Random Walks on Critical FK–Ising Maps and Liouville Brownian Motion

Proof. Since \mu_n(x)=\deg(x)/(2n), every incident half-edge has \mu_n-weighted accelerated rate 1/2. A nonloop edge with endpoints x,y therefore contributes

to \ip{u}{G_nv}_{L^2(\mu_n)}. Both half-edges of a loop contribute zero. Summing proves (2.1), including multiplicities. Zero energy forces equality across each edge, hence constancy by connectedness.□

Proof. For every finite real Y, bounded convergence shows that F_Y(r)=\EE[\arctan(Y-r)] is continuous, strictly decreasing, and has limits \pi/2 and -\pi/2 at the two ends of the real line. Its zero therefore exists and is unique. Directly, \mathfrak m(Y+c)=\mathfrak m(Y)+c. If Y_j converges in law to Y, then F_{Y_j}(r)\to F_Y(r) for each fixed r. For each \varepsilon>0, the values at \mathfrak m(Y)-\varepsilon and \mathfrak m(Y)+\varepsilon have opposite strict signs. The same eventually holds for F_{Y_j}, trapping its zero between these two points. Thus \mathfrak m is continuous under convergence in law.

Diameter is continuous in the Gromov–Hausdorff topology. Since a_n\to0, the assumed convergence gives

Applying the preceding continuity statement after taking logarithms and using the shift identity gives

□

Source of the inputs and comparison of conventions. The substantial assertions are imported from [12]. Its Proposition 2.4 supplies the common coordinate, the full flag mesh, and the tours; its Theorem 5.2 supplies the local recovery assertion; its Proposition 8.3 identifies the closed continuum form and centered inverse; and its Lemma 9.1 and Equation (9.11) give the Hilbert-space identification and Hilbert–Schmidt convergence. In the common extraction of its Section 9, retain also the countable family of local constrained-energy infima from its Lemma 5.1. These are infima of finite sums of squared edge differences under rational interval constraints on vertex values, viewed in the compact space [0,\infty]. Their joint limits supply the simultaneous recovery statement of Theorem 5.2, and persist on further subsequences. Proposition 2.4 preserves the geometric conclusions under this joint extraction, so all four assertions hold on one coupling.

The model conventions agree exactly: the companion uses the same rooted FK weight 2^{\ell/2}, counts all n primal edges, and pushes normalized corner counting to \deg(v)/(2n). Its all-edge network therefore has operator B_n=2G_n by Lemma 2.1. Its limiting surface is the ordinary unit-area sphere with the original marks forgotten. Auxiliary marks and rerootings used in the coupling change neither of these marginals.

For completeness, the companion first gives Hausdorff convergence of the distance graphs

in the product of the round topology and the ordinary topology on [0,\infty). Uniform continuity of D_h on the compact \Sph\times\Sph turns this into vanishing vertex distortion. Approximating the triples (x,x,0) in the opposite direction gives round density of the vertex locations, hence D_h-density. Lemma 2.2 permits replacement of \widehat a_n by a_n: along this coupling the \widehat a_n-scaled diameters are bounded, and the ratio of the two scales tends to one.

The corner multiplicities give the isometry of J_n, and the area pushforward gives that of J. The almost-everywhere singleton-fiber property makes J onto, as detailed in [12, Lemma 9.1]. Uniform tour tracking and uniform continuity of f give (2.7). The centered inverses in the companion are those of the unhalved operators, which yields exactly (2.8).

Finally, the chart families in [12, Proposition 2.4(iv)] use two distinct exploration roots and separate auxiliary mark pairs. Their interiors cover the entire sphere, not merely the complement of one root. The finite-sphere clause of its Theorem 5.2 gives (2.10) in the common coordinate, simultaneously for continuous finite-energy inputs and buffered domains. We have stated only the smooth supported case needed here. The vanishing flag mesh and the strict buffers ensure that edges incident at the compact support are indeed among the edges counted in W. Conformal and anticonformal coordinate changes preserve the Dirichlet integral, and hence the diffusion and its clock.□

Proof. Write \Pi for the orthogonal projection onto the constants in H. On the mean-zero part of J_nL^2(\mu_n), the operator 2K_n is the inverse of G_n. On the constants and on (J_nL^2(\mu_n))^\perp, it is zero. Therefore

In particular, both sides vanish on the orthogonal complement of the discrete space, while both act as the identity on constants. The same identity holds for G and K, with no extra orthogonal complement since J is unitary. Hilbert–Schmidt convergence of K_n implies operator-norm convergence and places all their spectra in a common bounded interval. Since \psi_s is continuous on that interval, continuous functional calculus proves (3.1). Similarly,

and the corresponding identity for G proves (3.2). Neither argument asserts operator-norm convergence at time zero.

To identify finite-dimensional laws, take continuous real functions f_0,\ldots,f_k on \Sph and times 0\le t_0<t_1<\cdots<t_k. Put g_{n,i}=f_i\circ z_n and let M_a denote multiplication by a bounded function a on H. By stationarity and the Markov property,

where S_n(s)=J_ne^{-sG_n}J_n^*. Indeed, multiplication by J_ng_{n,i} preserves the range of J_n and represents multiplication by g_{n,i} there. The tour convergence in Proposition 2.3 gives J_ng_{n,i}\to Jf_i uniformly, hence convergence of these multiplication operators in norm. Equation (3.1) now passes every factor in (3.4) to the limit. The resulting expression is the same product expectation for B^h: its semigroup is e^{-sG} and its initial law \mu_h is stationary because G1=0. Products of continuous functions determine the laws on each finite product of the compact sphere. Repeated observation times can be combined into a single test function, so all finite-dimensional distributions converge.□

Proof. Apply local recovery to \chi on U, obtaining \widetilde\chi_n uniformly close to \chi\circ z_n at vertices in W. Choose e_n\downarrow0 bounding these uniform errors; decreasingness can be achieved by taking tail suprema. Define

The clipping map is 1-Lipschitz, and \chi vanishes outside W, so \norm{\chi_n-\chi\circ z_n}_\infty\le2e_n. More importantly, \chi_n(v)>0 implies \chi(z_n(v))>0. Thus an edge with a nonzero difference of \chi_n has an endpoint in \operatorname{supp}\chi\Subset V. The vanishing edge mesh and the buffer ensure that every such edge is eventually wholly in W, with both endpoints there. Its difference is then bounded in absolute value by the corresponding difference of \widetilde\chi_n. Local recovery therefore gives

The last equality uses the support condition and conformal invariance of the two-dimensional Dirichlet integral. This proves (4.1), including the factor 1/2.□

Proof. Write q(x,y) for the off-diagonal jump rates. The process

is a square-integrable martingale starting from zero. Its quadratic variation and stationarity give

Over [0,T], reverse the path using its right-continuous modification in reversed time. Reversibility and stationarity give the same chain law, so its corresponding martingale N' satisfies the same second-moment identity. At every fixed t\in[0,T], almost surely there is no jump at t or at either deterministic endpoint. Subtracting the forward and backward formulas yields

Use this identity first at a countable dense set of times, together with T, and then use right continuity. It follows that

If the two martingale suprema are a and b, the square of the right side is at most 2\abs{v(Y_0)}^2+a^2+4b^2. Doob’s L^2 inequality bounds each of \EE a^2 and \EE b^2 by 8T\langle v,Av\rangle_{L^2(\pi)}, proving (4.2).□

Proof. Take the recoveries w_n from Lemma 4.1. Their uniform approximation, together with the tour convergence, gives J_nw_n\to Jf in H. For fixed \alpha, Equation (3.6) therefore implies

To see the energy assertion explicitly, expand

The last two terms converge by (3.6), and the first has limsup at most \E_h(f). Since u_\alpha lies in the operator domain of G, the resulting expression is the form expansion of \E_h(f-u_\alpha).

The right side of (4.5) tends to zero. Indeed, if \nu_f is the spectral measure of G at f, then it equals

Here f belongs to the form domain, so 1+\lambda is integrable against \nu_f; dominated convergence applies. Lemma 4.2, applied with A=G_n to w_n-u_{n,\alpha}, now gives

Cauchy–Schwarz and the uniform bound on w_n-g_n yield (4.3).

Finally, the Markov semigroup representation

shows that 0\le u_{n,\alpha}\le1. The resolvent identity G_nu_{n,\alpha}=\alpha(g_n-u_{n,\alpha}) proves the remaining bound in (4.4).□

Proof. Fix d>0. Choose finitely many round disks O_1,\ldots,O_m covering \Sph and smooth cutoffs f_i=1-\chi_i to which Lemma 4.1 applies, such that

For example, take sufficiently small inner disks and slightly larger support disks, all compactly contained in chart regions and all with diameter smaller than d. Compactness gives a finite subcover, and there is room for the recovery buffers outside each support disk.

Let \tau,\sigma be stopping times for Y^n, with 0\le\tau\le\sigma\le T and \sigma-\tau\le s. Let I be the first index for which \xi^n_\tau\in O_I. This index is measurable with respect to the information \mathcal F_\tau at time \tau. Since 0\le f_i\le1, (4.6) implies

Denote by u^{(i)}_{n,\alpha} the resolvent approximation of f_i\circ z_n. Optional sampling of its generator martingale gives

The equality is valid because each event \{I=i\} lies in \mathcal F_\tau; no independence of I and the path is required. The last inequality uses (4.4) and the fact that the events \{I=i\} partition the sample space. Consequently

This bound is uniform over all the stated stopping-time pairs. First take \limsup_n, then let s\downarrow0 with \alpha fixed, and finally let \alpha\to\infty. Lemma 4.3 gives

This is the stopping-time increment condition in Aldous’s tightness criterion [1]. Compact containment is automatic on \Sph, so the laws are tight in D([0,T],\Sph) with its J_1 topology.

Every jump of \xi^n joins the endpoints of an embedded map edge. Proposition 2.3 therefore makes its largest jump tend to zero. A jump of a J_1 limit must be approximated by jumps of comparable size; hence every subsequential limit is continuous. Evaluation at fixed times is continuous at continuous paths, so Proposition 3.1 identifies the finite-dimensional laws of every such limit with those of B^h. These finite-dimensional laws determine the continuous path law. Tightness and uniqueness of the subsequential limit prove the proposition.□

Proof. Start with the disjoint union of S and the C_n. Retain the metric within each piece, and allow a crossing from v\in V_n to z_n(v)\in S at cost c_n. Define distance as the infimum of the costs of finite chains made of such crossings and movements within individual pieces.

This construction does not shorten any of the original metrics. Indeed, an excursion from S through C_n, entering at z_n(u) and returning at z_n(v), costs at least

Replacing all such excursions by movements in S proves the assertion for pairs of points of S. For points of a fixed C_n, first remove excursions through the other cable graphs in this way. Each remaining excursion out of C_n, from a vertex u to a vertex v, costs at least

so it too can be removed. This also proves the assertion for points in edge interiors. Distinct pieces remain separated: every chain leaving C_n pays at least the positive cost c_n. Thus the chain construction defines a metric, not merely a pseudometric.

Every point of C_n is at distance at most a_n from a vertex, even when its edge is a loop. It is consequently at distance at most a_n+c_n from S. Conversely, every point of S is at distance at most \varepsilon_n+c_n from C_n. These observations prove (5.1)–(5.2) on the union. The union is totally bounded: its tail lies arbitrarily close to the compact space S, and each of the finitely many remaining cable graphs is compact. Its completion is the required compact space Z.

The coupling v\mapsto(v,z_n(v)) of \nu_n and (z_n)_*\nu_n has all its mass at pairs at distance at most c_n, whence

For the last convergence, weak convergence on the compact subspace S is also weak convergence in Z.□

Proof. Attempts occur at rate n and apply the transition kernel P_n, so the generator is n(P_n-I)=-G_n. This remains true when an attempt traverses a loop. The identity at S_{n,k} follows from the right-continuous definition of N_n.

Fix T and write

The centered partial sums form a martingale. Since \operatorname{Var}(e_{n,k})=n^{-2}, the L^2 maximal inequality gives

For k=\lfloor nt\rfloor, the point \overline X^n_{nt} lies on the cable edge traversed from X^n_k, at distance at most a_n from that vertex. Thus

The spatial bound holds for loop edges as well as nonloop edges. On the event \Delta_{n,T}+1/n\le s\le1, all the times appearing here lie in [0,T+1]. Consequently, for a_n<\eta/2,

First take the upper limit in n, using (5.6), and then let s\downarrow0, using (5.4) on [0,T+1]. This proves (5.5).□

Proof of Theorem 1.1. Start from an arbitrary subsequence and take the coupled further subsequence of Proposition 2.3. Fix a realization in its probability-one event and a finite integer r\ge1. Conditional on this realization, the r cable paths are independent with the common law in (5.8). Their joint law therefore converges to that of r independent copies of B^h in C([0,\infty),Z)^r. For example, a Skorokhod representation of (5.8), repeated on r independent probability spaces, couples the product laws with almost sure convergence of every coordinate.

Lemma 5.1 also gives Hausdorff convergence of the spaces and Prokhorov convergence of their measures in the same Z. Under the preceding product coupling, the curves converge uniformly on every compact time interval. By the definition of the curve-decorated Gromov–Hausdorff–Prokhorov topology, the decorated spaces consequently converge almost surely under that coupling. In particular, for any bounded continuous function F on the space of such decorated objects, the expectation of F over the r walks on this fixed environment converges to its expectation over r independent stationary Liouville Brownian motions on the fixed limiting sphere.

These expectations are the conditional expectations obtained by sampling walks with fresh randomness after the environment coupling. The walk kernel on a finite graph depends only on the map, not on the auxiliary coordinates, marks, or tours used in that coupling; the limiting kernel is the diffusion law specified by the quantum sphere. We have proved pointwise convergence of these conditional expectations on the probability-one event. The compact space Z, the path couplings, and the recovery functions used earlier are witnesses for this pointwise statement. They need not be chosen measurably as the environment varies. The conditional expectations from these sampling kernels are measurable, and bounded convergence now gives convergence of the unconditional expectations of F along the coupled further subsequence.

Thus every subsequence admits a further subsequence with the asserted joint limit. This proves convergence along the full sequence, for each fixed r. Lemma 5.2 used the deterministic acceleration b_n=n, which has exactly the time normalization in the theorem.□