The sharp factor-of-IID threshold for the free Ising model on regular trees

1 Introduction

Can the free Ising state on a regular tree be generated from independent vertex labels without choosing a root? The broadcast description of the state begins at a vertex and propagates a random spin along the edges. Its law does not depend on that vertex. A factor construction must also remove the distinguished vertex from the rule that generates the spins.

Let T_d=(\mathcal V,\mathcal E) be the infinite d-regular tree, d\ge3, and put b=d-1 and \theta=\tanh\beta for \beta\ge0. The free zero-field Ising law \mu_{d,\beta} is obtained by choosing a fair spin at one vertex and, independently along edges directed away from it, letting a child agree with its parent with probability (1+\theta)/2. Its marginal on any finite connected subtree D is

This expression also verifies that the law is invariant under all tree automorphisms and independent of the chosen broadcast root.

We call a spin law a factor of IID if it is the law of \Phi(U) for independent uniform labels (U_v)_{v\in\mathcal V} and a measurable map \Phi:[0,1]^{\mathcal V}\to\{-1,1\}^{\mathcal V} satisfying

The exceptional null set may depend on g. For spin laws this convention is equivalent to requiring a Borel factor equivariant on every input: Lemma 8.5 constructs such a version. The input contains no distinguished root or end. Measurability does not require a spin to be determined by finitely many labels; no finitary coding or coding-radius estimate is asserted.

The construction proves the positive implication, including equality. The strict converse is known: the obstruction beyond the reconstruction threshold is due to Sly, as recorded and proved in Lyons [12, Section 3 and Corollary 3.2]. We use the quantitative correlation bound of Backhausz, Szegedy and Virág [2, Theorem 3.1 in arXiv:1305.6784v1] to give that implication below. The map is obtained for each fixed d and \beta in the permitted range; no parameter-uniform or computational bound is asserted.

For the free state on T_d, reconstruction from distant levels is possible exactly when b\theta^2>1; equivalently, the free Gibbs state is extremal exactly when b\theta^2\le1. For the regular-tree result, including equality, see the Bethe-lattice work of Bleher, Ruiz, and Zagrebnov [3] and Ioffe [10]. Ioffe [9] and Evans, Kenyon, Peres, and Schulman [6] developed branching-number criteria on general trees away from equality. Pemantle and Peres [17, Theorem 2.1] later gave an electrical-capacity criterion for free-state extremality for couplings bounded above and away from zero, which also covers critical cases.

Reconstruction asks whether distant spins retain information about a fixed spin. The factor question asks for a rule that generates the entire law from independent labels. Nonreconstruction supplies no such rule, and a weak limit of factor laws need not itself be a factor [12, Corollary 3.3]. Thus equality requires a construction that retains measurable dependence on its independent input.

In the uniqueness range \theta\le b^{-1}, including equality, a finite-cluster construction gives the factor; see Lyons [12, Section 3]. Nam, Sly and Zhang [14, Theorem 1 and Section 2.2 in arXiv:2012.09484v2] used posterior-drift stochastic differential equations driven by IID Brownian motions to construct the free Ising law as a factor for d\ge d_0 and \theta\le c/\sqrt{d-1}, for unspecified absolute constants c,d_0>0. Their construction takes root-independent limits of finite-volume solutions. Theorem 1.1 proves their conjecture for every d\ge3, including equality and their pointwise convention for equivariance.

The known converse. For a centered variance-one real vertex factor, the BSV bound is

It applies also under (1.1). To check this, condition the observable at o on the labels in its radius-R ball and average a Borel version over the finite automorphism group of that rooted ball. Every such permutation extends to a tree automorphism fixing o, so uniqueness of conditional expectation leaves this average unchanged almost surely. It now defines an exactly invariant local rule. Apply the same rule at every vertex. These local factors converge in L^2 at each vertex to the original factor, because the balls exhaust the label coordinates. For all sufficiently large R their variances are positive. Center and normalize them, apply BSV, and pass to the limit in variances and covariances on this same IID probability space. Since the Ising spins have mean zero, variance one and correlation \theta^n, the bound would require (\theta\sqrt b)^n\le1+(d-2)n/d. This is impossible when \theta>b^{-1/2}.

1.1 Why independent innovations determine the spins

We follow the observation framework of Nam–Sly–Zhang. On an auxiliary probability space, sample the desired spin configuration \sigma and independent standard Brownian motions B_v, and observe

The finite-dimensional posterior evolution also belongs to stochastic localization. Eldan [5] developed the underlying localization scheme, and El Alaoui and Montanari [4, Section 3, Theorem 2] give the fixed-covariance Gaussian-observation formulation used here. Subtracting the conditional drift gives

These innovations are independent Brownian motions in the observation filtration. This property alone does not make X a function of W: the original construction still used the spins. Classical innovations theory separates the Brownian property from generation of the observation filtration [7, Lemma 2.2 and Remark 2.2]. Proving that the innovations determine X is the central step here.

A companion construction for the free uniform spanning forest uses a uniform Lipschitz response bound in the coordinate supremum norm to invert the innovations causally for every continuous input [15, Section 4]. Here recovery instead follows from equality of conditional copies under the observation law.

Take two observations X,X' independently conditional on the entire innovation array W. They have the same marginal law and share the same innovations. Conditioning on the full paths could in principle expose their future increments through an observation past. We prove that it does not: the conditional law of either observation past given all of W depends only on the past of W. Hence W remains Brownian in the joined filtration, and the equations for both observations use the same driving Brownian motions. This is the compatibility principle for conditional copies [11, Lemmas 2.11–2.12 in arXiv:1305.6747v2]; we prove the required statement and the transport of the stochastic integrals directly.

We then show that these particular copies agree. For each coordinate, conditional independence gives

Equality therefore forces the conditional law of X given W to be a point mass. Finally

recovers the spins from W. A common measurable sampler converts one uniform label at each vertex into its Brownian path. This gives a spin factor satisfying (1.1); finite-ball averaging then selects an everywhere equivariant version with the same law.

1.2 What changes at the critical point

The tree gives a concrete equation for comparing the copies. Set h_v=\operatorname{atanh}m_v. If T_{j\to i} is the component containing j after deleting ij, define its field by

using the unconditional prior on that component. For a path v=v_0,\ldots,v_r=u, the field-response coefficient is

The innovation equation is

Its infinite sum requires proof. At criticality we obtain \E A_v(t)\le C/\min(\sqrt t,1) for t>0, so the row is integrable down to time zero. An exact covariance-energy identity and a testing argument establish the equation without a residual martingale.

For the conditional copies, let V(t)=\E[(h_o(t)-h'_o(t))^2]; invariance makes the choice of o irrelevant. It starts at zero. Below the threshold, the squared factor \theta^{2r} and the shell size d b^{r-1} give a summable geometric series, yielding V'\le CV and thus equality of the copies. At b\theta^2=1 those two factors cancel exactly.

Three estimates replace the lost geometric decay. First, an intact side branch attenuates a path product on the length scale t^{-1/2} at small times. Second, spatial cancellation bounds the variance of a sum of differences between the two copies’ \log a(h_{j\to i}) factors by the path length times the variance at one edge. The spectral bound is proved for differences f(X)-f(X'), with the same square-integrable equivariant vertex observable f applied to each copy, and their closed linear span. No spectral bound for every function of the pair is assumed. The operator argument is related to the correlation methods of Backhausz–Szegedy–Virág [2] and Backhausz–Gerencsér–Harangi–Vizer [1]. Third, concentration of path averages makes the attenuation usable with discrepancy weights that depend on the same observations. Both path orientations are needed in this averaging argument.

These estimates leave one small-time difficulty: the leading noise term is of order S(t)/t, where

The elementary bound S\le CV would leave a nonintegrable coefficient. We prove instead a small-field tail estimate and use the vanishing derivative of \log a at zero to improve that comparison. The complete noise and drift calculation then gives an integrable Osgood-type inequality for V, forcing V=0. The moment estimate is proved beside this final comparison, where its purpose becomes explicit.

Notation and organization. The case \theta=0 is already IID, so the proof below assumes 0<\theta\le b^{-1/2}. Constants c,C>0 may depend on the fixed d,\theta and may change between occurrences; a subscript T permits additional dependence on a fixed finite time horizon. Every path is simple unless specified otherwise, and all logarithms are natural. Roots used to describe subtrees or compute norms are auxiliary choices in the proof, not inputs to the factor.

Sections 2–4 develop the posterior identities, critical information bounds, and infinite innovation equation. Section 5 constructs the conditional copies and proves their equality below the threshold. Sections 6 and 7 establish spatial cancellation and concentration of path averages. Section 8 proves the small-field tail estimate, completes the critical comparison, and recovers the factor from the innovations.

2 Posterior fields along the tree

The observation construction reduces the factor problem to a question about conditional spin laws. We first describe those laws on a finite subtree, where the Gaussian likelihood is explicit. The tree structure then expresses each covariance as a product along the connecting path. Passing these formulas to the whole tree will identify the response coefficients whose squared rows must be integrable in time.

For the affirmative implication, write b=d-1 and assume 0<\theta=\tanh\beta\le b^{-1/2}; the case \theta=0 is already IID. On an auxiliary probability space sample the free Ising configuration \sigma, and independently sample standard Brownian motions (B_v)_{v\in\mathcal V}, independent across vertices. Set

The B_v’s are the noise used to plant these observations. The innovations W_v, obtained later by subtracting the conditional drift from X_v, are different processes. In this section \mathcal F_t^0 is the raw observation filtration; we reserve \mathcal F_t for its usual augmentation. For any D\subseteq\mathcal V, write

The Gaussian observation framework follows Nam–Sly–Zhang [14]. In finite volume it is the observation form of stochastic localization described by El Alaoui–Montanari [4, Section 3]. We derive the finite posterior directly, since its derivative and its tree factorization will both be needed.

2.1 The finite posterior and its branch messages

Let D be a finite connected subtree. Its broadcast marginal is

Indeed, choose any vertex of D as the broadcast root and multiply its probability 1/2 by the transition probabilities on the |D|-1 edges; also 1+\theta s=e^{\beta s}/\cosh\beta for s\in\{-1,1\}. Conditional on \sigma_D=\eta, the likelihood of the observation paths through time t, relative to product Wiener measure, is

The last term is independent of \eta. Bayes’ formula gives, for every finite terminal vector x\in\mathbb R^D,

Here Z_D(x) is the positive finite partition function. This calculation shows that the terminal vector is sufficient for the spin posterior; it does not identify the terminal-vector sigma-field with the entire observation history.

Let

By Bayes’ formula, m_v^D(t) is a version of \E[\sigma_v\mid\mathcal F_t^D]. All posterior masses are positive, so |M_v^D(x)|<1 and the field is finite. The functions M_v^D are smooth. If u,v\in D, then differentiating the finite sum in (2.2) gives

At x=X_D(t) we denote this conditional covariance by C_{vu}^D(t).

For an oriented edge v\to w, let T_{v\to w} be the component containing v after deletion of \{v,w\}. If both vertices belong to D, define

Thus a component field uses only its own component’s observations and the unconditional free prior on that component. It is not the field at v after observing the opposite side or specifying the spin at w. The same distinction will be essential on the infinite tree.

The normalized response c_{vu}, rather than the covariance alone, is the coefficient that will occur in the field equation. At distance r, its deterministic bound is \theta^r. Below the threshold this already gives a summable squared row. At equality, the size of a distance shell cancels \theta^{2r}; the next section obtains the additional decay from the observations in side branches.

3 Critical information controls the response row

At the critical parameter, the squared edge factor in (2.6) is exactly balanced by the number of vertices at that distance. To sum a response row, we must therefore obtain decay from the factors a(h_{v_i\to v_{i-1}}). We will show that the observations in one intact infinite side branch supply a loss of order \sqrt t at each path step for small t. The resulting expected squared row is of order at most t^{-1/2}, which is integrable at the initial time.

Until the final row estimate, assume b=d-1\ge2 and \theta=b^{-1/2}. Constants in the critical estimates depend only on the fixed degree d. We work at a fixed deterministic time t>0 with the raw posterior versions from Lemma 2.1; their continuity is not needed. For an oriented edge v\to w, put

The law of Y is independent of the oriented edge. It is symmetric: reversing all spins and all observations preserves their joint law and negates the conditional field almost surely at this fixed time. We first determine the scale of M(t). Its lower bound gives path decay, while the whole-site upper bound will control the early-time energy in the filtering limit.

3.1 Combining symmetric binary signals

Combining two branches loses a definite amount of second-moment information compared with adding their separate contributions. This quadratic loss is related to Ioffe’s recursive second-moment estimates for Ising trees [9, Proposition 4]. The following elementary formulation will allow us to keep the loss explicit.

The subadditivity in this lemma can be iterated over the branches of a vertex. A collection of already combined signals remains independent of every unused signal conditional on the spin. Its posterior mean remains symmetric, and remains strictly inside (-1,1), since

Only the first two child signals below will require the sharper loss for means bounded by \theta.

3.2 The amount of information in one component

For a lower bound, we use a weighted linear estimator in the tradition of second-moment reconstruction estimates on trees [6]; see also Mossel–Peres [13, Section 5, Lemma 5.3]. Here it uses the Gaussian observations at all depths up to a cutoff. The cutoff balances the independent noise against the correlations among the spins. For the upper bound, the fusion loss gives the complementary estimate.

We have now identified the critical information scale. To turn its lower bound into decay along a path, we need the conditional variation of a side branch’s outgoing message, even when the attaching spin has been specified.

3.3 Attenuation from an intact side branch

The next argument uses the full infinite component beyond one side edge. Its conditioning specifies the spins on a deterministic path, while leaving the selected side branches unobserved. This is the setting in which the information lower bound supplies a loss uniformly over the contribution from the rest of the tree.

The loss above comes from the posterior information in an intact infinite branch. It is not a claim about arbitrarily truncated branches: a finite boundary will require a separate estimate using the target’s own Gaussian observation. For the present response row, the infinite-branch estimate is exactly what is needed.

3.4 Time-integrable response rows

Return to the full range 0<\theta\le b^{-1/2}. For the progressive coefficients in (2.6), define the extended nonnegative sum

It is measurable as a countable sum. The following estimate is the input that permits division by the posterior variance when the filtering equation is converted to a field equation.

In particular, A_v(t) is finite almost surely at each fixed positive time and is finite for almost every time almost surely on each compact horizon. At criticality the raw sum at t=0 is infinite: all fields vanish, a(0)=1, and every positive-distance shell contributes d/b. That single time has zero Lebesgue measure. The integrated estimate, rather than a bound at time zero, is the conclusion used in the construction of the infinite innovation equation.

4 Constructing the infinite innovation equation

The posterior fields describe how the observations affect the spins. We now turn that description into a stochastic equation driven by independent Brownian motions. This is the equation that will allow us to compare two observation processes driven by the same noise. Throughout the section,

We use the raw observation versions from Lemma 2.1. In particular, at this stage a posterior identity holds almost surely at each fixed time, and therefore outside a \dd t\,\dd\P-null set; continuity of the infinite filter has not yet been established.

For each vertex define its innovation by

The integrand is a bounded, progressively measurable observation function. Thus this formula defines continuous paths for every observation input. Their values through time t use only the observation history through t. The time integrals are Borel functions of that history, so X\mapsto W is a causal Borel map on the countable product of continuous path spaces. The centered-ball definitions of the posterior means also show that this map commutes with graph relabeling.

The innovation construction is classical in nonlinear filtering [7] and is the starting point of the Brownian Ising construction of Nam, Sly, and Zhang [14]. Here we must justify its equation for countably many observations. We will first prove that W is Brownian, then identify the entire posterior martingale using an exact energy limit. Neither step assumes that W determines X.

The raw and augmented filtrations both occur in the proof. We record the elementary passage between them once. If (\mathcal G_t^0) is a raw filtration, write \mathcal G_t=\bigcap_{r>t}\overline{\mathcal G_r^0} for its usual augmentation, where the bar includes the null sets of the ambient probability space.

4.1 Brownian innovations and finite posterior energy

We first prove the asserted Brownian property. For 0\le s<t, the future increments of the planted Brownian motions are independent of all spins and of their Brownian pasts. Conditioning on the smaller raw observation history therefore gives

For every deterministic r\ge s, the tower property for the bounded raw posteriors also gives \E[m_v(r)\mid\mathcal F_s^0]=m_v(s). Conditional Fubini is applicable because |m_v|\le1. Subtracting the conditional means of the two increments in (4.1) shows that W_v is a continuous square-integrable martingale in (\mathcal F_t^0).

This conclusion concerns the innovations, not the planted noises. In the filtration containing all spins at time zero and all Brownian histories, their relation is

The added path has finite variation. Hence the pathwise quadratic covariations are

The bounded drift and L^2 continuity of B_v give L^2 continuity of W_v. Lemma 4.2 makes each coordinate a martingale in (\mathcal F_t). Lévy’s characterization applied to each finite vector, with the displayed covariations, proves that vector is standard Brownian motion in (\mathcal F_t) and hence also in the smaller raw filtration. Its future increments are independent of the observation past. By a monotone-class argument over finite vertex sets and rational future times, the entire future increment field (W_v(t+s)-W_v(t))_{v\in\mathcal V,s\ge0} is independent of \mathcal F_t for every fixed t.

Finite observation sets have their own innovations. They will provide an exact scalar energy identity, but we will never identify their Brownian paths with the infinite innovations.

4.2 No covariance energy escapes to the boundary

The finite energy identity does not yet imply its infinite analogue: coordinatewise convergence could leave a positive amount of energy at ever more distant vertices. We control that possibility in two intervals. Away from time zero, each vertex’s own Gaussian noise attenuates its path factor, including at a finite-tree leaf. Near time zero, the whole-site information estimate bounds the entire finite energy at once.

Fix v and let D_R be its radius-R ball. Extend C_{vu}^{D_R} by zero for u\notin D_R. For each fixed u and T<\infty,

Indeed, at each deterministic s, increasing-sigma-field convergence applies to the bounded variables \sigma_v, \sigma_u, and \sigma_v\sigma_u. Their conditional means determine the covariance. The covariances are uniformly bounded, so dominated convergence also integrates the resulting L^2 convergence in time.

When b\theta^2<1, the pointwise bound C_{vu}^{D_R}(s)^2\le\theta^{2\dist(v,u)} controls all spatial tails: there are n_r=db^{r-1} vertices at distance r\ge1, and

The same deterministic estimate holds for the infinite coefficients.

At equality, fix 0<\delta\le\min\{T,1\}. The tower property gives m_v^{D_R}(\delta)= \E[m_v(\delta)\mid\mathcal F_\delta^{D_R}]. Conditional Jensen, the finite energy identity, and the whole-site estimate in (3.2) yield

On [\delta,T], Lemma 4.4 applies uniformly in R, and now n_r\theta^{2r}=d/b. Thus

This also bounds the infinite tail. To see this without assuming its finiteness, first restrict that tail to a finite annulus, pass R\to\infty using (4.6), and then increase the outer radius using monotone convergence. First send L\to\infty with \delta fixed, so that the geometric term vanishes. Then send \delta\downarrow0. The order matters: \kappa_\delta may shrink as \delta tends to zero. We have proved that the integrated spatial tails of both the finite and the infinite rows vanish uniformly.

For a fixed L, the sum over \dist(v,u)\le L is finite. Equation (4.6) passes its squared energy to the limit in L^1. The uniform tail estimates then pass the complete energy to the limit. At each fixed t we also have m_v^{D_R}(t)\to m_v(t) in L^2. Consequently (4.4) gives the exact infinite identity

At t=0 both sides are zero. Only the scalar energies have been passed to the limit; the distinct processes W^{D_R} have played no part in this limiting operation.

4.3 Identifying the entire posterior martingale

The energy identity gives the norm that a Brownian representation must have. A second identity identifies its inner product with the posterior. Together they determine the posterior itself and exclude any additional martingale orthogonal to the innovations.

We first record the approximation that permits coefficients to be used in this test. It will also let the subsequent conditional-copy construction transport individual stochastic integrals.

Fix v,u and t<\infty. For any bounded elementary process of the same form whose \xi_j is merely \mathcal F_{t_j}^0-measurable, we claim

For an elementary process, the integral is the finite sum of \xi_j(W_u(t_{j+1})-W_u(t_j)), hence is measurable in \mathcal F_t^0. In its expectation against m_v(t) we may therefore replace m_v(t) by \sigma_v. In the planted filtration,

Here \sigma_vG is a bounded predictable integrand: \sigma_v is known at time zero and every raw observation past is contained in the planted past. Thus \E[\sigma_v\int_0^tG\dd B_u]=0. For the finite-variation part, condition at time s on the raw observation history:

Boundedness permits Fubini, proving (4.10). This argument has used only the fact that W is a function of the observation past, not any converse measurability.

Apply Lemma 4.5 to C_{vu}. Itô’s isometry for W_u and Cauchy–Schwarz on the two sides of (4.10) pass to the limit, so the identity also holds with G=C_{vu}. For a finite vertex set F let

The integrals for different u are orthogonal, even though their integrands may depend on every observation: their driving Brownian coordinates have zero mutual quadratic covariation. The isometry and the testing identity give

Combining this with the exact energy (4.9) yields the useful squared-error identity

The right side tends to zero as F exhausts the vertices. This is precisely the step that eliminates a possible residual martingale.

There is also convergence of the processes, rather than just their values at a fixed time. For F\subseteq F', Doob’s maximal L^2 inequality gives

The total energy is finite by (4.9). The partial sums therefore converge in \mathcal M_T^2 to a continuous square-integrable martingale. The limits agree on overlapping horizons, and (4.11) identifies each fixed-time value with the raw posterior m_v(t). Countability of the vertices lets us choose these continuous versions simultaneously. This proves (4.2).

4.4 Continuous versions that retain component locality

The Brownian representation has supplied a continuous version of each site mean. We now select Borel observation functions realizing this version and continuous component fields. Constructing component fields from site fields could apparently use information from the opposite side of an edge. We first establish continuous component modifications, then choose them by explicit past-observation formulas that retain the required component restriction.

Write M_v for the continuous mean version temporarily. At every rational time it agrees almost surely with the raw mean from Lemma 2.1. Intersect these events over all vertices and rational times. The raw field bound and continuity of M_v and X_v imply, on this one event,

Indeed the inequality holds first at rational times and then passes to every time by continuity. On each compact interval,

Thus H_v=\operatorname{atanh}M_v is continuous. The stronger inequality |H_v-X_v|\le d\beta also passes from rational times to all times.

For adjacent vertices, consider for arbitrary real H,K the equations

They have a unique solution. In fact x is the unique fixed point of

a contraction of the complete space \mathbb R with constant at most \theta^2<1, and y=K-\psi(x). Comparing two solutions gives

and the analogous bound for y. The solution therefore depends continuously on (H,K).

Apply this construction to (H_v(t),H_w(t)) and denote the resulting continuous processes by (H^c_{v\to w},H^c_{w\to v}). At every fixed time the raw cavity identities identify this solution almost surely with the raw component fields, by uniqueness. After intersecting over rational times and oriented edges, continuity therefore extends both cavity identities and the component bound to all times. This proves existence of continuous component modifications. The construction by adjacent site fields is used only for that existence assertion.

We now specify the observation functions actually used in the rest of the proof. Let m_v^0(t,x) and h^0_{v\to w}(t,x) be the original raw functions obtained from the centered finite-ball limits in Lemma 2.1. For q_n(t)=2^{-n}\lfloor2^nt\rfloor, set

and set \widehat h_v=\operatorname{atanh}\widehat m_v. For every oriented edge, set

For each n the sampled process is constant on the half-open dyadic time intervals, and its value there is a Borel function of the appropriate observation history at the left endpoint. It is therefore progressively measurable. Real convergence is a countable Cauchy condition. For the site selector, membership of the resulting limit in (-1,1) is an additional Borel condition. Taking these limits with the displayed defaults therefore preserves Borel and progressive measurability. For the component expression both the entire sequence and its default decision use only observations in T_{v\to w} through time t. In particular, two inputs agreeing on that component history have the same sequence and the same chosen value. This is pointwise component locality, including on inputs where the limit fails.

At every dyadic time the raw site mean agrees almost surely with M_v, and the raw component field agrees almost surely with H^c_{v\to w}. Intersecting over the countable dyadic times and indices gives one event of probability one. On that event, for every real t the sampled values above are values of the respective continuous modifications at q_n(t). Since q_n(t)\to t, continuity makes their limits agree simultaneously for every t. Thus the chosen functions are indistinguishable from the continuous modifications. They retain all the all-time cavity identities and bounds just proved. The same dyadic rule and the same defaults at every index also preserve the graph-relabeling covariance of the raw functions.

We henceforth use these chosen versions and omit the hats. Define c_{vu} again by its finite path product and C_{vu}=(1-m_v^2)c_{vu}. These are continuous in time almost surely, and they agree at each fixed time with their former raw versions. Consequently every changed coefficient agrees with its former version outside a \dd t\,\dd\P-null set. Likewise, Fubini and boundedness give equality of the old and new mean integrals on each integer horizon, and hence, after a countable intersection,

Thus the innovation paths have not changed.

These versions are also the posteriors for the usual filtrations. For a fixed time t, apply the second part of Lemma 4.2 to the bounded continuous version of \E[\sigma_v\mid\mathcal F_t^0]. It gives m_v(t)=\E[\sigma_v\mid\mathcal F_t]. The process C_{vu}+m_vm_u is a bounded continuous version of the raw conditional mean of \sigma_v\sigma_u; applying the same argument identifies it with that conditional mean in \mathcal F_t. This identifies C_{vu} as the augmented conditional covariance. Applied instead to \tanh h_{v\to w} and the raw component filtration, the argument gives the corresponding assertion for its usual augmentation. These statements include time zero. The one-spin site and component means start at zero. The two-spin posterior starts at its prior value \E[\sigma_v\sigma_u]=\theta^{\dist(v,u)}, including value one when u=v.

4.5 The field equation

It remains to pass from the posterior mean to its field. This requires the response row A_v=\sum_u c_{vu}^2, rather than merely the covariance row: the transformation divides by 1-m_v^2. The estimate (3.11), proved before this section, gives

Its strict case follows from a deterministic geometric sum. At criticality it follows from the intact infinite-side-branch estimate \E A_v(s)\le C/\min\{\sqrt s,1\} for s>0. This is a different use of attenuation from the finite-boundary Gaussian estimate above. The inputs here are the second-moment information and forward infinite-branch attenuation already proved.

For clarity, define the version used in the drift directly from the chosen observation functions:

The extended nonnegative sum is a countable sum of Borel progressive functions, so this finite-or-zero version is again a causal Borel progressive observation function. Permuting the summation index and using the same default preserve graph relabeling. The integrated row estimate shows that this version agrees with the extended sum outside a \dd s\,\dd\P-null set. The predictable representatives used for Brownian integration agree with the chosen path functions outside one such null set for all u, by countability. Their squared sum therefore agrees with this A_v there. At criticality the unmodified row is infinite at time zero, as noted after (3.11); the displayed convention supplies the value used in the drift.

The Brownian representation and the covariance formula now imply, as an equality of densities up to \dd s\,\dd\P-null sets,

One may obtain the first equality from finite Brownian sums: their brackets converge in L^1 total variation on every finite horizon, by (4.9), while their martingales converge in \mathcal M_T^2. The defining bracket identity then passes to the limit.

For n\ge2 let

On the stopped interval the derivatives

are bounded. Itô’s formula for the stopped continuous martingale therefore gives

The Brownian coefficient is exactly c_{vu}. The equality of this countable Brownian sum with the transformed mean integral follows first for finite sums and then by the isometry; on the stopped interval the derivative is bounded.

The global Brownian sum with coefficients c_{vu} exists in \mathcal M_T^2. Indeed, for finite F\subseteq F', Doob’s inequality gives

The right side tends to zero along an exhaustion by the integrated row estimate. Also \E\int_0^T|m_v|A_v\dd s\le\E\int_0^T A_v\dd s<\infty. Finally \tau_n\uparrow\infty almost surely: the continuous mean stays strictly inside (-1,1) on every compact interval. The stopped identities hold simultaneously in time by continuity. On any fixed compact interval, almost every path therefore agrees with one of these identities for all sufficiently large n. Removing the stopping yields (4.3).

We now have the full innovation-driven field equation in the observation filtration, with both Brownian and drift terms integrable. The next comparison must prove that the same innovations remain Brownian when two conditional observation histories are put together; that compatibility is the next step of the argument.

5 Comparing observations with the same innovations

The innovations constructed in the preceding section are independent Brownian motions, but that fact alone does not show that they determine the observations. We now turn this question into a comparison of two observation processes. We sample them independently conditional on the same entire innovation array, prove that they can still be driven by the same Brownian motions in their joint filtration, and derive an equation for their discrepancy. This is the conditional-copy compatibility method developed by Kurtz [11]; we prove the specific compatibility and transport statements needed here.

Throughout this section, d\ge3, b=d-1, and 0<\theta=\tanh\beta\le b^{-1/2}. We use the continuous causal observation functions selected in Proposition 4.1. With the selected means, the innovation map

is a Borel function of X whose restriction to [0,t] uses only observations up to t. By Section 4 and a countable intersection over vertices, this selected-mean innovation map agrees with the original raw innovation map as a whole-path map almost surely under the observation law. We use this selected version throughout. The field equation (4.3) and the integrated row bound (3.11) have already been proved in the observation filtration. No inverse map from W to X is assumed in what follows. For drift integrals involving A_v, we use the finite-or-zero drift convention fixed in Section 4.

5.1 Conditioning on the complete innovations

Write

Give the path space its locally uniform Polish topology and the arrays the countable product topology. The subscript 0 specifies the value at time zero and imposes no condition at infinity. Let \mathsf Q be the joint law of (X,W) from the planted observation construction. Its W-marginal is product Wiener measure, denoted by \mathsf w. These are standard Borel spaces, so there is a regular conditional probability \mathsf K(w,\dd x) for X given W=w. Define a probability law on the three path arrays by

Write X^1,X^2,W for its coordinates. Each pair (X^j,W) has law \mathsf Q, while X^1 and X^2 are independent conditional on W. The conditioning in (5.1) uses every coordinate of W at every time, not only the portion already observed.

For t\ge0, let \mathcal H_t^0 be the raw sigma field generated by X^1_v(s),X^2_v(s),W_v(s) for v\in\mathcal V and 0\le s\le t, and let (\mathcal H_t) be its usual augmentation. All posterior fields and coefficients attached to copy j below are obtained by applying the same causal observation functions to X^j.

The purpose of this construction can already be stated precisely. For each vertex v and nonnegative rational q, \E_{\mathsf Q}X_v(q)^2=q+q^2<\infty. Conditional independence gives

Thus equality of the two observation paths would make every rational coordinate have zero conditional variance. These countably many coordinates determine a continuous path configuration. The final section will turn that implication into a Borel recovery map after the comparison has proved equality; no equality has been assumed here.

5.2 The field discrepancy and its variance

From now on unprimed quantities refer to the first copy, primed quantities to the second, and \Delta means their difference. All expectations in the comparison are under \widehat{\mathsf Q}. Every one-copy estimate proved earlier remains valid because the marginals have not changed.

Fix T>0. In each pair the defining innovation identity holds as an identity of continuous paths. Subtracting the two identities gives

Together with |h_v-X_v|,|h'_v-X'_v|\le d\beta, this gives

on one full-probability event. The field bounds and identities hold simultaneously because the chosen fields are continuous and the vertices are countable.

Choose a vertex o only for taking expectations, and put

Invariance of the joined law makes the value independent of the choice of o. It is nonnegative, bounded by H_T^2 on [0,T], and V(0)=0. We need the full variance equation, including its drift.

The coefficients in this equation are products along directed paths. We first control the difference of one factor. For a directed edge e=(i,j) define

Transitivity on directed edges makes S independent of e. The two cavity identities at the ends of ij give

Substituting the second inequality into the first yields

In particular, on [0,T] it is bounded by H_T/(1-\theta); Minkowski’s inequality and invariance also give \|\Delta h_{j\to i}\|_2\le\sqrt V/(1-\theta).

Direct differentiation gives

Here the first inequality uses 1-\theta^2\tanh^2z\ge1-\theta^2. We conclude that

Also (\log a)(0)=0, so |\log a(h_{j\to i}(t))|\le2|h_{j\to i}(t)| \le2(|X_j(t)|+b\beta). Since \E X_j(t)^2=t+t^2, each one-copy log factor is square integrable. Equality of the marginal laws then gives \E D_e(t)=0. These facts will permit the later spatial cancellation estimate to be applied to this particular edge observable.

5.3 Reducing the comparison to path estimates

For a directed simple path v_0,\ldots,v_r, r\ge1, let e_i=(v_{i-1},v_i). At a common time t, set

The orientation matters: every factor uses the component at the next vertex excluding the preceding one. Invariance makes these expectations depend only on r and t, so we suppress the particular path. Both P_r and P'_r lie in (0,1], and \log P_r-\log P'_r=Z_r. The mean-value theorem, first for e^x and then for e^{2x}, gives

For example, the derivative of e^x on the segment joining \log P_r and \log P'_r is at most P_r+P'_r; squaring and using (P_r+P'_r)^2\le2(P_r^2+(P'_r)^2) proves the first bound. The derivative of e^{2x} on the same segment is at most 2(P_r^2+(P'_r)^2), proving the second.

There are n_r=db^{r-1} vertices at distance r\ge1 from o. The diagonal coefficients are c_{oo}=c'_{oo}=1, and the coefficient along any positive-length path is \theta^rP_r. Tonelli’s theorem and (5.12) therefore bound the noise term in (5.6) by

The drift contributes a second, different path sum:

To prove this, write m_oA_o-m'_oA'_o=(m_o-m'_o)A_o+m'_o(A_o-A'_o). Since \tanh is 1-Lipschitz, the first part is bounded by

For the second, use |m'_o|\le1. The diagonal terms of A_o-A'_o cancel. For each remaining path, put w_r=P_r^2+(P'_r)^2\le2. The second inequality in (5.12) and weighted Cauchy–Schwarz give

Summing proves (5.14). These inequalities may first be read for finite spatial sums and then with nonnegative extended right sides; the estimates at their subsequent uses make those right sides finite. In particular no independence between a path product and its discrepancy is used. The square-root sum in the drift must be controlled along with the noise sum.

At equality, n_r\theta^{2r}=d/b, so the geometric decay in this proof disappears. The next two sections will replace the elementary r^2S loss by spatial cancellation of order rS and attenuation on the length scale \omega(t)^{-1}. Their leading noise estimate has the form Q_r\lesssim rS(t)e^{-c\omega(t)r}. Near time zero, where \omega(t)=\sqrt t, its sum has size

Thus even those two estimates, combined only with S\le CV, would leave a nonintegrable coefficient at time zero. The final comparison will improve the relation between S and V using small component fields, while also retaining the square-root drift sum. This is the remaining obstruction at the critical parameter.

6 Cancellation along a path

The comparison of the two observation copies led to sums of the edge differences D_e(t). Cauchy–Schwarz bounds the second moment of a sum over r edges by r^2S(t), where S(t)=\E D_e(t)^2. We now improve the factor r^2 to r by proving that the correlations of these differences are summable along a path. The proof begins with local correlations under one observation law and passes to the joining through the closed space generated by differences of marginal functions. Although only equality remains in the comparison, this path estimate holds throughout 0<\theta\le b^{-1/2}.

Here and below \nu is the observation marginal of the original joint law \mathsf Q, namely the law of the entire array

Here C_0 again means continuous paths starting at zero, with the topology of uniform convergence on compact time intervals. The vertex and nonbacktracking operator geometry used in this argument is related to the correlation arguments of Backhausz, Szegedy, and Virág [2, Theorem 2.2 in arXiv:1305.6784v1] and Backhausz, Gerencsér, Harangi, and Vizer [1, Theorems 4.1–4.2 in arXiv:1603.08423v2]. The proof below supplies the precise transfer to differences of marginal observation functions.

We prove the proposition in three steps. First, local observation functions have exponentially small correlations across a long connecting path. This bounds the self-adjoint neighbor average on centered vertex functions. Second, the bound transfers to a closed space of differences inside the joining. Finally, head and tail lifts of that vertex space control the nonbacktracking edge average. Its powers are summable, which gives the linear variance in (6.1). All equivariance identities in this section are understood separately for each fixed automorphism, as in the proposition.

6.1 The vertex bound for one observation law

We shall use the same Hilbert-space notation for several invariant configuration laws. For such a law \eta, a vertex assignment is a family f=(f_v)_{v\in\mathcal V} of measurable real functions satisfying

Let \mathcal H_{\mathrm v}(\eta) be the space of square-integrable vertex assignments, modulo almost sure equality at each vertex, with

Transitivity and invariance make this independent of the vertex o. The analogous space \mathcal H_{\mathrm e}(\eta) consists of assignments to directed edges, with inner product at any one directed edge. These spaces are Hilbert spaces: a Cauchy sequence has an L^2 limit at each index, and the equivariance identity for any fixed automorphism passes to the limits because that automorphism preserves \eta. There are countably many indices, so equality in the indicated norm is equality of assignments. We use real spaces; the usual spectral theorem applies after complexification if needed.

On \mathcal H_{\mathrm v}(\eta) define the neighbor average

Jensen’s inequality and invariance give \|\mathsf A_\eta\|\le1. For adjacent o,u, an automorphism interchanging them gives \E_\eta[f_ok_u]=\E_\eta[f_uk_o]. Averaging this identity over the neighbors of o proves that \mathsf A_\eta is self-adjoint. It preserves the closed subspace of centered assignments \E_\eta f_o=0. This subspace is reducing: if y is orthogonal to it, then \langle \mathsf A_\eta y,f\rangle =\langle y,\mathsf A_\eta f\rangle=0 for every centered f. Thus the restriction to that subspace is again self-adjoint.

6.2 Passing the bound to marginal differences

Fix the joining \pi in Proposition 6.1. For a marginal vertex or edge assignment define

This is well defined on the marginal L^2 equivalence classes: each coordinate under \pi has law \nu. The same observation and the triangle inequality give \|\delta f\|_\pi\le2\|f\|_\nu. In the joined assignment spaces put

where the closures use the respective L^2(\pi) norms. These are closed linear spaces.

The vertex neighbor averages for \nu and \pi commute with \delta, since they use the same finite sums over neighboring vertex indices. Thus \mathsf A_\pi\mathcal K\subseteq\mathcal K. The joined \mathsf A_\pi is self-adjoint, so the same orthogonality argument as above shows that \mathcal K is reducing and its restriction \mathsf A_{\mathcal K}:=\mathsf A_\pi|_{\mathcal K} is self-adjoint.

Subtracting the constant \E_\nu f_o from a marginal vertex assignment does not change its difference. Lemma 6.2 therefore gives

Applying the spectral-measure argument from (6.4) to the joined self-adjoint operator \mathsf A_\pi shows that the spectral measure of \delta f is supported on [-\rho,\rho]. The constant on the right of (6.5) may depend on the chosen preimage f; only its exponential rate matters as n\to\infty. In particular, no lower comparison between \|\delta f\|_\pi and \|f-\E_\nu f_o\|_\nu is used. The closed range of the joined spectral projection \mathbf 1_{[-\rho,\rho]}(\mathsf A_\pi) contains the closure of all these differences. Hence

This is the vertex estimate needed below. Its domain is \mathcal K, the closed space generated by marginal differences.

6.3 From vertices to nonbacktracking edge averages

A vertex assignment can be read at either endpoint of a directed edge. We compare these two lifts and then isolate the part of an edge assignment orthogonal to both. On the joined assignment spaces define the tail and head lifts \mathsf F,\mathsf H:\mathcal H_{\mathrm v}(\pi)\to \mathcal H_{\mathrm e}(\pi), the reversal \mathsf J, and the forward nonbacktracking average \mathsf N by

Invariance makes the two lifts isometries and makes \mathsf J a self-adjoint isometry with \mathsf J^2=I. Jensen’s inequality makes \mathsf N a contraction. The adjoints of the lifts are

Indeed, in the edge inner product we may average the fixed outgoing edge over all neighbors of its tail, because the tail stabilizer is transitive on those neighbors. This proves the first formula. Interchanging the endpoints first proves the second. In particular,

Direct substitution in the definitions gives

Thus (\mathsf N^*g)_{(i,j)} =b^{-1}\sum_{k\sim i,\ k\ne j}g_{(k,i)} and \mathsf N^*=\mathsf J\mathsf N\mathsf J.

The same formulas act on the marginal assignment spaces. They use finite sums over neighboring indices, selections of endpoint values, and edge reversal, so they commute with taking marginal differences. Apply this first on the ranges of \delta, and then use boundedness to pass to their closures. We obtain

Equation (6.8) now shows that both \mathsf N and \mathsf N^* preserve \mathcal D. Thus \mathcal D is a reducing subspace for the nonbacktracking operator.

Define

The adjoint identities and (6.6) give

Here 2|\langle f,\mathsf A_{\mathcal K}g\rangle| \le\rho(\|f\|^2+\|g\|^2). Because \rho<1, \mathsf R is injective with closed range

In fact, the lower bound makes the preimages of a convergent sequence in the range Cauchy. It also gives \|\mathsf R^{-1}\|\le(1-\rho)^{-1/2} on \mathcal L, while the upper bound gives \|\mathsf R\|\le(1+\rho)^{1/2}.

For y\in\mathcal D\ominus\mathcal L, the vectors \mathsf F^*y,\mathsf H^*y lie in \mathcal K and are orthogonal to every vector in \mathcal K; for example \langle \mathsf F^*y,f\rangle=\langle y,\mathsf Ff\rangle=0. Both therefore vanish. Reversal preserves \mathcal D and interchanges the two lifts in \mathcal L, so it preserves \mathcal D\ominus\mathcal L as well. On this complement, (6.8) reduces to

The space \mathcal L is also invariant under \mathsf N by (6.9). We have split \mathcal D into two orthogonal invariant spaces. The complement already decays geometrically; it remains to bound powers on the head and tail range.

On \mathcal K\oplus\mathcal K, (6.9) reads

This matrix need not be normal. We bound its powers by its second-order recurrence. Put

The operator \mathsf B is self-adjoint and \|\mathsf B\|\le1. Matrix multiplication gives

Let U_{-1}=0, U_0=1, and U_{m+1}(x)=2xU_m(x)-U_{m-1}(x) be the Chebyshev polynomials of the second kind. Induction using (6.12) gives

For m\ge0, the trigonometric identity

gives \sup_{[-1,1]}|U_m|\le m+1, including the endpoints. The spectral theorem applied to the self-adjoint contraction \mathsf B therefore yields

For completeness, the off-diagonal part of \mathsf Z has norm \sqrt b, while its only nonzero diagonal block is (d/\sqrt b)\mathsf A_{\mathcal K}, of norm at most 2. Thus \|\mathsf Z\|\le\sqrt b+2. At the allowed scalar endpoints \pm1 for \mathsf B, the characteristic polynomial in (6.12) has a double root. The linear factor in n accounts for this endpoint behavior.

Using \mathsf T^n=b^{-n/2}\mathsf Z^n and the two bounds for \mathsf R and \mathsf R^{-1}, we conclude that \|\mathsf N^n|_{\mathcal L}\| \le C_d(n+1)b^{-n/2}. Together with (6.11) and the orthogonal invariant decomposition, this gives

The same estimate holds for (\mathsf N^*)^n on \mathcal D, by reversal. The spectral theorem was used only for the self-adjoint vertex averages and \mathsf B; the edge estimate follows from the displayed recurrence.

6.4 Summing the edge correlations

Let D=\delta g\in\mathcal D be the difference in the proposition. Its mean at every directed edge is zero, since the two marginals are equal, and \|D\|^2=S. Fix a directed simple path and write e_i=(v_{i-1},v_i). If i<j, invariance and transitivity on directed simple paths give

Indeed, \mathsf N^{j-i} averages over the b^{j-i} nonbacktracking continuations of a fixed directed edge. On a tree each such continuation is simple. An automorphism carries it to the corresponding fixed subpath, so every term has the expectation on the left. This is a finite average of identities for fixed automorphisms.

By (6.13), the absolute value of the lag-n correlation is at most C_d(n+1)b^{-n/2}S. Expanding the square of the sum and grouping equal lags therefore gives

The series converges because b\ge2. If all edges are reversed, order them as (v_r,v_{r-1}),\ldots,(v_1,v_0). They are the forward edges of the reversed simple path, and their sum is unchanged by this reordering. The same argument proves the second inequality of (6.1).

Finally, take \pi to be the (X,X')-marginal of the joining in Proposition 5.1, forgetting its common innovation coordinate. At a fixed finite time t, use

The component versions constructed earlier make this an equivariant marginal assignment. The cavity field bound and (\log a)(0)=0, |(\log a)'|\le2, imply

Since \E_\nu X_j(t)^2=t+t^2, it is square integrable. Its difference is precisely D_{(i,j)}(t). Applying the general estimate completes the proof of Proposition 6.1. The bound is uniform in the time and in the joining because its constant depends only on d.

7 Attenuation with dependent path weights

The path variance estimate controls sums of edge discrepancies, but the critical comparison also needs decay of the products multiplying those sums. Those products and differences depend on the same observations. We therefore average over paths before using the variance estimate. Keeping the first half of a path fixed gives an outward continuation; keeping the second half fixed requires enumerating the first half backward. This is why we need concentration in both directions.

Throughout this section, b=d-1\ge2, \theta=b^{-1/2}, and \omega(t)=\min\{\sqrt t,1\}. We first fix t>0 and work with the spins and observations in the planted realization of one marginal. All fields below are evaluated at this time, which we usually suppress. In particular, h_{v\to w} is the previously chosen measurable function of the observation history in T_{v\to w} alone. We use the shifted estimate from one branch (3.7): for an intact infinite component attached to a spin \sigma_w, every real z, and either value s of that spin,

Here c_0 depends only on d. The shift may be fixed by conditioning on other variables as long as the selected component retains its law conditional on its attaching spin. This is the only quantitative information estimate used in the present section.

7.1 Products in both directions

For a vertex z and either the empty set or a singleton F\subseteq\{u:u\sim z\}, let \mathcal P_\ell(z;F) consist of the simple paths \gamma=(z_0,\ldots,z_\ell) that start at z and, when \ell\ge1, have z_1\notin F. Thus

Define the two squared products and their averages over paths by

Empty products are one, so G^+_0=G^-_0=1. We call the products outward and inward according to the coefficient traversal: R^+ is traversed from z_0 to z_\ell, and R^- from z_\ell to z_0. In each factor the field arrow names the component root and its excluded neighbor, and points opposite to that coefficient traversal.

We will split the path at an intermediate vertex and average its continuations inside disjoint descendant components. Outward continuation products already use only their descendant observations. An inward product can also see the observations above the split. The next lemma removes the one entering message that causes this dependence.

Here is the consequence that makes concentration useful for dependent weights. Let \zeta be any bounded nonnegative random variable on a probability space carrying an observation array with the planted marginal law. It may also depend on further variables coupled to that array. For \ell\ge\ell_*(t), split according to the event in (7.5); on its complement use the displayed bound for G^\varepsilon_\ell, and on the event use G^\varepsilon_\ell\le1. Then

Only the observation marginal enters the event probability. The principal term therefore pays the mean of the weight even when the weight depends on all those observations.

7.2 Averaging with the discrepancy weights

Return to the invariant joined law of the two observation copies. Recall from (5.11) that, on a directed path v_0,\ldots,v_r with e_i=(v_{i-1},v_i),

Here D_{(i,j)}=\log a(h_{j\to i})-\log a(h'_{j\to i}), S=\E D_e^2, and V=\E(\Delta h_v)^2. Invariance makes S,V,Q_r,L_r independent of the chosen edge, vertex, or directed path of that length. The path averages below use that invariance and leave all observation fields unconditioned.

8 Critical comparison and recovery

We now show that two conditional copies with the same innovations agree at the critical parameter \theta=b^{-1/2}, where b=d-1\ge2. The weighted path estimates reduce the question to the variance V(t)=\E[(\Delta h_o(t))^2] of the site-field difference. Their principal noise term has the scale

where S(t)=\E[D_e(t)^2] and D_{(i,j)}=\log a(h_{j\to i})-\log a(h'_{j\to i}). The earlier estimate S\le CV would leave a coefficient 1/t, which is not integrable at zero. We improve it by using the fact that (\log a)' vanishes at the origin. A field is usually small at early times; the first result below makes the exceptional probability small enough to compare it with any positive value of V.

8.1 All moments and a small-field tail

We use the fixed-time posterior identities and critical information estimate already proved for one observation process. At a deterministic time t>0, fix an oriented edge v\to w and write

The law of Y is independent of the choice of edge and is symmetric. Recall that M(t)\le C\sqrt t for 0<t\le1, that |Y-X_v(t)|\le b\beta, and that X_v(t)=t\sigma_v+\sqrt t\,G with G a standard normal variable independent of the spin. These are statements about the planted observation law. Each marginal of the conditional coupling has this same law.

The second moment is critical because b\theta^2=1, whereas b\theta^p<1 for every integer p\ge3; this strict contraction lets the recursion propagate the second-moment scale to every higher integer order.

8.2 Comparing log-factor and site-field differences

Fix a finite horizon T>0. The conditional-copy construction gives

and Lemma 5.2 makes V absolutely continuous, with V(0)=0. Choose a constant, fixed throughout this horizon,

In particular I(t)>1, including at times when V(t)\ge1.

8.3 A cutoff depending on the discrepancy

We next sum the weighted estimates. Recall \omega(t)=\min\{\sqrt t,1\}, and suppress t at a fixed time with 0<t\le T and V(t)>0. The preceding path results give

for fixed positive constants c,\alpha,\gamma,C_0' depending only on d. The last two lines follow from Proposition 7.3, with constants decreased or enlarged so that both exceptional terms have the indicated common form. These estimates include weights and discrepancies formed from the same observations.

Choose C_1 sufficiently large that

and take the integer

The weighted estimates apply for r>R. Since \omega^{-1}\ge1 and I>1, the choices above also give

Splitting the exceptional exponent into equal halves therefore gives

The last step uses K_T>1, and so remains valid when V\ge1.

8.4 An integrable comparison at time zero

8.5 Recovering the observations and the spin factor

The comparison has now proved V(t)=0 for the particular copies sampled in (5.1): Proposition 5.3 handles b\theta^2<1, and Proposition 8.4 handles b\theta^2=1. We first deduce equality of their observation paths and turn it into a Borel map from innovations to observations. The spins will then be read from the integer-time slopes of those paths.

The final passage to a spin map equivariant on every input uses local conditional expectations.