Loop-erased Random Walk Research Articles

Loop-Erased Random Walk on a Torus in Dimensions 4 and Above

We show that the statistics of loop erased random walks above the upper critical dimension, 4, are different between the torus and the full space. The typical length of the path connecting a pair of sites at distance L, which scales as L2 in the full space, changes under the periodic boundary conditions to L d /2. The results are precise for dimensions ≥5; for the dimension d=4 we prove an upper bound, conjecturally sharp up to subpolyonmial factors.

Estimates of Random Walk Exit Probabilities and Application to Loop-Erased Random Walk

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.

Stochastic Loewner evolution in doubly connected domains

This paper introduces the annulus SLEκ processes in doubly connected domains. Annulus SLE6 has the same law as stopped radial SLE6, up to a time-change. For κ ≠ 6, some weak equivalence relation exists between annulus SLEκ and radial SLEκ. Annulus SLE2 is the scaling limit of the corresponding loop-erased conditional random walk, which implies that a certain form of SLE2 satisfies the reversibility property. We also consider the disc SLEκ process defined as a limiting case of the annulus SLE’s. Disc SLE6 has the same law as stopped full plane SLE6, up to a time-change. Disc SLE2 is the scaling limit of loop-erased random walk, and is the reversal of radial SLE2.

Conformal invariance of planar loop-erased random walks and uniform spanning trees

This paper proves that the scaling limit of a loop-erased random walk in a simply connected domain $Dsubsetneqq\C$ is equal to the radial SLE$_2$ path. 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 $\p D$ 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 $A\subset\p D$, is the chordal SLE$_8$ path in $\overline D$ joining the endpoints of A. A by-product of this result is that SLE$_8$ is almost surely generated by a continuous path. The results and proofs are not restricted to a particular choice of lattice.

Winding angle variance of Fortuin-Kasteleyn contours.

The variance in the winding number of various random fractal curves, including the self-avoiding walk, the loop-erased random walk, contours of Fortuin-Kastelyn clusters, and stochastic Loewner evolution, has been studied by numerous researchers. Usually the focus has been on the winding at the end points. We measure the variance in winding number at typical points along the curve. More generally, we study the winding at points where k strands come together, and some adjacent strands may be conditioned not to hit each other. The measured values are consistent with an interesting conjecture.

Probability distribution of the sizes of the largest erased loops in loop-erased random walks.

We have studied the probability distribution of the perimeter and the area of the kth largest erased loop in loop-erased random walks in two dimensions for k=1 to 3. For a random walk of N steps, for large N, the average value of the kth largest perimeter and area scales as N(5/8) and N, respectively. The behavior of the scaled distribution functions is determined for very large and very small arguments. We have used exact enumeration for N< or =20 to determine the probability that no loop of size greater than l is erased. We show that correlations between loops have to be taken into account to describe the average size of the kth largest erased loops. We propose a one-dimensional Levy walk model that takes care of these correlations. The simulations of this simpler model compare very well with the simulations of the original problem.

Distribution of sizes of erased loops of loop-erased random walks in two and three dimensions.

We show that in the loop-erased random-walk problem, the exponent characterizing the probability distribution of areas of erased loops is superuniversal. In d dimensions, the probability that the erased loop has an area A varies as A(-2) for large A, independent of d, for 2< or =d< or =4. We estimate the exponents characterizing the distribution of perimeters and areas of erased loops in d=2 and 3 by large-scale Monte Carlo simulations. Our estimate of the fractal dimension z in two dimensions is consistent with the known exact value 5/4. In three dimensions, we get z=1.6183+/-0.0004. The exponent for the distribution of the durations of avalanches in the three-dimensional Abelian sandpile model is determined from this by using scaling relations.

Time-Inhomogeneous Fokker-Planck Equation for Wave Distributions in the Abelian Sandpile Model.

The time and size distributions of the waves of topplings in the Abelian sandpile model are expressed as the first arrival at the origin distribution for a scale invariant, time-inhomogeneous Fokker-Plank equation. Assuming a linear conjecture for the time inhomogeneity exponent as a function of a loop-erased random walk (LERW) critical exponent, suggested by numerical results, this approach allows one to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three dimensions. The avalanche size distribution in two dimensions is found to be the difference between two closed power laws.

Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are strongly 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 these 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 the plane.The scaling limits of these processes are conjectured to be conformally invariant in dimension 2. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Löwner differential equation $$\frac{{\partial f}}{{\partial t}} = z\frac{{\zeta (t) + z}}{{\zeta (t) - z}}\frac{{\partial f}}{{\partial z}}$$ (1) , with boundary valuesf(z,0)=z, in the rangez ∈U= {w ∈ ℂ : •w• < 1},t≤0. We choose ζ(t):=B(−2t), where B(t) is Brownian motion on ∂\( \mathbb{U} \) starting at a random-uniform point in ∂\( \mathbb{U} \). Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to ∂\( \mathbb{U} \) has the same law as that of the pathf(ζ(t),t) (wheref(z,t) is extended continuously to ∂\( \mathbb{U} \)) ×(−∞,0]). We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.

The Upper Critical Dimension of the Abelian Sandpile Model

The existing estimation of the upper critical dimension of the Abelian Sandpile Model is based on a qualitative consideration of avalanches as self-avoiding branching processes. We find an exact representation of an avalanche as a sequence of spanning subtrees of two-component spanning trees. Using equivalence between chemical paths on the spanning tree and loop-erased random walks, we reduce the problem to determination of the fractal dimension of spanning subtrees. Then the upper critical dimension du=4 follows from Lawler's theorems for intersection probabilities of random walks and loop-erased random walks.

Loop-Erased Random Walks, Spanning Trees and Hamiltonian Cycles

We establish a formula for the distribution of loop-erased random walks at certain random times. Several classical results on spanning trees, including Wilson's algorithm, follow easily, as well as a method to construct random Hamiltonian cycles.

The asymptotic determinant of the discrete Laplacian

We compute the asymptotic determinant of the discrete Laplacian on a simply-connected rectilinear region in R^2. As an application of this result, we prove that the growth exponent of the loop-erased random walk in Z^2 is 5/4.

Scaling limits for minimal and random spanning trees in two dimensions

A general formulation is presented for continuum scaling limits of stochastic spanning trees. A spanning tree is expressed in this limit through a consistent collection of subtrees, which includes a tree for every finite set of endpoints in $\R^d$. Tightness of the distribution, as $\delta \to 0$, is established for the following two-dimensional examples: the uniformly random spanning tree on $\delta \Z^2$, the minimal spanning tree on $\delta \Z^2$ (with random edge lengths), and the Euclidean minimal spanning tree on a Poisson process of points in $\R^2$ with density $\delta^{-2}$. In each case, sample trees are proven to have the following properties, with probability one with respect to any of the limiting measures: i) there is a single route to infinity (as was known for $\delta > 0$), ii) the tree branches are given by curves which are regular in the sense of H\"older continuity, iii) the branches are also rough, in the sense that their Hausdorff dimension exceeds one, iv) there is a random dense subset of $\R^2$, of dimension strictly between one and two, on the complement of which (and only there) the spanning subtrees are unique with continuous dependence on the endpoints, v) branching occurs at countably many points in $\R^2$, and vi) the branching numbers are uniformly bounded. The results include tightness for the loop erased random walk (LERW) in two dimensions. The proofs proceed through the derivation of scale-invariant power bounds on the probabilities of repeated crossings of annuli.

A Lower Bound on the Growth Exponent for Loop-Erased Random Walk in Two Dimensions

The growth exponent for loop-erased or Laplacian random walk on the integer lattice is dened by saying that the expected time to reach the sphere of radius n is of order n .W e 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. R esum e. L'exposant de croissance pour la marche al eatoire a boucles eac ees ou \laplacienne sur le r eseau Z d est d eni de la mani ere suivante : le nombre moyen de pas au moment o u la marche issue de l'origine atteint la sph ere de rayon nest d'ordre n lorsque n tend vers l'inni. Nous montrons que lorsque d = 2, l'exposant de croissance est strictement sup erieur a 1. La preuve utilise une estimation connue concernant le moment d'ordre trois de la probabilit e de fuite, ainsi qu'un ranement de la version discr etis ee du th eor eme de projection de Beurling. AMS Subject Classication. 60J15.

How to Get a Perfectly Random Sample from a Generic Markov Chain and Generate a Random Spanning Tree of a Directed Graph

A general problem in computational probability theory is that of generating a random sample from the state space of a Markov chain in accordance with the steady-state probability law of the chain. Another problem is that of generating a random spanning tree of a graph or spanning arborescence of a directed graph in accordance with the uniform distribution, or more generally in accordance with a distribution given by weights on the edges of the graph or digraph. This article gives algorithms for both of these problems, improving on earlier results and exploiting the duality between the two problems. Each of the new algorithms hinges on the recently introduced technique of coupling from the past or on the linked notions of loop-erased random walk and “cycle popping.”

Loop-Erased Walks Intersect Infinitely Often in Four Dimensions

In this short note we show that the paths two independent loop-erased random walks in four dimensions intersect infinitely often. We actually prove the stronger result that the cut-points of the two walks intersect infinitely often.

Distribution of sizes of erased loops for loop-erased random walks

We study the distribution of sizes of erased loops for loop-erased random walks on regular and fractal lattices. We show that for arbitrary graphs the probability $P(l)$ of generating a loop of perimeter $l$ is expressible in terms of the probability $P_{st}(l)$ of forming a loop of perimeter $l$ when a bond is added to a random spanning tree on the same graph by the simple relation $P(l)=P_{st}(l)/l$. On $d$-dimensional hypercubical lattices, $P(l)$ varies as $l^{-\sigma}$ for large $l$, where $\sigma=1+2/z$ for $1<d<4$, where z is the fractal dimension of the loop-erased walks on the graph. On recursively constructed fractals with $\tilde{d} < 2$ this relation is modified to $\sigma=1+2\bar{d}/{(\tilde{d}z)}$, where $\bar{d}$ is the hausdorff and $\tilde{d}$ is the spectral dimension of the fractal.

Loop-erased self-avoiding random walks in four and five dimensions

Monte Carlo simulations of loop-erased self-avoiding random walks in four and five dimensions were performed, using two distinct algorithms. We find consistency between these methods in their estimates of critical exponents. The upper critical dimension for this phenomenon is four, and it has been shown that the mean square end-to-end distance grows as n(log n)α. It has recently been established that the mean square end-to-end distance is asymptotically bounded by n(log n)1/3 (see Ref. 21). Our results show that asymptotic convergence to n(log n)1/3 in fact obtains and does so rather quickly. In five dimensions we examine the rate of asymptotic convergence to the mean-field model. © 1996 by John Wiley & Sons, Inc.

Dynamic algorithm for loop-erased self-avoiding random walks in two and three dimensions.

Dynamic algorithm for loop-erased self-avoiding random walks in two and three dimensions.

On the topology of loop-erased self-avoiding random walks

Loop-erased self-avoiding random walks were generated on a 5-choice cubic lattice. Knot formation in these walks were determined by counting the number of crossings in each walk. The results were fitted to a power law.