Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs
Yuval Peres, David Revelle
备注：
6/6/05 version is substantially reorganized, with the main proof being more clearly presented as a proof by induction and the individual lemmas are now more self-contained
Let x and y be chosen uniformly in a graph G. We find the limiting distribution of the length of a loop-erased random walk from x to y on a large class of graphs that include the discrete torus in dimensions 5 and above. Moreover, on this family of graphs we show that a suitably normalized finite-dimensional scaling limit of the uniform spanning tree is a Brownian continuum random tree.
The loop-erased random walk (LERW) in $ \Z^d, d \geq 2$, is obtained by erasing loops chronologically from simple random walk. In this paper we show the existence of the two-sided LERW which can be considered as the distribution of the LERW as seen by a point in the "middle" of the path.
These are notes based on a course that I gave at the University of Chicago in Fall 2016 on "Loop measures and the loop-erased random walk." This is not intended to be a comprehensive view but rather a personal selection of some key ideas including some recent results.
Let $β$ be the growth exponent of the loop-erased random walk (LERW) in three dimensions. We prove that the scaling limit of 3D LERW is $h$-Hölder continuous almost surely for all $h < 1/β$, while not $1/β$-Hölder continuous almost surely.
We prove that loop-erased random walks on finite pre-Sierpinski gaskets can be extended to the infinite pre-Sierpinski gasket by virtue of the `erasing-larger-loops-first' method, and obtain the asymptotic behavior of the walk as the number of steps increases, in particular, the displacement exponent and a law of the iterated logarithm.
We prove an estimate for the probability that a simple random walk in a simply connected subset A of Z^2 starting on the boundary exits A at another specified boundary point. The estimates are uniform over all domains of a given inradius. We apply these estimates to prove a conjecture of S. Fomin in 2001 concerning a relationship between crossing probabilities of loop-erased random walk and Brownian motion.
We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with parameter 2, and is valid for irreducible bounded symmetric random walks on any two-dimensional discrete lattice.
We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an n x n square is used in a loop-erased walk connecting opposites sides is of order n^{-3/4}. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.
Let ζbe the intersection exponent of random walks in Z^3 and αbe a positive real number. We construct a stochastic process from a simple random walk by erasing loops of length at most N^α. We will prove that for α< \frac{1}{1+2ζ}, the limiting distribution is Gaussian. For α> 2 the limiting distribution will be shown to be equal to the limiting distribution of the loop erased walk.
In recent work we have shown that loop-erased random walk (LERW) connecting two boundary points of a domain converges to the chordal Schramm-Loewner evolution (SLE(2)) in the sense of curves parametrized by Minkowski content. In this note we explain how to derive the analogous result for LERW from a boundary point to an interior point, converging towards radial SLE(2).
The growth exponent $α$ for loop-erased or Laplacian random walk on the integer lattice is defined by saying that the expected time to reach the sphere of radius $n$ is of order $n^α$. We prove that in two dimensions, the growth exponent is strictly greater than one. The proof uses a known estimate on the third moment of the escape probability and an improvement on the discrete Beurling projection theorem.
Let $M_{n}$ be the length (number of steps) of the loop-erasure of a simple random walk up to the first exit from a ball of radius $n$ centered at its starting point. It is shown in [18] that there exists $β\in (1, \frac{5}{3}]$ such that $E (M_{n} )$ is of order $n^β$ in 3 dimensions. In the present article, we show that the Hausdorff dimension of the scaling limit of the loop-erased random walk in 3 dimensions is equal to $β$ almost surely.
Let $(G_n)_{n=1}^{\infty}$ be a sequence of finite graphs, and let Y_t be the length of a loop-erased random walk on G_n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G_n is the d-dimensional torus of size-length n for $d \geq 4$, the process $(Y_t)_{t=0}^{\infty}$, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Bro…
We show that the `erasing-larger-loops-first' (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpinski gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpinski gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the `standard' self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The e…
We consider a model of loop-erased random walks on the finite pre-Sierpinski gasket which permits rigorous analysis. We prove the existence of the scaling limit and show that the path of the limiting process is almost surely self-avoiding, while having Hausdorff dimension strictly greater than 1. This result means that the path has infinitely fine creases, while having no self-intersection. Our loop-erasing procedure is formulated by a `larger-scale-loops-first' rule. It enables us to obtain exact recursion relations, making use of `self-similarity' of a fractal structure.
The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the `middle part' of an infinite LERW loop going through 0 and infinity. In this note we derive expressions for transition probabilities for this model in dimensions two and up. In the plane, the formula can be further expressed in terms of a Laplacian with signed weights acting on certain discrete harmonic functions at the tips of the walk, and taking a determinant. The discrete harmonic functions are closely related to a discrete version of the complex sq…
Let X and Y be independent transient Markov chains on the same state space that have the same transition probabilities. Let L denote the ``loop-erased path'' obtained from the path of X by erasing cycles when they are created. We prove that if the paths of X and Y have infinitely many intersections a.s., then L and Y also have infinitely many intersections a.s.
In this paper we consider the natural random walk on a planar graph and scale it by a small positive number $δ$. Given a simply connected domain $D$ and its two boundary points $a$ and $b$, we start the scaled walk at a vertex of the graph nearby $a$ and condition it on its exiting $D$ through a vertex nearby $b$, and prove that the loop erasure of the conditioned walk converges, as $δ\downarrow 0$, to the chordal SLE$_{2}$ that connects $a$ and $b$ in $D$, provided that an invariance principle is valid for both the random walk and the dual walk of it.
We outline a strategy for showing convergence of loop-erased random walk on the Z^2 square lattice to SLE(2), in the supremum norm topology that takes the time parametrization of the curves into account. The discrete curves are parametrized so that the walker moves at a constant speed determined by the lattice spacing, and the SLE(2) curve has the recently introduced natural time parametrization. Our strategy can be seen as an extension of the one used by Lawler, Schramm, and Werner to prove convergence modulo time parametrization. The crucial extra step is showing that the expected occupation…
We consider loop-erased random walk (LERW) running between two boundary points of a square grid approximation of a planar simply connected domain. The LERW Green's function is the probability that the LERW passes through a given edge in the domain. We prove that this probability, multiplied by the inverse mesh size to the power 3/4, converges in the lattice size scaling limit to (a constant times) an explicit conformally covariant quantity which coincides with the SLE(2) Green's function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem t…
Let x and y be points chosen uniformly at random from $\Z_n^4$, the four-dimensional discrete torus with side length n. We show that the length of the loop-erased random walk from x to y is of order $n^2 (\log n)^{1/6}$, resolving a conjecture of Benjamini and Kozma. We also show that the scaling limit of the uniform spanning tree on $\Z_n^4$ is the Brownian continuum random tree of Aldous. Our proofs use the techniques developed by Peres and Revelle, who studied the scaling limits of the uniform spanning tree on a large class of finite graphs that includes the d-dimensional discrete torus for…
We consider uniform spanning tree (UST) in topological polygons with $2N$ marked points on the boundary with alternating boundary conditions. In [LPW21], the authors derive the scaling limit of the Peano curve in the UST. They are variants of SLE$_8$. In this article, we derive the scaling limit of the loop-erased random walk branch (LERW) in the UST. They are variants of SLE$_2$. The conclusion is a generalization of [HLW20,Theorem 1.6] where the authors derive the scaling limit of the LERW branch of UST when $N=2$. When $N=2$, the limiting law is SLE$_2(-1,-1; -1, -1)$. However, the limiting…
The determination of the Hausdorff dimension of the scaling limit of loop-erased random walk is closely related to the study of the one-point function of loop-erased random walk, i.e., the probability a loop-erased random walk passes through a given vertex. Recent work in the theoretical physics literature has investigated the Hausdorff dimension of loop-erased random walk in three dimensions by applying field theory techniques to study spin systems that heuristically encode the one-point function of loop-erased random walk. Inspired by this, we introduce two different spin systems whose corre…
We show how to compute the probabilities of various connection topologies for uniformly random spanning trees on graphs embedded in surfaces. As an application, we show how to compute the "intensity" of the loop-erased random walk in ${\mathbb Z}^2$, that is, the probability that the walk from (0,0) to infinity passes through a given vertex or edge. For example, the probability that it passes through (1,0) is 5/16; this confirms a conjecture from 1994 about the stationary sandpile density on ${\mathbb Z}^2$. We do the analogous computation for the triangular lattice, honeycomb lattice and ${\m…
Starting from a sequence regarded as a walk through some set of values, we consider the associated loop-erased walk as a sequence of directed edges, with an edge from $i$ to $j$ if the loop erased walk makes a step from $i$ to $j$. We introduce a coloring of these edges by painting edges with a fixed color as long as the walk does not loop back on itself, then switching to a new color whenever a loop is erased, with each new color distinct from all previous colors. The pattern of colors along the edges of the loop-erased walk then displays stretches of consecutive steps of the walk left untouc…
Sharp estimates for the length of loop erased random walk between two vertices on the [n]^d -torus, d > 4, are established. The mean length is order n^{d/2} . In dimension 4 we have only an upper bound.
We consider matrices whose elements enumerate weights of walks in planar directed weighted graphs (not necessarily acyclic). These matrices are totally nonnegative; more precisely, all their minors are formal power series in edge weights with nonnegative coefficients. A combinatorial explanation of this phenomenon involves loop-erased walks. Applications include total positivity of hitting matrices of Brownian motion in planar domains.
Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent loop-erased random walk both started at the origin do not intersect up to leaving the ball of radius $n$. This allows us to show that there exists $C$ such that for all $n$ and all $k=1,2,...,\mathbf{E}[M_n^k]\leq C^kk!\mathbf{E}[M_n]^k$ and hence to establish exponential moment bounds for $M_n$. This implies that there exists $c>0$ such that for all $n$ and al…
In this work, we consider loop-erased random walk (LERW) in three dimensions and give an asymptotic estimate on the one-point function for LERW and the non-intersection probability of LERW and simple random walk in three dimensions for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.
We review some recently completed research that establishes the scaling limit of Fomin's identity for loop-erased random walk on Z^2 in terms of the chordal Schramm-Loewner evolution (SLE) with parameter 2. In the case of two paths, we provide a simplified proof of the identity for loop-erased random walk and simple random walk, and prove directly that the corresponding identity holds for chordal SLE(2) and Brownian motion. We also include a brief introduction to SLE and discussion of the relationship between SLE(2) and loop-erased random walk.
We give evidence that the functional renormalization group (FRG), developed to study disordered systems, may provide a field theoretic description for the loop-erased random walk (LERW), allowing to compute its fractal dimension in a systematic expansion in epsilon=4-d. Up to two loop, the FRG agrees with rigorous bounds, correctly reproduces the leading logarithmic corrections at the upper critical dimension d=4, and compares well with numerical studies. We obtain the universal subleading logarithmic correction in d=4, which can be used as a further test of the conjecture.
Loop-erased random walk, abbreviated LERW, is one of the most well-studied critical lattice models. It is the self-avoiding random walk one gets after erasing the loops from a simple random walk in order or alternatively by considering the branches in a uniformly chosen spanning tree. This paper proves that planar LERW parametrized by renormalized length converges in the lattice size scaling limit to SLE(2) parametrized by 5/4-dimensional Minkowski content. In doing this we also provide a method for proving similar convergence results for other models converging to SLE. Besides the main theore…
We revisit the convergence of loop-erased random walk, LERW, to SLE(2) when the curves are parametrized by capacity. We construct a coupling of the chordal version of LERW and chordal SLE(2) based on the Green's function for LERW as martingale observable and using an elementary discrete-time Loewner "difference" equation. This coupling is different than the ones previously considered in this context. Our recent work (arXiv:1603.05203) on the convergence of LERW parametrized by length to SLE(2) parameterized by Minkowski content uses specific features of the coupling constructed here.
We derive a rate of convergence of the Loewner driving function for planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE(2). The proof uses a new estimate of the difference between the discrete and continuous Green's functions that is an improvement over existing results for the class of domains we consider. Using the rate for the driving process convergence along with additional information about SLE(2), we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.
Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on $\mathbb{Z}^2$ is $\mathrm{SLE}_2$. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into $\mathbb{C}$ so that edges do not cross one another). We show that if the scaling limit of the random walk is planar Brownian motion, then the scaling limit of its loop-erasure is $\mathrm{SLE}_2$. Our main contribution is showing that for such graphs, the discrete Poisson kernel can be approximated by the continuous one. One example is the infinite component of …
Following the strategy proposed by Makarov and Smirnov in arXiv:0909.5377, we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up of arXiv:0909.5377 appeared since then, we believe that such a treatment might be of interest for the community. We do not require any regularity of the limiting planar domain $Ω$ near its degenerate prime ends $a$ and $b$ except that $(Ω^δ,a^δ,b^δ)$ are assumed to be `close discrete approximations' to $(Ω,a,b)$ near $a$ and $b$ in the sense of a recent work arXiv:1810.05608.
The loop-erased random walk (LERW) in $\mathbb{Z}^4$ is the process obtained by erasing loops chronologically for simple random walk. We prove that the escape probability of the LERW renormalized by $(\log n)^{\frac{1}{3}}$ converges almost surely and in $L^{p}$ for all $p>0$. Along the way, we extend previous results by the first author building on slowly recurrent sets. We provide two applications for the escape probability. We construct the two-sided LERW, and we construct a $\pm 1$ spin model coupled with the wired spanning forests on $\mathbb{Z}^4$ with the bi-Laplacian Gaussian field on …
We consider a random partition of the vertex set of an arbitrary graph that can be sampled using loop-erased random walks stopped at a random independent exponential time of parameter $q>0$, that we see as a tuning parameter.The related random blocks tend to cluster nodes visited by the random walk on time scale $1/q$. We explore the emerging macroscopic structure by analyzing 2-point correlations. To this aim, it is defined an interaction potential between pair of vertices, as the probability that they do not belong to the same block of the random partition. This interaction potential can be …
We introduce partial loop-erasing operators. We show that by applying a refinement sequence of partial loop-erasing operators to a finite Markov chain, we get a process equivalent to the chronological loop-erased Markov chain. As an application, we construct loop-erased random paths on bounded domains of resistance spaces as the weak limit of the loop erasure of the Markov chains on a sequence of finite sets approximating the space, and the limit is independent of the approximating sequences. The random paths we constructed are simple paths almost surely, and they can be viewed as the loop-era…
We prove that the scaling limit of loop-erased random walk in a simply connected domain $D$ is equal to the radial SLE(2) path in $D$. In particular, the limit exists and is conformally invariant. It follows that the scaling limit of the uniform spanning tree in a Jordan domain exists and is conformally invariant. Assuming that the boundary of the domain is a $C^1$ simple closed curve, the same method is applied to show that the scaling limit of the uniform spanning tree Peano curve, where the tree is wired along a proper arc $A$ on the boundary, is the chordal SLE(8) path in the closure of $D…
It is well known that there are close connections between non-intersecting processes in one dimension and random matrices, based on the reflection principle. There is a generalisation of the reflection principle for more general (e.g. planar) processes, due to S. Fomin, in which the non-intersection condition is replaced by a condition involving loop-erased paths. In the context of independent Brownian motions in suitable planar domains, this also has close connections to random matrices. An example of this was first observed by Sato and Katori (Phys. Rev. E, 83, 2011). We present further exam…
Loop-erased random walk and it's scaling limit, Schramm--Loewner evolution, have found numerous applications in mathematics and physics. We present a 2 dimensional analogue of LERW, the loop erased random surface. We do this by defining a 2 dimensional spanning tree and declaring that LERS should have the same relation to these 2 trees as LERW has to ordinary spanning trees. Furthermore we present numerical evidence that the growth rate for LERS on a $δ$ fine grid as $δ\to 0$ is $2.5269 \pm 0.0017$ and we hypothesize that it has an exact value of 48/19. This suggests the possibility of a fract…
We estimate convergence rates for curves generated by Loewner's differential equation under the basic assumption that a convergence rate for the driving terms is known. An important tool is what we call the tip structure modulus, a geometric measure of regularity for Loewner curves parameterized by capacity. It is analogous to Warschawski's boundary structure modulus and closely related to annuli crossings. The main application we have in mind is that of a random discrete-model curve approaching a Schramm-Loewner evolution (SLE) curve in the lattice size scaling limit. We carry out the approac…
We consider a continuous height version of the Abelian sandpile model with small amount of bulk dissipation gamma > 0 on each toppling, in dimensions d = 2, 3. In the limit gamma -> 0, we give a power law upper bound, based on coupling, on the rate at which the stationary measure converges to the discrete critical sandpile measure. The proofs are based on a coding of the stationary measure by weighted spanning trees, and an analysis of the latter via Wilson's algorithm. In the course of the proof, we prove an estimate on coupling a geometrically killed loop-erased random walk to an unkilled lo…
We provide a decomposition of the trace of the Brownian motion into a simple path and an independent Brownian soup of loops that intersect the simple path. More precisely, we prove that any subsequential scaling limit of the loop erased random walk is a simple path (a new result in three dimensions), which can be taken as the simple path of the decomposition. In three dimensions, we also prove that the Hausdorff dimension of any such subsequential scaling limit lies in $(1,\frac53]$. We conjecture that our decomposition characterizes uniquely the law of the simple path. If so, our results woul…
As an image of the many-to-one map of loop-erasing operation $\LE$ of random walks, a self-avoiding walk (SAW) is obtained. The loop-erased random walk (LERW) model is the statistical ensemble of SAWs such that the weight of each SAW $ζ$ is given by the total weight of all random walks $π$ which are inverse images of $ζ$, $\{π: \LE(π)=ζ\}$. We regard the Brownian paths as the continuum limits of random walks and consider the statistical ensemble of loop-erased Brownian paths (LEBPs) as the continuum limits of the LERW model. Following the theory of Fomin on nonintersecting LERWs, we introduce …
We study simple random walk on the uniform spanning tree on Z^2 . We obtain estimates for the transition probabilities of the random walk, the distance of the walk from its starting point after n steps, and exit times of both Euclidean balls and balls in the intrinsic graph metric. In particular, we prove that the spectral dimension of the uniform spanning tree on Z^2 is 16/13 almost surely.
The uniform spanning tree (UST) and the loop-erased random walk (LERW) are related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of the subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in t…
We study spanning trees on Sierpinski graphs (i.e., finite approximations to the Sierpinski gasket) that are chosen uniformly at random. We construct a joint probability space for uniform spanning trees on every finite Sierpinski graph and show that this construction gives rise to a multi-type Galton-Watson tree. We derive a number of structural results, for instance on the degree distribution. The connection between uniform spanning trees and loop-erased random walk is then exploited to prove convergence of the latter to a continuous stochastic process. Some geometric properties of this limit…
