}^kL_j, with H_0=0.

A family of likely finite templates. The width strong law gives constants c>0, C, and C_0 such that

holds with probability greater than .95. Fix an integer C_1 with C_1c>C+4, then b>1/c, and then \eta>0 so small that 4C_1\eta<.1. For large a, choose the least positive integer N with I(aN)\ge\eta a. Then N\to\infty, and minimality and concavity give

for all sufficiently large a.

With probability greater than .8, the first C_1N words satisfy the height bounds above and

Here is the maximal estimate behind this assertion. Truncate each radius at aC_1N. Its expectation is at most C_1I(aN)<2C_1\eta a. The union bound for a truncation among the first C_1N radii is therefore at most 2C_1\eta. For a stationary nonnegative sequence (Z_j),

To see this, mark the starts among 1,\ldots,M admitting a witness interval of length at most n with average greater than a. Select such intervals greedily from left to right. They are disjoint and cover all marked starts, so their total sum is greater than a times the number of marked starts. The selected intervals lie in 1,\ldots,M+n. Take expectations, divide by M, and let M\to\infty to remove the boundary contribution. Applied to the truncated radii, this costs at most another 2C_1\eta. Together with the height event this proves the asserted .8 bound. This is the nonnegative-sequence form of Hopf’s maximal ergodic inequality; see Garsia [9].

The polynomial lower bound. Fix a large B>1. Choose i_0 so that

Choose a sufficiently large that N\ge i_0 and I(Bai_0)<\eta a/4. Tail integration, followed by removal of the width exceptions, gives

For the lower bound, the unrestricted sum is at least [I(Ba(N+1))-I(Bai_0)]/(Ba)\ge3\eta/(4B); for the upper bound it is at most I(BaN)/(Ba)\le I(aN)/a.

Count the indices i at which this large-word event occurs and the preceding words satisfy the height and radius bounds through i-1. Independence gives mean at least .8\lambda. Dropping the preceding-word restrictions bounds the second moment by \lambda+\lambda^2. The count is therefore positive with probability at least c_2/B, where c_2>0 does not depend on B or a. At some point of that word,

The height bound holds throughout this prefix because every word ends at a strict record and has no earlier height above its endpoint. One of the 2d signed coordinate vectors u thus satisfies

for a fixed c_3>0 and sufficiently large fixed B. Consequently, under the raw law the union of these events over u has probability at least p_hc_2/B, before the path leaves \{h\ge0,\ M\le(C+2)N\}.

A fresh continuation caps the ratio. Fix u and put Q_n=Y(X_n)/[a(1+M_n)]. Let \zeta be the exit time just described. We claim that constants \epsilon_0>0 and K<\infty, independent of B,a and the threshold t>0, have the following property: at every finite prefix ending at the first crossing \sigma<\zeta of Q>t, a continuation of conditional raw probability at least \epsilon_0 exits the strip before Q exceeds t+K.

At the crossing,

Thus the arrival is a strict Y-record. A step changes Y by at most ba+1, so Q_\sigma\le t+b+1 when a\ge1. Choose an integer M_0'>C_0 with M_0'b>1+bC_0. First prescribe M_0' steps of height +1. Each increases Y by at least ba-1>0. At their endpoint impose a translated D_h-suffix whose first C_1N words satisfy the template above. In its word k+1, every site has Y-increment from the launch at least

This holds uniformly in 0\le k<C_1N for large a: the coefficient of k is a(bc-1)>0, and the constant term is positive by the choice of M_0'. These words therefore avoid every old departure row. Their terminal true cut has height at least

The entire remaining suffix stays above this cut, and so also avoids all old departures. The template stays above its own starting height, which is strictly above every script departure. Thus the whole infinite continuation uses no old departure row: first the Y barrier provides separation, and then the terminal height does. The fresh-row factorization gives probability at least

Figure 1 records these two separation mechanisms.

It remains to bound the ratio on this continuation. The positive part of the launch coordinate u\cdot X_\sigma, divided by any later denominator, is at most t+b+1, since h(X_\sigma)\ge0 and the denominator is nondecreasing. The later height term contributes at most b. The script adds at most M_0' to the ratio. In word k+1, the additional coordinate contribution is at most

The numerator uses the bound on the sum of radii through word k+1; the denominator uses the height of that word’s starting boundary. Thus any fixed

proves the claim. Exit occurs by the last prescribed template endpoint.

The exponential upper bound. Let \sigma_j be the first crossing before \zeta of t_j=1+jK. The continuation just constructed prevents \sigma_{j+1}<\zeta. Summing its bound over the finite prefixes at \sigma_j yields

This conditions only on the observed crossing prefix; it does not reveal whether an infinite template has failed. Hence the event in (3) has probability at most C'e^{-c'B} for this u. Taking the union over signed coordinates contradicts the lower bound p_hc_2/B when B is large. All estimates use the permitted order: fix the geometry constants, then B, then i_0, then take a large. This proves (1).□

Proof. Word endpoints obey the vector strong law, with mean displacement w satisfying h(w)=\mathbb E L>0. Proposition 2.2 also gives R_k/k\to0 almost surely: for each \varepsilon>0, the probabilities \Pr(R_k>\varepsilon k) are summable. Thus interpolation inside each word gives the same limiting direction r=w/\|w\|, without any duration moment. The completed-word count tends to infinity. Lemma 2.1 transfers the conclusion through the finite initial raw prefix.

Some signed coordinate of r is positive. Relabel it as the first coordinate, preserving the lattice, uniform ellipticity, and the iid row law. Then (X_n)_1\to+\infty almost surely. The proofs of Lemma 2.1 and Proposition 2.2 apply again with this coordinate height, using only its transience. We retain the same notation for the transformed walks and laws.□

Proof. Uniform ellipticity excludes confinement forever in a fixed finite-height strip, even quenched: blocks of sufficiently many consecutive upward steps have a fixed positive chance. Hence a_H(x)\downarrow q(x). Also q(je_1) uses only layers at or above j, so the event that q(je_1)>0 infinitely often is a tail event of the independent layers. For every m, the union of these events over j\ge m has probability at least P(q(0)>0)>0, by stationarity and Eq(0)=p>0. The tail zero–one law makes their limsup almost sure. A finite upward script connects zero to one such site and gives q(0)>0. Translation and countability give positivity at every site simultaneously.

At layer H, let \mu_H=K_H(0,\cdot)/a_H be the endpoint probability distribution. It is \mathcal F_H-measurable, and the quenched Markov property, restricted to a suffix never below layer H, gives

Each q(x) on that layer has the original marginal law independently of the lower rows. If q(0)/a_H<s, then more than half the \mu_H-mass has q(x)\le2s. Conditional expectation followed by Markov’s inequality therefore gives

The endpoint values q(x) need not be independent of one another. Partitioning by a finite stopping layer gives the same conditional bound there.

Choose 0<\delta<\kappa sufficiently small that \rho:=2P(q(0)\le2\delta/\kappa)<1, and set

These are stopping layers. One upward step after a successful crossing gives a_{H+1}\ge\kappa a_H, so on H_j<\infty, a_{H_j}\ge\kappa\delta^j. The event H_{j+1}<\infty implies q(0)<\delta^{j+1}, and hence q(0)/a_{H_j}<\delta/\kappa. The preceding conditional bound yields

Since a_H\downarrow q(0), the event H_j<\infty is exactly \{q(0)<\delta^j\}. Iteration gives its probability at most \rho^{j-1}. Summing over \delta^{j+1}\le q(0)<\delta^j, and choosing \lambda>0 with \rho\delta^{-\lambda}<1, proves (4).□

Proof. All conclusions below using (5) are conditional on that estimate.

Spatial confinement before exit. The raw path before leaving the height interval [0,N] stays in \|\underline X\|_\infty\le CN^2, except with probability at most Ce^{-cN}. To prove this, fix a signed lateral coordinate u. By Corollary 2.3, a sufficiently large fixed b>0 makes Y(x)=u\cdot x+bx_1 transient in its positive direction. Thus the event of staying forever at Y\ge0 has positive raw probability, by the minimum argument of Lemma 2.1.

For a sufficiently large fixed C_2, the path through its first relative height hit N+1 has norm at most C_2N with probability tending to one. Indeed the ray gives \|X_n\|\le(C_2/2)(X_n)_1 eventually, and the earlier path has a finite maximum norm. The intersection of this event with the preceding no-backtracking event therefore has probability bounded below uniformly for large N.

At a first crossing of Y>jC_3N, before strip exit, the walk is at a strict Y-record. Impose that favorable translated suffix using the fresh rows at or above the record. If C_3 is sufficiently large, the suffix precludes crossing (j+1)C_3N before strip exit: its Y-displacement through relative height N+1 is at most (1+b)C_2N, the launch overshoot is bounded, and absolute strip exit occurs no later than this relative height hit. The finite-prefix threshold recursion used in Proposition 2.2, now with spacing C_3N, gives exponential decay through N thresholds. Since u\cdot X\le Y before exit, the finite union over signed lateral coordinates proves the confinement assertion.

Super-polynomial width tails.

Now \Delta u_N:=u_N-u_{N+1}\ge0 is the annealed probability of reaching N before the drop but dropping before N+1. In environments good at all sites of the strip and box (where a_{N+1}(x)\ge(N+1)^{-\beta}), the portion of this event inside the box has quenched probability at most

Indeed, after reaching N, at first descending-level visits to each of N,N-1,\ldots,0 still in the box, there is at least the indicated chance to hit top N+1 before the next descent by one. Condition (5), translation, a polynomial union bound, and the box claim thus give super-polynomial decay of \Delta u_N.

The true levels under P_0^+ form the renewal process of iid widths L_{\rm s}, with renewal masses exactly u_n: impose the fresh D-suffix at the first reach of n before drop, and cancel p. If G_j=P_0^+(L_{\rm s}>j), then \sum_{j=0}^n G_j u_{n-j}=1 by the last renewal up to n, so

Put f_k=\Delta u_{k-1} for k\ge1. This nonnegative kernel has total mass \theta=1-p<1 and all polynomial moments. The recurrence, with G_0=1, gives G=\sum_{r\ge0}f^{*r}, including unit mass at zero for r=0. If \theta>0, normalize f by \theta. For every positive integer m, the m-th moment of its r-fold convolution is at most r^m times its m-th moment, by (z_1+\cdots+z_r)^m\le r^{m-1}\sum_i z_i^m. Summing with the geometric weights \theta^r proves \sum_n n^mG_n<\infty. The case \theta=0 is immediate. In particular the width tail G_n decays super-polynomially.

Time integrability and velocity. The raw exit time T_N\wedge T_{-1} exceeds N^A with super-polynomially small probability, for a sufficiently large fixed A. To see this, use confinement and (5) on a slightly enlarged box and strip, requiring a_N(x+e_1)\ge N^{-\beta} for every relevant site with 0\le x_1<N. A polynomial union bound controls the exceptional environments.

From any visit to such a site x, the quenched probability of no further visit before exit is at least \kappa a_N(x+e_1): take one upward step and then advance N before dropping below the new layer. This event forces original strip exit before revisiting x, even though the auxiliary target may be higher than N. The quenched Markov property consequently bounds each site’s visit count by a geometric tail with success probability at least \kappa N^{-\beta}. There are at most CN^{2d-1} sites in the confined box. If the exit time exceeds N^A, one site has at least N^A/(CN^{2d-1}) visits. For A>2d-1+\beta, the union of these geometric-tail events is super-polynomially small, as are the bad-box and bad-environment events.

On D, if L_{\rm s}\le N, the first word ends at its first arrival to height L_{\rm s}, and hence \tau_{\rm s}\le T_N. Thus

Both terms have super-polynomial decay, proving \mathbb E\tau_{\rm s}<\infty. The iid-word strong laws now give the velocity as mean displacement divided by mean duration; its first coordinate is \mathbb E L_{\rm s}/\mathbb E\tau_{\rm s}>0. The integrable spatial radius controls the inter-endpoint deviations, so this limit holds at every path time. The finite raw initial prefix does not change it. Finally the velocity is a positive multiple of the ray in Corollary 2.3, which has positive projection on the original \ell. Undoing the signed coordinate permutation proves the stated conclusion. As an almost-sure deterministic limit, this velocity is unique for the law \nu.□

Proof. Work first with the raw pair laws. Test a new integer level above both explored path pasts, pausing each walk at its first arrival there. All departure rows revealed by these prefixes lie strictly below the candidate level. The conditional probability that both arrivals are true is p^2 in independent environments and p_2 at the arrival sites in a shared environment. Both probabilities have a uniform positive lower bound.

If the candidate fails, resume the two walks, for example one step of each per round, until one drops below the candidate. This failure is detected in finite time. Then test a level strictly above both explored pasts. Each walk tends to positive infinite height almost surely; this also holds for the shared raw pair by its marginals. Thus, after every detected failure, another candidate can be reached. The uniform success bound proves the existence of a common true level above any prescribed height. Conditioning on the two initial no-drop events preserves this almost-sure conclusion.

For the continuation law, specify a pair of finite words from x,y to their first hits x',y' of a common terminal level L>0. Call this word pair admissible when both prefixes avoid the initial drop and every intermediate positive level has a witnessed failure of common truth: at least one prefix has dropped below that level after its first hit. These are exactly the required prefix conditions. Indeed, once truth is imposed at L, any failure of truth at an earlier level must already have occurred in the specified prefixes.

Let A_{x,y}^{\rm sh}(w_1,w_2) be the raw annealed prefix weight of an admissible word pair. Its quenched weight is the product of the two path weights in the shared environment, including any repeated use of a row. Every departure row lies below the terminal layer. The suffixes constrained by truth use only rows at or above that layer, so site-row independence gives the conditional word probability

More generally, for any event B of the suffix pair, its joint probability with this word pair is

Here B is read with the new sites as starting sites. In particular, the factor p_2(x',y')/p_2(x,y) is part of the shared word law; it does not cancel in general.

In independent environments the corresponding terminal and initial conditioning probabilities are both p^2, so their ratio is one. Translation of each walk separately then makes the relative suffix law P_0^+\otimes P_0^+, independent of the word just traversed. Enumerate the countably many admissible finite word pairs and iterate these factorizations to obtain the stated continuation rules. Summing over a finite stopping index gives the same rules there. The information at such an index does not include any unseen upper environment.□

Proof. The probability that a deterministic level h is true under the single conditioned law is u_h, by the exact-truth identity in Section 2. Independence therefore makes the renewal mass for common words equal to u_h^2\ge p^2. The renewal-count strong law, including its infinite-mean version obtained by truncation, implies \bar L\le p^{-2}. The lower bound follows from L\ge1.

For each of the two paths, let N_i be the number of its single words used before the first positive common boundary. Since single widths are positive integers, N_i\le L. The event N_i\ge j is determined by that path’s first j-1 words and the independent boundary sequence of the other path. It is consequently independent of the radius of its j-th word. The finite mean single-word radius from Section 2 gives

The absolute value of S is bounded by the sum of these two total radii. Finally, uniform ellipticity permits two short paths with different second-coordinate displacements to level one. Imposing fresh truth there gives positive probability of an exact common cut with nonzero S.

We record the renewal estimate that supplies local tightness. For either the single renewal sequence or the independent common sequence, let L' be its word width and A any almost surely finite size of the word, such as its maximal displacement. If A_{\rm word} is the size of the word based at the last boundary at or below a deterministic level h, then

To obtain the first inequality, sum over the possible preceding boundary levels h-j, use independence of the next word, and bound each renewal mass by one. Since \mathbb E L'<\infty, the final quantity tends to zero as s\to\infty. The same reasoning with A=L' gives tightness of the covering width. Thus first-hit positions at a deterministic layer differ from their preceding boundary positions by tight amounts. A fixed interval of integer levels meets only the words covering those levels, so it also has tight total displacement, uniformly in its base. Shared marginals inherit this statement from their uniform domination by the single conditioned law.

It remains to prove (6). Test the m levels

Call failure at h visible if, after first hitting h, some walk drops below it before its first hit of h+B. In the raw law, the probability that all m tests have visible failure is at most (1-\underline p_2)^m. Indeed, at each pair of first-hit arrivals to h, all previous visible-failure tests are already determined, while both fresh truths at h would preclude the next failure. Passing to the conditioned pair law costs at most \underline p_2^{-1}.

If h is not common true but has no visible failure, some walk drops below h after its first hit of h+B. That walk has no single true boundary between h and h+B, inclusive; its word covering h therefore has width greater than B. The covering-word estimate and marginal domination bound this alternative by

A union bound over the m tests now gives

Choose m large and then B large. This proves uniform tightness of the height overshoot. The fixed-interval displacement bound then proves the other assertions in (6). Finally, the first common boundary at or above H is a stopping index of the common-word chain, so its suffix rule follows from Lemma 3.1.□

Proof. The ratio m(r)/r^2=\mathbb E[(S^2/r^2)\wedge1] is continuous and nonincreasing. There is a bounded interval of strictly positive values of |S| having positive probability; its contribution is strictly decreasing once r exceeds that interval. This proves the monotonicity and eventual strict monotonicity of n. Monotonicity of m proves the first inequality in (7), and S^2\wedge r^2\le r|S| proves the second, hence n(r)\to\infty.

For (8), replace S_j by S_j\mathbf1_{\{|S_j|\le z\}}. These truncated variables have mean zero by symmetry. The probability of any replacement among the first H variables is at most H\Pr(|S|>z). Off this event the original partial sums agree with the truncated sums, whose expected squared maximum is at most 4H\mathbb E[S^2\mathbf1_{\{|S|\le z\}}] by the square maximal inequality. On the replacement event, the capped squared maximum is at most z^2. Adding these bounds gives the first inequality in (8); Markov’s inequality for the capped square gives the second.□

Proof. In a raw first walk, select the first level h\le H, if any, such that T_h<T_{-1} and |Y_h-b_h|>z. Let A be the event that such a level exists. Conditional on this prefix and its selected level, an independent raw second walk reaches h before dropping, with its displacement on the opposite side of the median b_h, with probability at least p/2. Here the conditioned no-drop law only lower bounds a prefix probability; it is not imposed as a separate test on that second walk.

At the two first-hit endpoints, impose both fresh truths. Their probability factor is p^2. The resulting event implies the two initial no-drop events and has h as an exact common true level, with |\Delta_h|>z. Dividing its raw probability by p^2, its probability under the independent conditioned pair law is at least P_0(A)p/2. At a common cut of height at most H, the difference is a partial sum of at most H common-word increments, so

This proves (9). All of these adaptive prefix transfers can also be obtained by summing the corresponding finite stopped-word identities separately over the possible levels h.□

Proof. By (7)–(9), choose a fixed large C_0, then t small enough that C_0\sqrt t\le1, to obtain

For i+j\le2H, the event that i is exactly true under P_0^+ has probability u_i\ge p, and its suffix has the fresh conditioned law. There is therefore positive probability on this event that Y_i, Y_{i+j}, and Y_{i+j}-Y_i are all within C_0\sqrt t\,r of b_i, b_{i+j}, and b_j, respectively. For the first two requirements use the preceding maximal bound, and for the third use the same bound on the fresh suffix. Consequently,

uniformly in the indices.

Set d_j=b_j-jb_H/H for 0\le j<H. Apply the last relation to j+j, and, when 2j\ge H, also to H+(2j-H). This gives

with a uniform error. Taking the maximum absolute value over 0\le j<H bounds that maximum by half itself plus C\sqrt t\,r. Since the error at j=H is zero, this proves (10).□

Proof. On the joint environment-and-path space,

This density is integrable but need not be bounded. For any prescribed error, truncate it and then approximate the truncation by a bounded nonnegative environment function g depending on finitely many rows. Thus g approximates p/q(0) in L^1(P_0^+), and E_0^+g tends to one as the approximation error tends to zero. We will remove the finite initial segment that can be affected by g.

First check the deterministic centering error caused by such a removal. If \limsup P_0^+(W_H>z/10)\ge0.3, Equation (11) is immediate with a sufficiently large fixed C. Otherwise these probabilities are eventually less than 0.3. For every fixed integer s_0, local tightness then gives

for all sufficiently large indices in the sequence. Indeed, each of |Y_h-b_h|\le z/10 and |Y_{h+s}-b_{h+s}|\le z/10 has probability greater than 0.7. By Lemma 3.2, the probability of |Y_{h+s}-Y_h|\le z/10 tends to one uniformly over these indices. The three events have a nonempty intersection, which proves the claimed deterministic inequality by the triangle inequality.

Let s be the first positive single true boundary strictly above every row used by g. Its index in the single-word chain is a finite stopping index. For a sufficiently accurate approximation, E_0^+g>0. Under the probability measure with density g/(E_0^+g) relative to P_0^+, the translated suffix at s has law P_0^+, independently of the prefix. To verify this, apply exact-word factorization separately to each finite prefix through the selected boundary. Both its path weight and g use only rows below that boundary; all rows for the conditioned suffix are fresh. This remains true for rows used by g that the prefix has not visited. The separation is by height, not by which rows the walk happened to reveal.

For fixed g, first choose a deterministic s_0 so large that s\le s_0 except for an arbitrarily small g-weighted error. As z\to\infty, we may further restrict to

with a further error tending to zero. On these restrictions no offense |Y_j-b_j|>z can occur before s. After s, write the relative suffix displacement as Y'_u=Y_{s+u}-Y_s. The centering estimate above gives, for s<j\le H,

Thus W_H>z requires the fresh suffix to have its own W_H>z/10. If H\le s, no offense remains at all.

The g-weighted probability is consequently bounded in the limsup by E_0^+g times \limsup P_0^+(W_H>z/10), plus the restriction errors. Return to \widetilde P_0 using the L^1 density approximation, and send all approximation and restriction errors to zero. Enlarging the same fixed C to include the earlier trivial case proves Equation (11).

Finally, in a fixed environment the event D is a subevent of \{T_H<T_{-1}\}. On first-hit prefixes, changing between these two conditional laws costs total variation at most 1-q(0)/a_H. Therefore the distance between \widetilde P_0 on those prefixes and the P-average of K_H/a_H is at most

since a_H\downarrow q(0)>0 almost surely. This proves the last assertion.□

Proof. Fix v_0>0, set w_0=v_0/8, and write \theta=b_H/H. The proof uses one estimate for loss of mass in a narrower slope tube, then restarts that estimate after its first failure. All stopping tests will use finite-height kernels.

One possible loss of mass. Choose t_{v_0} small enough that Equation (10) gives, for sufficiently large r, uniformly over the stated H,

The second inequality follows from the first at j=1, since b_1 is fixed. Equations (7)–(9) and (11) also give

uniformly in 1\le H\le tn(r) for large r. Indeed, a violating sequence of H,r would contradict the limsup transfer in Equation (11), whose right side is bounded by a constant times H/n(r). The scale inequality in Equation (7) absorbs the fixed change in threshold.

Choose \delta_0\in(0,1/4) so that P(q(0)<4\delta_0)\le t. Call a starting site x favorable when

The expected \pi-fraction of unfavorable sites is at most C_{v_0}t. Markov’s inequality shows that, except on an environment event of probability at most C_{v_0}t, favorable sites carry at least half the initial mass. From each favorable site the raw mass of paths on D in the median tube is at least 2\delta_0. Their first-hit prefixes lie in the slope tube of width w_0r at every shorter horizon. Hence, except on that one environment event,

In particular the probability that any of these inequalities fails is at most C_{v_0}t, uniformly in the initial probability \pi. No independence among the environments seen from different starting sites was used.

A fresh restart after the first failure. Test the masses M_u^\pi in increasing order, and let s be the first level at which the displayed inequality fails. There is no failure at level zero, since M_0^\pi=1. Each test is measurable from rows strictly below its tested layer, so s\le H, when finite, is an environment stopping layer.

The passing submeasure at layer s-1 has mass at least \delta_0. Extend every one of its prefixes by one direct +e_1 step. Uniform ellipticity leaves mass at least \kappa\delta_0. The extension has slope error at most 2w_0r, since the earlier error is at most w_0r and |\theta|\le w_0r. Both detection of the failure and this bridge use only rows below layer s.

Normalize the bridged endpoint submeasure to a probability \pi_s and restart at layer s. Conditional on the exposed lower rows, \pi_s is fixed and the upper field has its original product law. The one-failure estimate therefore applies again. Use the same slope \theta=b_H/H and its original horizon H; its simultaneous conclusions imply the required ones through the shorter remaining horizon H-s. Replacing H by H-s in the slope would not be justified and is unnecessary.

Thus, conditional on the first failure, the probability of a second failure before the remaining horizon is at most C_{v_0}t. The probability of two failures is at most C_{v_0}t^2, after enlarging the constant. This multiplication uses fresh upper rows at the stopping layer, not independence of two events in the original slab.

Retained mass and its tube. If no first failure occurs, at least \delta_0 of the original kernel mass reaches the top in the slope tube. If there is a first failure but no second one, concatenate the bridged mass and the restarted successful prefixes. Retain each prefix’s original starting site when doing so; restarted displacements are measured from the new endpoint. The total slope error is at most 2w_0r+w_0r=3w_0r, and the retained mass is at least \kappa\delta_0^2. Local no-drop successes are also successes above the original floor, so the concatenated kernels are submeasures of the original K_H.

Adding the deterministic median-line error w_0r/2 puts these prefixes in the required median tube as well. Both tube widths are less than v_0r. Take \delta=\min\{\delta_0,\kappa\delta_0^2\} to obtain Equation (12). The infinite no-drop event served only to bound failure probabilities; every test, bridge, and restart used the finite-height prefix kernels.□

Proof. Fix s>0, put H=\lfloor sn(r)\rfloor, and let k=\lfloor H/\bar L\rfloor. First we justify the approximation

By Lemma 3.2, the displacement from the preceding common boundary to the first-hit position at H is tight. The renewal strong law puts the number of completed common words within o_{\Pr}(n(r)) of k. No independence between this count and the increments is required: on the event that its distance from k is at most an(r), bound the sum error by the maximal sums in the deterministic forward and backward windows of that length around k. For every fixed \rho>0, (7)–(8) bound the probability of such an error exceeding \rho r by O_\rho(a)+o(1). Let a\downarrow0, and include the tight decoration error, to obtain the approximation.

For s=1, set

Its mean is comparable to r^2, since k\sim n(r)/\bar L, and

The second-moment inequality therefore makes V exceed a fixed positive multiple of r^2 with uniformly positive probability. Conditional on the magnitudes (|S_i|), the nonzero signs are independent fair signs. The sum \sum_{i=1}^kS_i has conditional variance Q=\sum_{i=1}^kS_i^2\ge V and conditional fourth moment at most 3Q^2. A further second-moment inequality gives a uniformly positive conditional probability of a displacement of order \sqrt Q, hence of order r on the preceding event. The approximation, with a smaller displacement constant, proves the first line of (13).

For the second line, take s=t and set p_r=\Pr(|S|>lr). The assumptions and the definition of m give

Among k\sim tn(r)/\bar L summands, the probability that exactly one has magnitude greater than lr is therefore at least c_{l,\epsilon}t for each sufficiently small fixed t>0 and large r. Conditional on the index of that exceptional summand, the others are independent symmetric variables conditioned to have magnitude at most lr. Their sum has variance at most

For t sufficiently small, Chebyshev’s inequality gives probability at least 1/2 that their sum has magnitude at most lr/2. On this event they cannot cancel more than half the exceptional summand. Thus the deterministic sum has magnitude greater than lr/2 with probability at least a constant times t. Keep t fixed while r\to\infty; the o_{\Pr}(r) approximation then proves the claimed bound at threshold lr/4.

Finally, if two iid endpoint displacements differ by more than a, at least one differs from any given deterministic center by more than a/2. The probability of this single-walk absolute deviation is at least half the two-walk separation probability. One of its two signs has at least half that mass. This proves the final assertion after adjusting constants.□

Proof. Write (L_i,S_i) for the successive independent common-word widths and difference increments, and put

The difference at the preceding common cut is \widehat\Delta_h=A_{N(h)}. As with the first-hit process, we interpolate its values linearly on the height grid h/n(r).

The limit at common cuts. First consider A_m/r, with n(r) summands per unit time. The Gaussian-sequence assumption permits a symmetric truncation

for which

Indeed, for every fixed 0<\eta<1,

a sufficiently slow diagonal choice of \eta_r gives both requirements. The probability that any summand is removed on a fixed time interval tends to zero.

The variables S_i^{(r)}/r have mean zero, variance (1+o(1))/n(r), and fourth moment at most \eta_r^2 times that variance. Taylor expansion of their characteristic functions therefore gives the standard Brownian finite-dimensional laws. For tightness, the fourth-moment computation for independent centered sums, followed by the fourth-moment maximal inequality, bounds the expected maximal fourth power on a block of \lfloor\delta n(r)\rfloor summands by

A union bound over the blocks in a fixed time interval, followed by \delta\downarrow0, proves tightness. Thus A_m/r has the standard Brownian functional limit.

The width strong law gives

Applying this time change to the tight sum process, whose limits are continuous, proves the Brownian limit for \widehat\Delta_h/r, with variance parameter 1/\bar L. The counts and the sums need not be independent. The maximal normalized jump tends to zero as well, so the stated interpolation convention has the same limit.

Passing from cuts to every first hit. Lemma 3.2 makes \Delta_h-\widehat\Delta_h tight uniformly at deterministic levels h. This already gives the required finite-dimensional limits. To prove path tightness, fix \delta>0 and partition the height range into consecutive blocks of \lfloor\delta n(r)\rfloor levels. There are only finitely many block starts for fixed T,\delta.

Suppose that within one of these blocks, with start a,

for some level h in the block. Under the raw independent pair law, select the first such offending level among prefixes for which both walks have avoided the initial drop. The selection is determined by the paired prefixes through that level. The raw probability of finding an offense is at least p^2 times its probability under the independent conditioned pair law. Imposing both fresh truths at the selected level multiplies each selected prefix weight by p^2, leaves the offense unchanged, and implies both initial no-drop events. After division by the initial normalizer p^2, both probabilities below are under the conditioned pair law, and

The same statement holds for the union of the blocks by selecting its first offense.

At a common cut h, the offending difference satisfies

The maximum of the second term over the finitely many deterministic block starts is o_{\Pr}(r). The first term is controlled by the modulus of the common-cut process, which has a continuous Brownian limit. Consequently, for each \rho>0, the limsup probability of an offense tends to zero as \delta\downarrow0. Adjacent-block bounds and interpolation give tightness of \Delta_h/r, proving the first assertion.

Recovering a single path. Let \mu_r be the law in continuous path space of the interpolated single-path process Y_h/r, on a fixed compact time interval. Its iid differences are tight by the first assertion. We first show that \mu_r is tight after suitable deterministic path translations. Choose compact sets K_j capturing the difference laws with errors \varepsilon_j^2, where \varepsilon_j\downarrow0 and \sum_j\varepsilon_j<1. If a prospective center f is sampled from \mu_r, then

Markov’s inequality shows that the centers for which \mu_r(f+K_j)<1-\varepsilon_j have \mu_r-probability at most \varepsilon_j. Some center path f_r therefore works for every j, and the translated laws are tight. We may choose f_r with the same interpolation as the original paths.

Along a subsequence on which these translated laws converge, let G be the continuous limiting process. Its iid two-copy difference has the Brownian law just proved. Cramér’s normal decomposition principle [6], in the iid-difference form verified below, applied to every scalar linear combination of finitely many coordinates of G, shows that G has Gaussian finite-dimensional laws. Its covariance is half that of the difference process, namely

Only a deterministic mean function remains to be identified.

For completeness, suppose that Z-Z' is normal for iid real variables Z,Z'. Choose C_0 with \Pr(|Z'|\le C_0)>0. Intersecting either tail of Z with this event bounds that tail by a constant times a Gaussian tail. Hence f(z)=\mathbb E e^{zZ} is entire and obeys |f(z)|\le\exp(C(1+|z|^2)). The identity f(z)f(-z)=\mathbb E e^{z(Z-Z')}, first on the real axis and then everywhere, shows that f has no zeros. Its entire logarithm g, chosen with g(0)=0, satisfies \operatorname{Re}g(z)\le C(1+|z|^2). On every circle the mean of \operatorname{Re}g is zero, so its absolute integral mean is also O(1+|z|^2). The Fourier coefficient formula on circles of arbitrarily large radius then makes every Taylor coefficient of g of order greater than two vanish. Thus g is quadratic, real on the real axis, with g''(0)=\operatorname{Var}Z\ge0. It follows that Z is normal, allowing a point mass.

Write m_G(t)=\mathbb E G(t). Sample continuity and convergence in law of Gaussian marginals at convergent times imply that m_G is continuous. Each marginal has the unique median m_G(t), including at time zero. At a grid time h/n(r), the number

is a median of the translated marginal. Along every sequence of grid times converging to t, weak path convergence and uniqueness of the limiting median make these medians converge to m_G(t). Compactness of the time interval makes the convergence uniform over grid times; interpolation gives uniform convergence on the whole interval. Use a slightly longer interval when interpolating at the endpoint. Subtracting the interpolated medians from the original process therefore removes precisely the limiting mean function. Every subsequential limit is Brownian motion of variance parameter \sigma_*^2, proving the single-path assertion.□

Proof. We first prove the additive relation. It suffices to consider arbitrary sequences of indices for which k/n(r)\to s and j/n(r)\to t, and to pass further so that k is fixed or tends to infinity. Let \mathcal C_k be exact single truth at level k. Under P_0^+, this event has probability u_k\ge p. Conditional on \mathcal C_k, the prefix has the raw law conditioned on T_k<T_{-1}, and its relative suffix is an independent fresh P_0^+ path.

If k\to\infty, the total variation distance between this prefix law and the P_0^+ prefix law is at most 1-p/u_k\to0. If k is fixed, its displacement and median are fixed random and deterministic quantities, respectively, and their normalized difference tends to zero. Lemma 4.2 therefore gives, under the conditional law,

The same lemma gives this normal limit under the unconditional P_0^+ law when the centering is b_{k+j}.

Set d_r=(b_k+b_j-b_{k+j})/r. The unconditional distribution centered by b_{k+j} dominates p times the conditional distribution with that same centering. Tightness first forces d_r to be bounded. Along a further subsequence with d_r\to d, the limiting domination would read

as measures. For v>0, a nonzero shift makes the ratio of the shifted normal density to the unshifted one unbounded. For v=0, the two point masses have different supports unless d=0. Thus d=0 in every case. Compactness of the ranges of k/n(r) and j/n(r) proves the uniform relation (15).

To turn approximate additivity into a linear approximation, put N=N_r, \theta=b_N/N, and d_j=b_j-j\theta for 0\le j<N. Apply (15) to j+j, and also to N+(2j-N) when 2j\ge N. Uniformly over these indices,

The maximum of |d_j| is therefore at most half itself plus o(r), so it is o(r). For h\le Tn(r), write h=qN+j, where q remains bounded, and apply (15) a bounded number of times. This proves \max_{h\le Tn(r)}|b_h-h\theta_r|=o(r). Finally, at h=1 this gives |b_1-\theta_r|=o(r); since b_1 is fixed, \theta_r=o(r).□

Proof. Relative to the single conditioned environment-and-path law from x, the first marginal has density

For a fixed integer J, replace q(y) by a_J(y), then normalize. The resulting probability law differs from the original one in total variation by at most C(u_J-p), uniformly in x,y. Indeed a_J\ge q, and the added mass before normalization is

All these normalized laws have density bounded by a common constant relative to the single conditioned law.

The weight a_J(y) uses only rows between the common base and the layer J above it, excluding the latter. Stop the single-word chain of the first walk at its first true boundary at or above that layer. Every row used by the weight lies below this boundary. Exact-word factorization therefore gives a fresh P^+ suffix, independent of the removed path prefix under the weighted law. The weight may depend on infinitely many sites in those J layers; only their heights matter for this factorization.

For fixed J, the height and size of this initial prefix are tight under P_x^+, and hence under every one of the bounded weighted laws. Its height divided by n(r) tends to zero. Its displacement divided by r also tends to zero, as does its centering correction, since \theta_r/r\to0. The suffix limit and its tight continuous modulus thus give the asserted limit under the approximating law. Send r\to\infty first and then J\to\infty. The same proof applies to the second marginal, and tightness of the two marginals gives joint tightness.□

Proof. In the top layer choose the windows

They are disjoint. Lemma 4.4 and the normal tail bound show that the shared conditioned probability of either endpoint error has limsup at most Ce^{-ca^2/t}, uniformly in the starts. Average the contact bound under the environment law with density q(x)q(y)/p_2(x,y). This gives the same bound for trace contact under P_{x,y}^{++}.

To compare laws off contact, first use raw successful prefixes: both walks reach level k before dropping below the common base. For any two disjoint finite traces, their raw prefix weights agree after averaging in shared and independent environments, because their departure-row sets are disjoint. Passing from two finite successes to two infinite no-drop events removes raw mass at most 2(u_k-p) under either pair law. Moreover p_2(x,y)-p^2\to0 uniformly for the present starts, since their separation tends to infinity. Thus the conditioned prefix laws agree off contact up to a variation error that tends to zero. Their total masses are both one, so the contact mass on the independent side has the same upper bound up to this error. The contact estimate proves (17).□

Proof. Choose 1<\alpha<\gamma<2. The potential will be a flattened and capped version of u^\alpha, where u is the absolute gap at an observed common cut. Its expected increase will pay for the number of near-diagonal cuts before the next observation. We need one estimate at nearly Gaussian radii and another below the largest remaining exceptional radius.

Drift at nearly Gaussian radii. Fix a sufficiently small t_0>0. There exist l\in(0,1), \epsilon>0, and u_0<\infty such that every radius satisfying

has, uniformly over shared starts with |Z_0|=u,

Call these radii good. The constants c_*,C_* can be fixed independently of small t_0; l,\epsilon,u_0 may depend on it.

Here is the compactness argument giving this uniform assertion. Along any Gaussian sequence of radii u, the independent gap at height k(u), divided by u, converges to \pm1+G, where G\sim N(0,2\sigma_*^2t_0). For small t_0,

Indeed |z|^\alpha has strictly positive second derivative near one, while the Gaussian probability of leaving that neighborhood is exponentially small in 1/t_0. The off-diagonal comparison changes the expectation by at most Ce^{-c'/t_0} in the limit. Replacing k(u) by \sigma(k(u)) costs o(1): by (6) the unscaled gap difference is uniformly tight, and the tested function is bounded and uniformly continuous. No moment of the overshoot is used.

For the second estimate, shrinking the gap by u/2 requires at least one centered marginal to move by u/4. The shared marginal limit and the Brownian maximal bound

give the required limit bound. This maximal bound follows by applying the nonnegative submartingale inequality to the two exponentials of Brownian motion on finite grids and then using continuity.

Choose strict slack in both limiting inequalities. If no triple (l,\epsilon,u_0) gave (19), choose violating radii while l,\epsilon\downarrow0 and u_0\to\infty. These radii would form a Gaussian sequence, contradicting the preceding limits. Finally, decrease \epsilon until 4/(1+4\epsilon)\ge2^\gamma, and increase u_0 until k(u)\ge1.

A cutoff above all exceptional radii. Fix A>1. Along the given Gaussian sequence r, the nongood radii in [u_0,Ar] are all o(r). For every fixed \xi>0, this follows uniformly on [\xi r,Ar] from the Gaussian tail tests, monotonicity of n, and n(Ar)\le A^2n(r).

On any subsequence pass further as follows. If the nongood radii stay bounded, choose a fixed cutoff b_\circ above them and above u_0. Otherwise choose a nongood radius D_*\to\infty within a factor two of their supremum, and set b_\circ=4D_*. In either case b_\circ=o(r), every radius between b_\circ and Ar is good, and

For this last bound, at a good radius

so n(2u)\ge2^\gamma n(u). Iterate while the doubling stays below r. For r\le u\le Ar, monotonicity of m gives the remaining estimate. The constant C_A need not depend on t_0.

Escape from the cutoff region. Uniformly over |Z_0|\le b_\circ, there is a positive probability of reaching an exact common true level h\le K with gap at least 2b_\circ, where K\le Cn(b_\circ). The constants are uniform along the chosen further sequence; when b_\circ is fixed they may depend on that fixed value.

For a fixed cutoff, bounded uniformly elliptic scripts to level one can enlarge the gap beyond 2b_\circ; fresh joint truth completes the claim. Consider therefore b_\circ=4D_* with D_*\to\infty. Since D_* is nongood, the second line of (13), at H=\lfloor t_1n(D_*)\rfloor, supplies one deterministic sign of single-path deviation from b_H, of size at least c_0D_* and P^+-probability at least c_1t_1. Here t_1>0 is fixed and sufficiently small, while c_0,c_1 do not depend on this small choice. The raw quenched mass of that deviation therefore exceeds pc_1t_1/2 on an environment event of probability at least pc_1t_1/2.

Assign this deviation to the rightmost walk for a positive sign, or to the leftmost walk for a negative sign. For the other walk, (12) supplies fixed positive raw mass within c_0D_*/3 of b_H, except on an environment event of probability O_{c_0}(t_1^2). Subtracting that error from the preceding \Omega(t_1) probability gives a uniformly positive probability that both masses are available in the same environment. Multiplying the quenched masses then gives a uniformly positive raw probability of two successes whose absolute gap has increased by at least c_0D_*/2. Their prefixes use only rows below the top. Fresh joint truth there has probability at least \underline p_2, so the top is an exact common cut and both initial no-drop events hold.

Repeat a fixed number of these blocks, choosing the outward-moving walk at each new common cut. The shared continuation rule applies at every repetition. Enough repetitions enlarge any initial gap in [0,4D_*] beyond 8D_*, with uniformly positive probability, and their total height is a fixed multiple of H. This proves the claim with K\le Cn(b_\circ).

Observation times and potential increase. Starting at the common cut zero, observe only the following subset of common cuts. Stop when an observed gap is at least Ar. From an observed gap u<Ar, choose the next observation by the rule

These are stopping indices of the common-word chain. At each observation the conditional continuation is the fresh shared conditioned pair law from the observed sites.

Use the bounded potential

For b_\circ<u<Ar, (19) implies an expected increase at least c_*t_0u^\alpha. Flattening below b_\circ only increases the terminal value, and the upper cap is at least 2u. For u\le b_\circ, the potential cannot decrease; the escape event just proved makes it increase by at least (2^\alpha-1)b_\circ^\alpha with a uniformly positive probability. Telescope up to observed exit, first with a deterministic truncation of the number of observations. Since \Phi\le(2Ar)^\alpha, monotone convergence gives

The first constant is independent of small t_0.

Counting all common cuts. Allocate each common cut to the last observation at or before it, excluding the next observation. An upper-regime observation at gap u receives at most k(u) cuts: each allocated offset is an integer strictly below k(u). A lower-regime observation receives at most K cuts. These bounds do not require any bound on the height overshoot of the next observation.

Let N_{\rho}^{\rm pre} count allocated common cuts with gap at most \rho r, before observed exit. Split observations into three sets. For fixed \rho>0, take r large enough that b_\circ\le2\rho r. Equation (20) and the drift bounds give

For the first row use K\le Cn(b_\circ). For the second use k(u)\le t_0n(u) and u^{\gamma-\alpha}\le(2\rho r)^{\gamma-\alpha}. For the last row, an allocated near-diagonal cut requires shrinking the gap from u to at most u/2 before offset k(u). Its conditional probability is bounded by (19); the number of allocated cuts is at most k(u). In both upper-regime rows, the factor t_0 from k(u) cancels the one in the drift budget. Their constants can therefore be independent of small t_0.

Under the same-site initial law, joint marginal tightness makes the probability of an observed gap \ge Ar before height Tn(r) tend to zero as A\to\infty, uniformly in the limiting upper bound in r. All common cuts after that exit and below the target height contribute at most Tn(r)+1 times its indicator.

From common cuts to deterministic layers. Let \mathcal C_h denote common truth at height h, and let A_h be the raw probability that both paths reach h before the initial drop with |Z_h|\le\rho r. Imposing the initial shared conditioning gives a numerator at most A_h. Instead, impose both fresh truths at height h. Their conditional probability is at least \underline p_2, and the resulting event implies the initial conditioning and \mathcal C_h. Dividing by the same initial normalizer gives

Sum over h\le Tn(r) and use the common-cut bounds above. On the chosen further sequence the lower-regime term vanishes because b_\circ=o(r). The remaining pre-exit bound is C_A\rho^{\gamma-\alpha}+C_Ae^{-c_*/t_0}. Every subsequence admits this further-sequence construction, so the bound controls the original limsup. Send \rho\downarrow0, then t_0\downarrow0, and finally A\to\infty. This proves (18).□

Proof. Take any subsequential continuous-path joint limit B_1,B_2 of (16) under P_{0,0}^{++}, on fixed horizons (enlarge a horizon if needed at endpoints), and let \mathcal G_s be the joint past. By (18)

One can integrate continuous cutoffs of the gap supported in shrinking neighborhoods of zero; their grid sums give the same limiting expectations by joint tightness with continuous limits.

Conditional marginal increments. At a deterministic time s, put H_r=\lfloor sn(r)\rfloor and restart the prelimit pair at \sigma(H_r). This is a stopping index of the chain of common words, not a stopping time for an individual walk. All first-hit coordinates of both paths at levels at most H_r are determined by the words traversed through this boundary. Its conditional suffix law is the fresh shared conditioned law from the terminal pair of sites.

By (6), \sigma(H_r)-H_r is uniformly tight. Consequently its normalized time shift vanishes. Joint tightness with continuous limits, on a slightly enlarged horizon, makes the values at H_r+\lfloor tn(r)\rfloor and at \sigma(H_r)+\lfloor tn(r)\rfloor differ by o_{\Pr}(r) after subtracting the same linear centering. At the initial endpoints this also follows directly from (6); the centering correction is a tight integer times \theta_r/r, which vanishes by Lemma 4.3.

Test against bounded continuous functions of finitely many joint past coordinates, represented at floor grid times in the prelimit. Condition at the common boundary and test the fresh normalized increment of length \lfloor tn(r)\rfloor. Lemma 4.4 makes its conditional bounded continuous test expectation converge to the normal value uniformly in the random terminal pair of sites. The time replacements just justified do not change the limit. Passing to the joint limit, then using a monotone-class argument for the joint past, proves that for each t>0,

Off-diagonal cross increments. Moreover if g is a continuous [0,1]-valued cutoff vanishing on [-2a,2a], a>0, and \delta_i=B_i(s+t)-B_i(s), then for bounded \mathcal G_s-measurable F,

The bound can be uniform in s. To see this, again restart as above; on the supported past gaps the restart state has absolute gap \ge a r except with vanishing error. Clip each future normalized increment symmetrically to the interval [-w\sqrt t,w\sqrt t], for the moment fixed w,t. By (17) at the fresh state their product expectation differs from the independent-law one by at most Cw^2 t\exp(-c a^2/t) in limsup; the independent-law value tends to zero by the single Brownian limits. Test first against continuous bounded past functions and pass to the joint limit. Removing the clipping there costs at most t\,o_{w\to\infty}(1)\|F\|_\infty, by the conditional marginal normals and Cauchy–Schwarz. Continuous past tests extend to bounded measurable ones, for instance by bounded approximation using E(|\delta_1\delta_2|\mid\mathcal G_s)\le \sigma_*^2 t. Choosing w\to\infty slowly gives (22).

Identification of the joint law. These facts force B=(B_1,B_2) to consist of independent Brownian motions. For a direct verification fix s_0, bounded F of the joint past there and a real vector \xi\in\mathbb R^2. Telescope the exponential of i\xi\cdot(B(s)-B(s_0)) on an equal mesh of [s_0,v], multiply by F and take expectation. Taylor remainders through order two total o(1) by the marginal Gaussian third absolute moment bounds. First-order terms vanish by the past-conditional laws, and diagonal second-order terms give the Riemann sum using their variances. Cross terms total o(1): away from gap zero by (22), and the complementary terms are bounded by a constant times the mesh sum of step lengths times expected continuous cutoff near gap zero (by the past-conditional absolute product bound), which is negligible by continuity and (21) as the cutoff shrinks. Thus f(v)=E[F\exp(i\xi\cdot(B(v)-B(s_0)))] satisfies

This gives the Gaussian increment law independent of the joint past, proving the assertion.□

Proof. Let F be a bounded real continuous test on path space, and let M_r(\omega) be its expectation under the quenched conditioned path. Give the environment the probability law

Under this weighting, two conditionally independent quenched conditioned paths have exactly the annealed law P_{0,0}^{++}. Proposition 4.7 and its marginal limits therefore give, with b=E[F(B)] for the limiting Brownian path,

Thus M_r\to b in \widehat P-probability. Since q(0)>0 almost surely, P is absolutely continuous with respect to \widehat P; convergence in probability also holds under P.

For (23), use Lemma 4.3 to replace the linear centering by the medians. Choose a nonnegative continuous path test below the desired event, with a fixed bounded supremum and with the endpoint, throughout a small neighborhood of time one, strictly inside the window of radius \rho. Such a test can be chosen with positive Brownian expectation: the time-one centered normal law assigns positive mass to a smaller window, and continuity permits the nearby-time and sufficiently large fixed supremum restrictions. Eventually this fixed supremum bound is smaller than b_*(r). The quenched no-drop-conditioned probability is consequently bounded below by a fixed positive constant with P-probability tending to one. Finally, on successful prefixes the total variation cost of replacing no-drop conditioning by conditioning on T_H<T_{-1} is at most 1-q(0)/a_H, which tends to zero almost surely. This proves the stated normalization by a_H. Translation invariance gives the same conclusion at any translated start.□

Proof. Take a subsequence realizing the left \limsup and extract compactified weak limits \mu,\nu' in [0,\infty] for the two laws. At each u>0, \mu([u,\infty]) is bounded by the limit-law probability of the open event above u/2, hence by the corresponding prelimit \liminf. By (11) and (9), this is at most C_1 times a prelimit \limsup on the V side above u/(2C_1); bounding by the limiting closed tail and enlarging it once more gives

For the first inequality of (25) simply integrate with weight 2u up to L, since capped squares are bounded continuous. For the second, first upper bound the left using X\ge A on the limiting law (upper semicontinuous test). This gives A^2\mu([A,\infty]) plus the tail integral between A and L. Transfer each term to bound the sum by a constant times the \nu'-expectation of the capped square on the open tail above A/(4C_1). By lower semicontinuity that last expectation is bounded by the prelimit \liminf on our subsequence. Enlarging constants gives (25).□

Proof of Proposition 5.1. We will concatenate a fixed number of short blocks. In each block we choose a probability law on a restricted set of successful prefixes, except on a rare failure event. The two requirements are different: the chosen endpoint displacements must have small variance, and the restriction must retain a fixed positive amount of the original kernel mass. The first requirement keeps the auxiliary first-hit process in a narrow tube; the second will transfer that conclusion back to the quenched walk.

Work along an arbitrary sequence r\to\infty, and pass to a subsequence on which

exists. This extraction is possible because the ratios are monotone in a, and changing a by a factor changes m(ar) by at most its square. Thus they are equicontinuous on compact logarithmic intervals. In particular F(1)=1, and the limit F_0=F(0+) exists in [0,1]. It suffices to prove the clipping assertion on every such further subsequence with some fixed positive mass bound. Otherwise there would be a sequence of scales at which every deterministic slope fails even for mass thresholds tending to zero.

The common block rule. Fix a small tube width b>0, and consider a block of integer height h\le tn(r), with t>0 small. Its length will be chosen separately in the cases F_0=0 and F_0>0. At its current starting site x, put

Short-block survival, Equation (12), gives a number d_b\in(0,1), depending on the fixed b,t, such that

for all sufficiently large r. On \{G<d_b\}, mark a failure and make an auxiliary jump to the block top with second coordinate displacement b_h and all other horizontal displacements zero. Otherwise sample a prefix from K_h(x,\cdot\cap N_b)/G. In the case F_0>0, we will sometimes further restrict this sampling to endpoints close to b_h.

The block test and every sampled prefix use only departure rows below the block top. Their endpoint and auxiliary choices therefore leave fresh iid layers above the next base. Repeating any one of these translated rules over equal blocks makes the second-coordinate endpoint displacements U_h iid under environment averaging and auxiliary sampling. Write \lambda_h=\mathbb E U_h. Both genuine and dummy endpoints satisfy

The dummy jumps keep the auxiliary process defined after a failure; only trajectories without a failure will later contribute genuine quenched path mass.

We next choose the block length and, when useful, its further endpoint restriction. In both cases the goal is to make the variance accumulated over O(n(r)/h) blocks small compared with r^2.

Case F_0=0: the broad restriction has small variance. Take h=\lfloor tn(r)\rfloor with t>0 fixed and small, and use the common block rule without further restriction. The truncated-moment transfer and Equation (8) give

The normalization in the block rule costs only a fixed factor in this moment estimate:

Indeed, on \{f\ge1/2\}, sampling the restriction costs at most a factor two relative to K_h/a_h(x). On \{f<1/2\}, use (br)^2 and

Dummy jumps have zero centered displacement. Since there will be O_T(1/t) blocks, Equation (26) makes their total variance at most C_TF(b)r^2 in the limit. This can be made as small as needed by choosing b small.

Case F_0>0: select a Gaussian subscale. Put z=\varepsilon r and h=\lfloor n(z)\rfloor. Limits in this case are taken first as r\to\infty along the chosen subsequence, and then as \varepsilon\downarrow0. We have

For every fixed a>0,

To see this, bound the expression by a constant depending on a times (m(az)-m(az/2))/m(z); its iterated limit is zero because F(a\varepsilon),F(a\varepsilon/2),F(\varepsilon)\to F_0>0.

The quenched Gaussian estimate (23) consequently implies that, for each fixed \rho,b>0, there is \alpha_\rho>0 such that

Here and below a block kernel without a stated start is at the origin. For completeness, a failure of this implication would give \varepsilon_j\downarrow0 and sufficiently large r_j for which the first j large-jump tests at thresholds a=1,1/2,\ldots,1/j are less than 1/j, whereas the probability in (27) stays bounded away from zero. Taking z_j=\varepsilon_jr_j\to\infty gives a Gaussian sequence, and b/\varepsilon_j\to\infty is an allowed tube multiple in (23). This is a contradiction.

We also need a moment estimate for the fallback to the broad restriction:

Fix A,b,\varepsilon with b>A\varepsilon and b>\varepsilon, and apply Equation (25) with L=b/\varepsilon. For the resulting iid sums, split each symmetric increment into its parts on

All three parts are centered. The maximal sum U of the first parts has fourth moment at most Cz^4, since hm(z)\le z^2. The second maximal sum U' has second moment bounded by Ch\mathbb E[S^2;z<|S|\le br]. The probability that any third part occurs, multiplied by the cap (br)^2, is bounded by h(br)^2\Pr(|S|>br). These last two bounds add to at most

Without a third jump, the original maximal sum is at most U+U'. On the event that it exceeds Az, its capped square is bounded by a fixed constant times U^2\mathbf1_{\{U>Az/C'\}}+(U')^2. Divide by z^2, use the fourth-moment bound for the first term, and take the indicated iterated limits. The last large-jump term vanishes; this proves (28).

We now refine the common block rule. Let

On a nonfailure, when g\ge\alpha_\rho, sample this stricter restriction, normalized. When g<\alpha_\rho, use the broad restriction as before. Take \alpha_\rho\le1. The raw mass of whichever restriction is selected is then at least \alpha_\rho d_b: in the stricter case this follows from a_h(x)\ge G\ge d_b.

For Az<br, the endpoint displacement of this refined rule satisfies

The stricter restriction contributes at most \rho^2. In the fallback case with f\ge1/2, split at W_h=Az and use the factor-two normalization bound. When f<1/2, use the same bound on its probability as in the first case, noting that \{W_h>br\}\subset\{W_h>Az\}. Again the dummy displacement costs zero. Equations (27)–(28) show that the iterated limit superior in (29) can be made arbitrarily small: first choose \rho,b small and A large, then choose \varepsilon small.

Concatenation and transfer to genuine path mass. In either case, run J=\lceil H_{\rm tot}/h\rceil blocks with the selected rule. In the second case set t=2\varepsilon^2/F_0, which bounds h/n(r) for large r. For small fixed \varepsilon, also F(\varepsilon)\le2F_0, so in both cases J=O_T(1/t). The probability of any marked failure is at most

For the centered sums of the iid endpoint displacements, the maximal inequality for independent centered sums gives

In the first case the limit superior of the numerator divided by r^2 is at most C_TF(b). In the second case Jz^2/r^2\le C_T, and Equation (29) gives the same required smallness. Choose the parameters in the orders stated above. In the first case choose t sufficiently small after b; in the second choose \varepsilon sufficiently small after \rho,b,A, with t=2\varepsilon^2/F_0. In both cases require the failure probability to be small and

where C is the constant in Equation (10).

Within a genuine block, its median-tube restriction and Equation (10) imply

Thus, outside the failure and maximal-deviation events, the concatenated first-hit positions through Jh stay within \eta r of the deterministic line of slope s_r=\lambda_h/h.

Choose the two annealed error bounds small enough that, with environment probability greater than 1-p_{\rm err}, the conditional auxiliary law at the specified start assigns at least one half to these genuine tube paths. On a nonfailure, its density with respect to the local kernel is at most d_*^{-1}, where d_*=d_b in the first case and d_*=\alpha_\rho d_b in the second. The density of a genuine concatenation is therefore at most d_*^{-J}. The concatenated local kernels are submeasures of the original-floor kernel, by the quenched Markov property at first hits: each local no-drop condition merely adds a restriction. All parameters are now fixed, so J\le J_*<\infty for large r. The original kernel through Jh consequently gives the tube mass at least d_*^{J_*}/2. Taking its first-hit prefix at H_{\rm tot} proves Equation (24).□

Proof. Apply (24) with tolerance \eta/2 and a sitewise failure probability \epsilon. For every tuple, all k clipped kernels are good except with probability at most k\epsilon. Hence the expected \pi-fraction of bad tuples is at most k\epsilon, and with probability at least 1-2k\epsilon at least half of the tuple mass is good. On that event the product of clipped masses gives d=d_0^k/2. Choose 2k\epsilon<p'. The single-walk assertion is the same argument with one coordinate.□

Proof. Splitting one starting mass. First consider a single prescribed start. We will produce k ordered endpoint groups, with support gaps of order r, each carrying a fixed positive mass. Use J_0 blocks of length h=\lfloor n(r)\rfloor, followed by a remainder, to reach H=C_*n(r), where J_0 is a large fixed integer and C_*=2J_0+2. Clipping through C_*n(r) supplies a common deterministic slope s_r and, with arbitrarily high environment probability, fixed positive mass within \eta r of that line. The tolerance \eta will be small compared with 1/J_0.

Endpoint spread, Equation (13), supplies a designated sign of a deviation of size at least cr from hs_r, with single-walk P^+-probability at least c>0. These constants are independent of the clipping parameters. The corresponding raw K_h-mass has expectation at least pc, by using paths on D, and is bounded by one. It therefore exceeds a fixed positive threshold on an environment event of fixed positive probability. The same reasoning applies after averaging over any starting probability.

Maintain the groups as submeasures of the walked mass, normalizing within each group for its next test. If there are fewer than k groups and the designated sign is positive, take the rightmost group and a second-coordinate median. Preserve its half at or below the median by central clipping. From the half at or above the median, retain the designated jumping mass whenever it exceeds its fixed threshold; these endpoints form a new rightmost group. Preserve all other groups by central clipping as well. For a negative designated sign, perform the reflected operation at the leftmost group.

Both median halves have mass fraction at least one half. A median atom may be used in both tests. In the positive-sign case, the retained central endpoints have an upper bound given by the median plus hs_r+\eta r, while the jumping endpoints have a lower bound given by the median plus hs_r+cr. The negative-sign case has the reflected inequalities. Thus the new groups have disjoint supports with a gap of at least (c-\eta)r. Every old gap loses at most 2\eta r under central propagation. Once there are k groups, only their central preservation is needed.

The initial-distribution clipping lemma makes all the central preservations succeed simultaneously with arbitrarily high conditional probability. Abort if a preservation fails. Subtracting this small error from the jump event shows that, whenever a split is still needed, preservation together with a split has conditional probability at least a fixed s_*>0, independent of \eta,J_0. The operations are measurable: take full restrictions to the indicated events and choose, for example, lower integer medians.

Choose J_0 large, and then choose the preservation errors small compared with 1/J_0. The probability of no abort but fewer than k-1 splits in these blocks is at most

Indeed, on such a pattern a split is needed at every step, and each prescribed nonsplit without abortion costs at most 1-s_* by conditioning on the previous blocks. Apply central clipping once more over the remaining height, which is at most C_*n(r). Taking \eta sufficiently small after fixing c,J_0 leaves final gaps at least c'r>0. The product of the finitely many retained mass fractions gives a fixed positive lower bound for each group in the original single-walk kernel K_H. Every test and continuation uses only layers below its actual endpoint.

Choosing separated endpoints. For any starting tuple whose k starts each have these k options, the product kernel gives fixed positive mass to separated endpoint tuples. Choose 2c_0<c' and expose endpoints successively. Every earlier endpoint excludes at most one group at the next start, since two distinct groups are separated by more than 2c_0r. Thus at the ith choice at least one of the k groups remains available. Each choice costs only its fixed positive group-mass lower bound. Averaging the favorable fraction over \pi, as in Lemma 6.1, proves (30).□

Proof. Truncate each q upward to q\vee e^{-B_*}, changing the product by at most e^{-B_*}: if a factor changes, the entire new product is at most this quantity. By bounded product-field mixing at mutually separated sites, the expectation of the inverse \lambda-power of the truncated product is bounded by 2(Eq(0)^{-\lambda})^k for sufficiently large separation. To see the required uniformity, at fixed k,B_* approximate the bounded single-site functional in L^1(P) by one using finitely many rows and having the same bound. Sufficient separation makes all its translates disjoint. On the event on the left of (31), the truncated product integral is <3s. Jensen’s inequality for its negative power and Markov’s inequality give (31).□

Proof. From the current layer in a subdivision, monitor the local path mass with budget d'/L', stopping at the first height where it is <s:=\kappa^{-k}e^{-x/L'}. The assumptions give e^{-B_*}\le s\le1. At any fresh start whose separation is still at least G-d', the probability of such a threat is bounded by the parenthesis in (32). Indeed the raw mass of all k infinite D-successes is exactly the current q-product integral. Its part violating the budget within H layers has expectation at most CkH/n(d'/(2L')), by (8)–(9) and a union over single walks on D with a median deviation greater than d'/(2L'). The other path factors are bounded by one and can be dropped. If the product integral is at least 2s and the violating mass at most s, enough mass satisfies the budget at every first hit to prevent a threat. Lemma 6.3 and Markov’s inequality now give the claimed bound; the factor \kappa^{-k\lambda} is absorbed into e^{Ck}. Neither D nor q is used to choose the stopping layer: the actual test is the finite-height mass, measurable from rows below that layer.

At a threat, extend the passing mass from one layer earlier by a direct upward step for every particle; at offset zero the passing mass is one. This preserves relative displacement errors and retains mass at least \kappa^ks=e^{-x/L'}. Normalize at the actual stopping layer and restart with the fresh upper environment. A final piece with no threat also has mass at least e^{-x/L'}.

If the horizon finishes within L' pieces, concatenation gives original mass at least e^{-x}, with total relative-motion budget at most d'. Otherwise each of the first L' pieces has detected a threat before completion. The conditional threat estimate applies at every restart: only lower rows have been revealed, and separation remains at least G-d'. Multiplying these conditional bounds gives Equation (32).□

Proof. We construct one sequence of retained endpoint laws and use it at every target height. A stage starts with an extra signed gap, or buffer, between the walks. It usually increases this buffer by a fixed factor; on failure it reduces the buffer by a smaller factor. The resulting positive logarithmic growth limits the number of stages through height H to order \log(1+H), apart from an exponentially integrable random error. Each stage costs only a fixed amount of logarithmic mass.

Endpoint tests and retention. Keep the original signed gap z_0=y_2-x_2 fixed, even when it is negative. First prescribe \lceil R_0\rceil positive lateral steps for walk 2 and one upward step for each walk. This bounded uniformly elliptic script supplies a buffer R_0, to be fixed large enough below, at a fixed positive mass cost.

At a stage base, the current pair probability \pi is supported on signed gaps at least z_0+R, where R\ge R_0. Plan h=\lfloor n(R)\rfloor layers. For constants a>0, 0<\ell'<1, and g_*>0 to be chosen, test

Stop at the first failing test. If all pass, also test at h whether mass at least g_* has gap at least z_0+(1+a)R. These are tests of the full kernels from the same stage base. We do not intersect their endpoint events as path restrictions: a completed stage uses its retained final law, while a target height inside that stage uses only the test at that one height.

On success retain and normalize the growing endpoint mass, and replace R by (1+a)R. On failure replace R by (1-\ell')R. At a first traversal failure, take the passing mass from the preceding layer and extend each walk by one direct upward step. This preserves its signed gap and supplies mass at least \kappa^2g_* at the stopping layer; at offset zero the passing mass is one. If only the final growth test fails, its passing traversal test already supplies this much mass. Normalize the retained law. Whenever the new buffer is below R_0, restore it by bounded positive lateral steps for walk 2 followed by an upward step for both walks. The buffer before restoration is at least (1-\ell')R_0, so the script has uniformly bounded length.

Every test, retained law, and restoration uses only departure rows below its terminal layer. Conditional on this information, the next stage therefore starts in fresh iid layers.

A uniform chance of buffer growth. We choose a and a success probability lower bound s_*>0 independently of small \ell'. Clip both walks at scale R about the common deterministic slope in Equation (24), with tolerance \delta R, where 2\delta<\ell'. The endpoint spread estimate (13), applied to the same deterministic center hs_R, gives one sign of a deviation of size at least c_1R with single-walk annealed P^+-probability at least a fixed positive constant. The constants do not depend on the clipping parameters. The favorable sign may depend on R: use a positive deviation for walk 2, or a negative deviation for walk 1. Either choice increases their signed gap when the other walk stays near the center.

The raw jump mass has a fixed positive expectation, by restricting to D-paths, and is bounded by one. Thus it is at least some j_0>0 on an environment event of probability at least c_0>0, uniformly over starts and large R. Choose the sitewise clipping errors with sum less than c_0/2, and let d>0 be their common retained mass bound. For each deterministic starting pair \mathbf x, the event A_{\mathbf x} that this jump mass is at least j_0 and both clipped masses are at least d then satisfies

This uses subtraction of the two clipping errors, with no independence between the events.

For an arbitrary current pair law, set F=\int\mathbf1_{A_{\mathbf x}}\,\pi(d\mathbf x). Since EF\ge c_0/2 and 0\le F\le1,

On this event, central clipping for the favorable pairs supplies at least Fd^2 to every traversal test. At the terminal height, the favorable jump and the central restriction for the other walk supply at least Fj_0d to the growth test. The terminal paths need not be the paths used to prove the traversal tests; the tests were defined separately for this reason. Choose \delta<c_1/2, a<c_1/2, and

Then every stage succeeds with conditional probability at least a fixed s_*>0, independently of small \ell'. The constants g_* and R_0 may depend on the chosen \ell'. All choices can be made deterministic at the countable set of buffer values reached by the procedure.

The number of stages through a given height. Fix \ell' small enough that a proportion s_*/2 of successes gives positive logarithmic growth. Let S_m count successes in the first m main stages, and put

The conditional success lower bound implies a uniform exponential tail for W. Indeed, for sufficiently small \theta>0,

is a nonnegative supermartingale: the conditional multiplier is at most e^{\theta s_*/2}(1-s_*+s_*e^{-\theta})\le1. Its maximal inequality gives P(W>u)\le e^{-\theta u}.

Success multiplies the buffer by 1+a, failure by 1-\ell', and restoration only increases it. Consequently, after m completed main stages and their restorations,

for fixed positive constants c_2,C_2. If H' is the height already consumed, we also have R\le C(H'+1). A successful increase from R_{\rm old} uses its full planned length \lfloor n(R_{\rm old})\rfloor\ge cR_{\rm old}, by Equation (7) and a sufficiently large R_0. Failures reduce the buffer, and restorations reset it to R_0. Thus at any integer target height H, at most

main stages have been completed, and at most one more is unfinished.

Concatenate the retained masses through H. If the last stage is unfinished, use its traversal test or its failure retention at the actual offset. If the target is the endpoint of a one-layer restoration, its bounded script supplies the continuation instead. The endpoint gap stays above the original baseline, so these paths contribute to p_H^\to(x,y). Every completed stage, including a possible restoration, and the last partial stage cost a fixed logarithmic amount. Hence, simultaneously for all H\ge1,

with a constant uniform in the starting pair.

An exponentially integrable local remainder. Take A_\to=2C_3, and define

The preceding bound gives \mathcal Z\le C_3(W+1), proving the exponential moment in Equation (33). It also gives

On \{W\le M\}, only a deterministic bounded range of heights can therefore contribute to the supremum. The probability of its complement tends to zero uniformly in x,y.

For a bounded height range, kernels can be uniformly approximated by bounded-length paths. Indeed, from any site inside a finite-height strip, a fixed number of consecutive upward steps forces exit and has uniformly positive probability. The strip exit time therefore has a geometric block bound, uniformly in the environment and horizontal start. Restricting both paths to a bounded number of steps changes their pair kernel mass by a uniformly small amount and uses only fixed-radius upper-row neighborhoods of the two starts. Those neighborhoods may overlap; the truncation estimate is unchanged. Finally,

by the two straight upward paths. Logarithms are therefore uniformly continuous on this bounded range of heights. Take the finite maximum, then cap at B', and let the two truncation errors tend to zero. This proves the uniform local approximation of \mathcal Z\wedge B'.□

Proof. The main construction gives several chances to create separated mass before any prescribed short interval of target heights. A failed chance reveals only layers below the next available chance, so the estimate can be repeated there. We first describe these chances and their probability bounds, and then choose the constants to cover all target heights. Throughout these estimates A\ge1 and the integer k\ge2 are provisional fixed parameters. Write

We take b large enough that m\ge1. All uses of the rapid-loss bound (32) will have cap B_*=2kb; after fixing b, increasing s_{\rm fl} will ensure its separation hypotheses.

A nested schedule before one target interval. Cover [H_e,H] by integer cells [v,v+w_m], where

and v ranges through H_e,H_e+w_m,\ldots up to H. For the growth test at H, add one cell with v=H, using H in place of H_e in the definition of w_0. After increasing s_{\rm fl}, each w_0 lies between one eighth and one quarter of its corresponding horizon. A certificate for the last cell may use paths above H. These certificates will only bound failure probabilities. The actual kernel tests in (35) alone determine the operational stopping layers.

Fix one cell. Its m opportunities have starting heights

All these heights are nonnegative and increase with i. When an attempt begins at opportunity i, use the normalized endpoint law of the full original stream \Gamma_{v-2w_i}^{\pi}. Uniform ellipticity makes this mass positive, and it uses only rows below v-2w_i. In particular, after an earlier failure we restart from this full stream, not from the mass retained by the failed attempt.

The attempt first runs a seed kernel over w_i layers. Apply (30) with error e^{-6k}, and put

Except with that error, it supplies mass at least g_0 whose endpoint separation is at least c_0\rho_i, with c_0\le1. On failure we consume opportunity i and try i+1, if one remains. The failure is known at height v-w_i, which is below v-2w_{i+1}.

On seed success, normalize its separated mass and protect it through the whole target cell. Number the protection steps l=0,\ldots,m-i-1. Step zero starts at v-w_i; a nonfinal step l ends at v-2w_{i+l+1}, and the final step ends at v+w_m. Every step has length at most 3w_i16^{-l}. Thus, if failure is detected through a nonfinal step l, opportunity i+l+1 begins exactly at its detection height. The final step covers the entire cell and leaves no retry.

Retaining the separated seed. Choose an integer cutoff l_0, sufficiently large as specified below. Combine the first \min(l_0,m-i) protection steps into a single kernel, ending at the same height as that last combined step. Its length is at most 3w_i=3C_*n(\rho_i). Apply Lemma 6.1 with

Except with probability at most e^{-6kl_0}, this retains mass g_1>0 with relative-motion budget c_0\rho_i/4 throughout the combined protection. On success normalize this mass. On failure consume the \min(l_0,m-i) opportunities and restart at the next opportunity if one remains. The test uses the kernel only through its actual endpoint, including when fewer than l_0 steps remain.

At every remaining step l\ge l_0, allow relative-motion budget

Let Z_l be minus the logarithm of the mass respecting this budget, from the current normalized law. Continue the attempt if

retaining and normalizing that budget-respecting mass. Otherwise declare failure at the endpoint of this step. The early budget is c_0\rho_i/4, and all late budgets together are at most c_0\rho_i/8. Consequently all current laws have separation at least c_0\rho_i/2. Every restriction and normalization uses only rows strictly below its endpoint. Figure 2 depicts the underlying height schedule.

Costs of the late protection steps. Set a'=2Ak and \lambda'=\lambda/4. We will obtain, conditionally at every active late step,

where C_{\rm mg} is independent of A\ge1 and k\ge2. The cutoff l_0, unlike C_{\rm mg}, may depend on these parameters, the seed constants, and fixed retry counts. The rapid-loss estimate is uniform over normalized starting laws with the required separation; it does not depend on the early retained mass g_1.

Indeed, (7) gives, for each fixed retry count L',

Choose a fixed c_x>0 with (1+\lambda)c_x<\theta and c_x<1. For 0\le x\le c_xl, use (32) with L'=1. After increasing l_0, its second term is at most e^{-\lambda x}. Below the applicability threshold k\log(1/\kappa), use the trivial bound one. Both ranges therefore give

with C' independent of A,k. For c_xl\le x\le a'(l+1), take a fixed integer L' sufficiently large that \theta L'>(1+2\lambda)a', and increase l_0 again. The same rapid-loss estimate then gives

Here l_0 absorbs both the constants and the ellipticity threshold for this fixed retry count. Also a'(l+1)\le2Akm\le kb/4<B_*. Finally, integrating these tails with \lambda'=\lambda/4 gives

This proves (36) with the required uniformity.

Why all opportunities rarely fail. Now impose \lambda'A>C_{\rm mg}+8. If the first cumulative late failure occurs at l, either Z_l>a'(l+1) or the sum of the truncated costs exceeds Ak(l+1). Earlier passing costs are below their own cutoffs, since they are nonnegative. Setting inactive costs to zero, conditional iteration of (36) bounds the second event by

Together with the cutoff tail, this is at most e^{-5k(l+1)}. A failed attempt consuming exactly n opportunities consequently has probability at most e^{-2kn}, measured at its start. The possibilities are seed failure (n=1), early-protection failure (n=\min(l_0,m-i)), and a first late failure (n=l+1); the stronger bounds just proved absorb any coincidence of these counts.

Each such failure is known before the next available opportunity, so revealing the full stream up to the new start leaves its upper layers fresh. Exhaustion partitions the m opportunities into positive consumed blocks. For a specified composition of m, conditional iteration gives probability at most e^{-2km}. There are 2^{m-1} compositions, so the probability that this cell has no certificate is at most

Choosing the constants. We can now choose constants in an order justified by the preceding estimates. Fix C_{\rm mg}, then take A\ge1 with \lambda'A>C_{\rm mg}+8, and define

Choose g,K with 1/g<\beta/64 and (1+f/g)D_0/K<\beta/64, and then choose k\ge2 so large that

Next fix the seed constants and retry counts, choose l_0 to validate (36), and then obtain g_1 from early clipping with its specified error e^{-6kl_0}. All these constants precede b. Thus even this very small early-protection error causes no circularity: its possibly small retained mass g_1 is fixed before we enlarge b. Finally enlarge s_{\rm fl}, after b, to meet all scale and separation thresholds.

The full stream and the short heights. By (31), outside an event of probability at most e^{-(2K+2)b} for sufficiently large b,

We combine this exception with the cell failures by a union bound; none of the cell-attempt estimates is conditioned on this full-stream event.

To cover heights up to H_e, apply (32) at horizon H_e, with d'=r_s/2, x=B/2, and a fixed retry count L'>k/2+2K+4. By (7),

Raising the rapid-loss bound to its indicated power gives an upper bound of the form

for large b, by (37). Off this event, taking prefixes gives mass at least e^{-B/2} at every height through H_e with relative-motion budget r_s/2. Initial separation is at least r_s, so these prefixes have separation at least r_s/2\ge r_{s_-}, again by (7).

Assembling the cell certificates. The total number of cells, including the separate growth cell, is at most

On successful completion of the attempt from opportunity i, separation throughout that cell is at least c_0\rho_i/2. The scale inequality gives

For the traversal cells,

which exceeds the required scale s_-=se^{-fb} after multiplication by the preceding fixed constant. For the growth cell,

which similarly exceeds s_+=se^{gb}. In each comparison the remaining factor e^b absorbs the fixed constants for large b. Thus the certificate supplies the separation required in (35) at every height of its cell.

On the separately controlled event q_\pi\ge e^{-B/4}, concatenate the full stream to the attempt start with the restricted seed and protection kernels. Their mass, and therefore their prefix mass at each height of the cell, is at least

because Akm\le B/8 and g_0,g_1 are fixed before b. The cell lies entirely in the protected part of the construction.

A union bound over cells, using (37), bounds the probability of any missing cell certificate by e^{-(2K+2)b} for large b. Adding the full-stream and short-height exceptions proves (35). At fixed b, all the smallest scales tend uniformly to infinity with s_{\rm fl}. Enlarging that floor therefore satisfies every scale and separation threshold used above.□

Proof. Mass and height within an episode. The unrestricted quenched k-survival mass through t_i is a_{t_i}^k. Starting from its normalized endpoint law, our retained continuation costs at most B_0+(kb+k\log(1/\kappa))J_i in logarithmic mass. Its concatenation is contained in the original survival kernel through t_{i+1}, so

This proves the upper bound for c_i in (38) once b\ge\log(1/\kappa); its nonnegativity follows from the monotonicity of a_t. Each stage length is at most se^{\chi b}, and the scale increases from the fixed floor by at most a factor e^{gb} per stage. Summing these geometric bounds, together with the one-layer injection, proves the height bound in (38). Every episode except possibly the last contains a failed stage, so M\le J+1. The last episode may consist only of its injection.

Counting stages by their scale changes. For this accounting only, update the scale after the last completed or failed stage as if the operation continued, and clamp it upward to the floor after a reset. Every completed success consumes its full planned height H\ge s_+, so its updated scale is at most \max(s_{\rm fl},N). Failures decrease the scale, and the clamping returns it only to s_{\rm fl}. If S is the number of completed successes, their increases of gb, minus the losses of at most fb per failure, therefore satisfy

for large N. There is at most one passing incomplete stage, so J\le S+F+1. This proves (39).

Transferring the failure bound to Q_N. Under P, conditional on the past at any actual stage start, its failure probability is at most e^{-2Kb}. If I is its failure indicator, then

There are at most N actual stages, since every stage advances at least one layer. Pad the list to N slots, with active indicator A_r, and put I_r=0 in inactive slots. Before each slot its activity is known, and

The inactive factor is one. Conditional iteration, using \sum_r I_r=F and \sum_r A_r=J, gives

Applying the entropy inequality with (34) yields

Only the exponential estimate was taken under the fresh law; this last inequality requires no independence under Q_N.

Write c_f=1+f/g. Combining the last inequality with (39) gives

The coefficient in parentheses is less than \beta/32. After absorbing the exponentially small term, for large fixed b and then large N,

Positive loss in episodes with bounded stage count. Under Q_N, \sum_{i<M}c_i=-\log a_N>\beta\log N. Summing (38) and using the last bound shows

This proves the lower bound on E_{Q_N}M; its upper bound, together with that on E_{Q_N}J, follows from M\le J+1. For any L_0>0, the loss in the remaining episodes satisfies

Although the injection cost B_0 may be large, it is fixed. Choose L_0 after it so that the first term’s expected contribution is at most (\beta/4)\log N. Episodes with more than L_0 stages then account for at most (\beta/2)\log N of expected loss. Subtracting this from the total proves the last assertion of (40).

Finally, the operational tests use only rows below their tested layer, so all episode boundaries are bounded layer stopping times. With the stated padding, \{i<M\}=\{t_i<N\} is known at t_i.□

Proof. Compactness gives a subsequential weak limit. The shift is continuous, and the occupation sum telescopes: for every bounded continuous F,

At the origin, (40) bounds the occupation mean of J_i. Truncation passes this bound to the limit, so J_0<\infty almost surely. The bounds (38) then pass and give the stated integrability and finiteness. For the positive truncated-drop bound in (40), put C_{L_0}=B_0/k+2bL_0 and use the globally bounded continuous test

It agrees with the required drop on every supported array, by (38). Its expectation therefore passes to the limit and gives E_{\mathbb L}c_0>0. The occupation law gives h_0\ge1, and shift invariance gives all assertions at every integer index. Uniform ellipticity is a closed condition on the row coordinates and therefore also passes to the limit.□

Proof. We first record the elementary entropy facts used here. Relative entropy is {\rm KL}(Q\mid P')=\int f_0\log f_0\,dP', where f_0=dQ/dP', and is infinite without absolute continuity. It has the variational formula

The upper bound follows from Jensen applied to e^F/f_0. For equality, truncate \log f_0 between \pm B and send B\to\infty, or test a singular set. Thus projection cannot increase entropy. On compact metric spaces, bounded continuous tests suffice by bounded approximation in L^1(Q+P'); the variational formula therefore gives lower semicontinuity under joint weak convergence of both laws.

For an event of respective probabilities s,t, where 0<t<1, projection to its indicator gives {\rm KL}(Q\mid P')\ge s\log(1/t)-\log2. If a reference law is a data marginal times an independent fixed law, replacing that marginal by the controlled law’s own data marginal only decreases finite entropy. Indeed the new density is the old density divided by its conditional mean given the data, and the entropy difference is the nonnegative marginal entropy. These log-density identities are taken on sets of positive controlled density; negative parts are integrable, and conditional Jensen for f_0\log f_0 justifies the marginal term.

Before the limit, at deterministic i\ge0, put \mathcal H_i=\sigma(\mathcal F_{t_i},(U_{i,a})_a) and \Phi_i=(\Omega^{i,m,a})_a; \mathcal A_i is \mathcal H_i-measurable. Include the same auxiliary sampling under P. Given \mathcal H_i, the spliced field above t_i, in absolute horizontal coordinates, is still iid under P. Indeed, at both bounded layer stopping times the layers at and above the stopping layer are fresh, while the current anchors use only lower-layer data. Keeping the slab up to the second stopping layer and replacing its tail by an independent copy therefore preserves the iid law. Thus \Phi_i has conditional law \mathbf R(\mathcal A_i,d\phi).

Let L_t=dQ_N|_{\mathcal F_t}/dP|_{\mathcal F_t}. Under Q_N, the conditional density of the information stopped at t_{i+m}, together with the independent extension, relative to the corresponding P-law given \mathcal H_i, is L_{t_{i+m}}/L_{t_i}. The anchors at i use the same lower-data sampling kernel under both laws, and the extension costs no density. Conditional Jensen therefore gives, for every bounded F,

Multiply by \mathbf1_{\{i<M\}}, which is past-known, and average as for \mathbb L_N. Jensen to pull the log outside this average gives the \mathbb L_N entropy bound at index zero (against its own current-data marginal and the kernel) of

Increments from i\ge M are zero. The displayed sum telescopes over deterministic indices, since t_i=N for i\ge N; its subtracted initial expected log densities are nonnegative relative entropies. For passage to the limit, finite-class consistency is a closed condition: self-offsets vanish, finite offsets are symmetric with sign change, and specified finite offsets add on triples. Violations are detected by finitely many clopen coordinate events. On this support the reference kernel is weakly continuous. For any finitely many field coordinates, coincidence of the input sites is specified by a finite-offset equality, while different sites have independent rows. Finite-coordinate tests suffice by compactness. Consequently the reference laws converge weakly with their current-data marginals. Lower semicontinuity and (40) yield the bound C_E m; shift invariance gives every i.□

Proof. Before the limit, the anchor draws at j, excluding a possible own label, are conditionally independent samples given the environment and U_{i,a}. The occupation indices and weights depend only on the environment, not on these draws. The mean-square error in (42) is therefore O(2^{-n}), including the possible own-label contribution. The finite-offset indicators are continuous, so these summable bounds pass to the limit. Borel–Cantelli proves (42).

For distinct labels and any continuous function g on [0,1], the same conditional sampling identity gives

Approximate the indicator of zero by bounded continuous functions to obtain the second assertion. All the indices and sites are countable, so the assertions hold on one common probability-one event.□

Proof. Before the limit, local no-drop continuation is contained in the global survival stream. Dividing its endpoint inequality by the new global mass gives (43), with C=\log(a_{t_i}/a_{t_{i+m}}). At finite limiting marks, pass first the inequalities involving finitely many starting sites and bounded-length successful kernel paths. Strict violations of these finite-path inequalities are open, since each path weight is a finite product of row coordinates. Increasing the finite restrictions gives (43). Successful paths to layer H use no departure row at or above H, so the auxiliary tail is irrelevant.

Before the limit, at a fixed index the average in (44) has conditional mean at most e^{-C} given the environment, again by the global mass inequality. Its terms are independent bounded functions of the independently sampled current anchors; they do not use appended rows. Chebyshev bounds the conditional probability of exceeding e^{-C}+\epsilon by 2^{-n}\epsilon^{-2}, and this also holds after occupation averaging. For fixed finite H, the function a_H is lower semicontinuous, being a sum of nonnegative continuous finite-path weights. Thus the strict upper-violation event is open at finite marks, and the same summable probability bound passes to the limit. Borel–Cantelli, followed by a countable sequence \epsilon\downarrow0, proves (44).□

Proof. Dust is isolated. If a dust label (i,a) had a distinct finite-offset neighbor (j,b), with offset z, profile consistency would give w_i^{jb}(-z)=w_i^{ia}(0)=0. Apply the second assertion of Lemma 8.3 with (j,b) as the reference label and (i,a) as the sampled target. This is impossible. Thus dust labels are singleton classes even when all times are included.

At any fixed time, the dust indicators are exchangeable, because anchor-permutation symmetry passes to the limit. They have an asymptotic frequency \delta_i, which is positive if any dust occurs. To see the latter assertion, apply the ergodic theorem to the stationary label sequence of dust indicators. On the invariant zero-frequency event, taking expectations shows that each indicator vanishes.

Fix i,m. Under the reference law of Lemma 8.2, conditional on \mathcal A_i, the singleton dust classes have independent upper fields. Their values q(0;\Omega^{i,m,a}) are therefore iid with mean p. Finite entropy gives absolute continuity, so the strong law on the dust subsequence also holds under \mathbb L whenever that subsequence is infinite. Since a_H\ge q, including in every spliced field, (44) yields

This holds simultaneously for every m. On \mathcal E, its left side tends to zero, by (41) and stationarity. Dust is therefore absent at all times on \mathcal E.

There are finitely many proper classes. For fixed n and r_0>0, whenever possible choose from \mathcal A_i labels in n distinct classes with self masses at least r_0. On this selection event, when C_{i,m}>\zeta m, (43) and the total-mass bound for each output profile give

The event of successful selection together with this deterministic product bound is measurable from the current data and the extension fields. Its reference probability is at most

by (4) and class independence. The binary entropy bound and Lemma 8.2 show that its \mathbb L-probability has \limsup_{m\to\infty} at most C_E/(\lambda n\zeta). On the selection event intersected with \mathcal E, the product bound holds for all sufficiently large m. This bounds the probability of that intersection by the same quantity. Send r_0\downarrow0, then n\to\infty, to exclude infinitely many proper classes.

The classes persist and have full support. Over any positive integer height, the local kernel has positive weight between any two horizontal sites: move horizontally at the starting layer and then ascend directly. Uniform ellipticity makes this finite script positive. Thus (43) gives a full-support next profile for every current proper class. By (42), every positive coordinate is represented by next-time labels. Distinct classes cannot merge, by the global finite-offset equivalence relation.

Conditional on the shift-invariant event \mathcal E, the current number of classes is therefore nondecreasing, finite, positive, and stationary. It is consequently constant almost surely. Indeed, writing N_i for this count, stationarity and monotonicity give

for every integer K. The nonnegative difference vanishes almost surely; letting K increase proves N_{i+1}=N_i. This uses no moment of the number of classes. The class sets transport bijectively through time, and every profile has full support by the update from the preceding time.□

Proof. We use the two-tail statistic introduced at the start of this section. For an integer R\ge1, recall that

Here z_2 means the second full-space coordinate of a horizontal site. Full support and finite mass imply 0<T_R(w)\le1, and a maximizing integer j_R exists. Choose one measurably. For each fixed full-support w, its maximizing cuts satisfy

For example, if j_R were bounded below along a subsequence, the left mass at both j_R and j_R-1 would stay bounded below whereas the right mass would tend to zero. Moving the cut one unit left would strictly increase the smaller mass, contradicting maximality. The other endpoint is treated by moving the cut right. Thus the normalized restrictions \alpha_- and \alpha_+ of w_0 to these two tails eventually sample sites outside every fixed neighborhood of the representative origin.

Fix m throughout the next steps. Let Q_R be the mass of endpoint pairs (u,v) with v_2-u_2\ge R under

Every starting pair already has signed gap at least R. Its outward order event is therefore contained in the event just tested. Finite entropy transfers the probability-one event in (33) to \Omega_m, simultaneously for the countably many starting pairs and integer heights, so it applies at the random height H_m. That estimate and Jensen’s inequality yield

Write \nu_\pm=\alpha_\pm K_{H_m}^{\Omega_m}. There is an integer j with \nu_-(u_2\le j)\ge Q_R/2 and \nu_+(v_2\ge j+R)\ge Q_R/2. Choose j minimal for the first inequality, so \nu_-(u_2<j)<Q_R/2. If the second inequality failed, split ordered pairs into u_2<j and u_2\ge j. The latter condition forces v_2\ge j+R. Each part would then have mass strictly less than Q_R/2, since both endpoint measures have mass at most one, contrary to the definition of Q_R.

Each initial tail has mass at least T_R(w_0). Applying (43) to the two tails at this cut gives

This is the decisive inequality: normalization contributes C_m, whereas preserving the order of the two tails costs only a logarithm of the traversed height and the local environmental remainder.

We next remove the environmental bias from that remainder. We claim

Write f_m(D,\Omega) for the density relative to (45). Under that product reference, additionally sample x,y from \alpha_-\otimes\alpha_+ conditionally on D alone. Then EZ_R is the reference expectation of f_m(D,\Omega)\mathcal Z((0,x),(0,y);\Omega).

The exponential moment in (33) is uniform in the sampled pair. Consequently the density and the remainder can both be truncated with errors uniform in R. To see the only delicate part, on \{f_m>B\} Young’s inequality gives, for sufficiently small \lambda>0,

The entropy term has vanishing tail. The reference probability of \{f_m>B\} is at most 1/B, and Cauchy–Schwarz bounds the exponential term on that event by O(B^{-1/2}). Once f_m is bounded, truncating \mathcal Z is justified directly by its exponential moment.

Approximate the bounded density by a bounded nonnegative function g(D,\Omega) using only a fixed finite box of field rows around the origin, allowing arbitrary dependence on D. With L^1 error at most \varepsilon, its reference integral is at most 1+\varepsilon. Approximate the capped remainder, uniformly in x,y, by a bounded function using fixed-radius upper-row neighborhoods of those two sites. This follows from the local approximation in (33); boundedness converts convergence in probability to convergence in L^1. As R increases, the two tail supports and their neighborhoods leave the fixed origin box. Under the product reference, conditional on D,x,y, the two approximating factors then use disjoint iid row sets. Their product expectation is bounded by (M_\mathcal Z+\varepsilon)E_{P_\uparrow}g(D,\cdot). The threshold in R may depend on D; boundedness permits integration by reverse Fatou. Sending the approximation errors to zero and removing the truncations proves (48). This entire step keeps m fixed, so its approximation boxes need not be uniform in m.

For fixed R,m, the right side of (47) is integrable. The entropy inequality and the same exponential moment show that EZ_R<\infty. Let X=\log T_R(w_0) and Y=\log T_R(w_m). Translation invariance of T_R and stationarity make these finite nonpositive variables identically distributed. Their difference has integrable negative part by (47). Clipping both variables below at -B gives an integrable zero-mean difference; its positive and negative parts increase to the corresponding parts of Y-X. Monotone convergence therefore shows that Y-X is integrable and E(Y-X)=0. Taking expectations in (47), and now sending R to infinity, yields

Finally let m increase. With Z_i=\log(1+h_i),

Divide by m, send m to infinity and then T to infinity. The assumed logarithmic moment gives E\log(1+H_m)=o(m), hence Ec_0\le0. Nonnegativity of c_0 proves the proposition.□

Completion of the proof of Theorem 1.1. Apply Proposition 8.6 to \mathbb L_*. The profile construction supplies its update and moment hypotheses, and the marked field argument above supplies the finite entropy hypothesis. But on the shift-invariant event \mathcal E, the stationary ergodic theorem gives E_{\mathbb L_*}c_0>\zeta. This contradicts the proposition and excludes the bad-event sequence Q_N. Thus (5) holds. Proposition 2.5 gives finite mean regeneration duration and an almost-sure deterministic vector velocity with positive first coordinate. This vector lies on the deterministic ray of Corollary 2.3, whose projection on the original transient direction \ell is positive. Undoing the signed permutation of the coordinate axes therefore gives X_n/n\to v with v\cdot\ell>0, as asserted in Theorem 1.1.□