Loop-erased Random Walk Research Articles

Fractal frontiers of bursts and cracks in a fiber bundle model of creep rupture.

We investigate the geometrical structure of breaking bursts generated during the creep rupture of heterogeneous materials. Using a fiber bundle model with localized load sharing we show that bursts are compact geometrical objects; however, their external frontiers have a fractal structure which reflects their growth dynamics. The perimeter fractal dimension of bursts proved to have the universal value 1.25 independent of the external load and of the amount of disorder in the system. We conjecture that according to their geometrical features, breaking bursts fall in the universality class of loop-erased self-avoiding random walks with perimeter fractal dimension 5/4 similar to the avalanches of Abelian sand pile models. The fractal dimension of the growing crack front along which bursts occur proved to increase from 1 to 1.25 as bursts gradually cover the entire front.

Scaling limit of the loop-erased random walk Green’s function

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 $$\hbox {SLE}_2$$ Green’s function. The proof does not use SLE techniques and is based on a combinatorial identity which reduces the problem to obtaining sharp asymptotics for two quantities: the loop measure of random walk loops of odd winding number about a branch point near the marked edge and a “spinor” observable for random walk started from one of the vertices of the marked edge.

Convergence rates for loop-erased random walk and other Loewner curves

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 approach in the case of loop-erased random walk (LERW) in a simply connected domain. Under mild assumptions of boundary regularity, we obtain an explicit power-law rate for the convergence of the LERW path toward the radial SLE$_2$ path in the supremum norm, the curves being parameterized by capacity. On the deterministic side, we show that the tip structure modulus gives a sufficient geometric condition for a Loewner curve to be H\"{o}lder continuous in the capacity parameterization, assuming its driving term is H\"{o}lder continuous. We also briefly discuss the case when the curves are a priori known to be H\"{o}lder continuous in the capacity parameterization and we obtain a power-law convergence rate depending only on the regularity of the curves.

Spanning trees of graphs on surfaces and the intensity of loop-erased random walk on planar graphs

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 Z 2 \mathbb {Z}^2 , that is, the probability that the walk from ( 0 , 0 ) (0,0) to ∞ \infty passes through a given vertex or edge. For example, the probability that it passes through ( 1 , 0 ) (1,0) is 5 / 16 5/16 ; this confirms a conjecture from 1994 about the stationary sandpile density on Z 2 \mathbb {Z}^2 . We do the analogous computation for the triangular lattice, honeycomb lattice, and Z × R \mathbb {Z}\times \mathbb {R} , for which the probabilities are 5 / 18 5/18 , 13 / 36 13/36 , and 1 / 4 − 1 / π 2 1/4-1/\pi ^2 respectively.

Loop-erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution.

We study the scaling limit of a planar loop-erased random walk (LERW) on the percolation cluster, with occupation probability p≥p(c). We numerically demonstrate that the scaling limit of planar LERW(p) curves, for all p>p(c), can be described by Schramm-Loewner evolution (SLE) with a single parameter κ that is close to the normal LERW in a Euclidean lattice. However, our results reveal that the LERW on critical incipient percolation clusters is compatible with SLE, but with another diffusivity coefficient κ. Several geometrical tests are applied to ascertain this. All calculations are consistent with SLE(κ), where κ=1.732±0.016. This value of the diffusivity coefficient is outside the well-known duality range 2≤κ≤8. We also investigate how the winding angle of the LERW(p) crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p(c). For finite systems, two crossover exponents and a scaling relation can be derived. This finding should, to some degree, help us understand and predict the existence of conformal invariance in disordered and fractal landscapes.

Publisher's Note: Loop-erased random walk on a percolation cluster: Crossover from Euclidean to fractal geometry [Phys. Rev. E89, 062101 (2014)

Publisher's Note: Loop-erased random walk on a percolation cluster: Crossover from Euclidean to fractal geometry [Phys. Rev. E<b>89</b>, 062101 (2014)

Loop-erased random walk on a percolation cluster: crossover from Euclidean to fractal geometry.

We study loop-erased random walk (LERW) on the percolation cluster, with occupation probability p ≥ p_{c}, in two and three dimensions. We find that the fractal dimensions of LERW_{p} are close to normal LERW in a Euclidean lattice, for all p>p_{c}. However, our results reveal that LERW on critical incipient percolation clusters is fractal with d_{f}=1.217 ± 0.002 for d=2 and 1.43 ± 0.02 for d=3, independent of the coordination number of the lattice. These values are consistent with the known values for optimal path exponents in strongly disordered media. We investigate how the behavior of the LERW_{p} crosses over from Euclidean to fractal geometry by gradually decreasing the value of the parameter p from 1 to p_{c}. For finite systems, two crossover exponents and a scaling relation can be derived. This work opens up a theoretical window regarding the diffusion process on fractal and random landscapes.

Enumeration of spanning trees in planar unclustered networks

Among a variety of subgraphs, spanning trees are one of the most important and fundamental categories. They are relevant to diverse aspects of networks, including reliability, transport, self-organized criticality, loop-erased random walks and so on. In this paper, we introduce a family of modular, self-similar planar networks with zero clustering. Relevant properties of this family are comparable to those networks associated with technological systems having low clustering, like power grids, some electronic circuits, the Internet and some biological systems. So, it is very significant to research on spanning trees of planar networks. However, for a large network, evaluating the relevant determinant is intractable. In this paper, we propose a fairly generic linear algorithm for counting the number of spanning trees of a planar network. Using the algorithm, we derive analytically the exact numbers of spanning trees in planar networks. Our result shows that the computational complexity is O(t), which is better than that of the matrix tree theorem with O(m2t2), where t is the number of steps and m is the girth of the planar network. We also obtain the entropy for the spanning trees of a given planar network. We find that the entropy of spanning trees in the studied network is small, which is in sharp contrast to the previous result for planar networks with the same average degree. We also determine an upper bound and a lower bound for the numbers of spanning trees in the family of planar networks by the algorithm. As another application of the algorithm, we give a formula for the number of spanning trees in an outerplanar network with small-world features.

UNIFORM SPANNING TREES ON SIERPI NSKI GRAPHS

We study spanning trees on Sierpi nski graphs (i.e., nite approxima- tions to the Sierpi nski gasket) that are chosen uniformly at random. We prove existence of a limit measure and 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 akin to the Brownian motion or the Schramm- Loewner evolution. Some geometric properties of this limit process, such as the Hausdor dimension, are investigated as well. The method is also applicable to other self-similar graphs with a sucient degree of symmetry.

The probability that planar loop-erased random walk uses a given edge

We give a new proof of a result of Rick Kenyon that the probability that an edge in the middle of an $n \times n$ square is used in a loop-erased walk connecting opposite sides is of order $n^{-3/4}$. We, in fact, improve the result by showing that this estimate is correct up to multiplicative constants.

Loop-erased random walk on the Sierpinski gasket

In this paper the loop-erased random walk on the finite pre-Sierpiński gasket is studied. It is proved that the scaling limit exists and is a continuous process. It is also shown that the path of the limiting process is almost surely self-avoiding, while having Hausdorff dimension strictly greater than 1. The loop-erasing procedure proposed in this paper is formulated by erasing loops, in a sense, in descending order of size. It enables us to obtain exact recursion relations, making direct use of ‘self-similarity’ of a fractal structure, instead of the relation to the uniform spanning tree. This procedure is proved to be equivalent to the standard procedure of chronological loop-erasure.

Some Partial Results on the Convergence of Loop-Erased Random Walk to SLE(2) in the Natural Parametrization

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 measure of the discrete curve, properly renormalized by the chosen time parametrization, converges to the occupation density of the SLE(2) curve, the so-called SLE Green's function. Although we do not prove this convergence, we rigorously establish some partial results in this direction including a new loop-erased random walk estimate.

Left passage probability of Schramm-Loewner Evolution

SLE(κ,ρ[over arrow]) is a variant of Schramm-Loewner Evolution (SLE) which describes the curves which are not conformal invariant, but are self-similar due to the presence of some other preferred points on the boundary. In this paper we study the left passage probability (LPP) of SLE(κ,ρ[over arrow]) through field theoretical framework and find the differential equation governing this probability. This equation is numerically solved for the special case κ=2 and h(ρ)=0 in which h(ρ) is the conformal weight of the boundary changing (bcc) operator. It may be referred to loop erased random walk (LERW) and Abelian sandpile model (ASM) with a sink on its boundary. For the curve which starts from ξ(0) and conditioned by a change of boundary conditions at x(0), we find that this probability depends significantly on the factor x(0)-ξ(0). We also present the perturbative general solution for large x(0). As a prototype, we apply this formalism to SLE(κ,κ-6) which governs the curves that start from and end on the real axis.

On the Rate of Convergence of Loop-Erased Random Walk to SLE2

We derive a rate of convergence of the Loewner driving function for a planar loop-erased random walk to Brownian motion with speed 2 on the unit circle, the Loewner driving function for radial SLE2. 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 SLE2, we also obtain a rate of convergence for the paths with respect to the Hausdorff distance.

From elongated spanning trees to vicious random walks

Given a spanning forest on a large square lattice, we consider by combinatorial methods a correlation function of k paths (k is odd) along branches of trees or, equivalently, k loop-erased random walks. Starting and ending points of the paths are grouped such that they form a k-leg watermelon. For large distance r between groups of starting and ending points, the ratio of the number of watermelon configurations to the total number of spanning trees behaves as r−νlogr with ν=(k2−1)/2. Considering the spanning forest stretched along the meridian of this watermelon, we show that the two-dimensional k-leg loop-erased watermelon exponent ν is converting into the scaling exponent for the reunion probability (at a given point) of k(1+1)-dimensional vicious walkers, ν˜=k2/2. At the end, we express the conjectures about the possible relation to integrable systems.

Generalized loop‐erased random walks and approximate reachability

AbstractIn this paper we extend the loop‐erased random walk (LERW) to the directed hypergraph setting. We then generalize Wilson's algorithm for uniform sampling of spanning trees to directed hypergraphs. In several special cases, this algorithm perfectly samples spanning hypertrees in expected polynomial time. Our main application is to the reachability problem, also known as the directed all‐terminal network reliability problem. This classical problem is known to be # P‐complete, hence is most likely intractable (Ball and Provan, SIAM J Comput 12 (1983) 777–788). We show that in the case of bi‐directed graphs, a conjectured polynomial bound for the expected running time of the generalized Wilson algorithm implies a FPRAS for approximating reachability. Copyright © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 44, 201‐223, 2014

An expansion for self-interacting random walks

We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions.

Return probability for the loop-erased random walk and mean height in the Abeliansandpile model: a proof**We are delighted to dedicate this work to Deepak Dhar on the occasion of his 60thbirthday.

Single site height probabilities in the Abelian sandpile model, and the corresponding mean height⟨h⟩, are directly relatedto the probability Pret that a loop-erased random walk passes through a nearest neighbourof the starting site (return probability). The exact values of thesequantities on the square lattice have been conjectured, in particular⟨h⟩ = 25/8 andPret = 5/16. We provide a rigorous proof of this conjecture by using a local monomer–dimerformulation of these questions.

Spanning forests and the vector bundle Laplacian

The classical matrix-tree theorem relates the determinant of the combinatorial Laplacian on a graph to the number of spanning trees. We generalize this result to Laplacians on one- and two-dimensional vector bundles, giving a combinatorial interpretation of their determinants in terms of so-called cycle rooted spanning forests (CRSFs). We construct natural measures on CRSFs for which the edges form a determinantal process. This theory gives a natural generalization of the spanning tree process adapted to graphs embedded on surfaces. We give a number of other applications, for example, we compute the probability that a loop-erased random walk on a planar graph between two vertices on the outer boundary passes left of two given faces. This probability cannot be computed using the standard Laplacian alone.

Loop-erased random walk and Poisson kernel on planar graphs

Lawler, Schramm and Werner showed that the scaling limit of the loop-erased random walk on ℤ2 is SLE2. We consider scaling limits of the loop-erasure of random walks on other planar graphs (graphs embedded into 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 SLE2. 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 super-critical percolation on ℤ2. Berger and Biskup showed that the scaling limit of the random walk on this graph is planar Brownian motion. Our results imply that the scaling limit of the loop-erased random walk on the super-critical percolation cluster is SLE2.