Related papers: LERW as an example of off-critical SLEs

We study the behavior of Random Walk in Random Environment (RWRE) on trees in the critical case left open in previous work. Representing the random walk by an electrical network, we assume that the ratios of resistances of neighboring edges…

The purpose of this paper is to establish, via a martingale approach, some refinements on the asymptotic behavior of the one-dimensional elephant random walk (ERW). The asymptotic behavior of the ERW mainly depends on a memory parameter $p$…

We consider a non-Markovian discrete-time random walk on $\mathbb{Z}$ with unbounded memory called the elephant random walk (ERW). We prove a strong invariance principle for the ERW. More specifically, we prove that, under a suitable…

The Elephant Random Walk (ERW), first introduced by Sch\"utz and Trimper (2004), is a one-dimensional simple random walk on $ \mathbb{Z} $ having a memory about the whole past. We study the Shark Random Swim, a random walk whose steps are $…

Excited random walks (ERWs) are a self-interacting non-Markovian random walk in which the future behavior of the walk is influenced by the number of times the walk has previously visited its current site. We study the speed of the walk,…

Stochastic Loewner evolution (SLE) is a differential equation driven by a one-dimensional Brownian motion (BM), whose solution gives a stochastic process of conformal transformation on the upper half complex-plane $\H$. As an evolutionary…

This paper investigates whether two independent Elephant Random Walks (ERWs) on $\mathbb{Z}$, each with a different memory parameter, can meet infinitely often, extending the work of Roy, Takei, and Tanemura. We also study the asymptotic…

We consider a model for random walks on random environments (RWRE) with random subset of Z^d as the vertices, and uniform transition probabilities on 2d points (two "coordinate nearest points" in each of the d coordinate directions). We…

In this work, we consider the scaling limit of loop-erased random walk (LERW) in three dimensions and prove that the limiting occupation measure is equivalent to its $\beta$-dimensional Minkowski content, where $\beta \in (1, 5/3]$ is its…

Stochastic Loewner Evolution (SLE_kappa) has been introduced as a description of the continuum limit of cluster boundaries in two-dimensional critical systems. We show that the problem of N radial SLEs in the unit disc is equivalent to…

We study a one parameter family of discrete Loewner evolutions driven by a random walk on the real line. We show that it converges to the stochastic Loewner evolution (SLE) under rescaling. We show that the discrete Loewner evolution…

The goal of this paper is to investigate the asymptotic behavior of the multidimensional elephant random walk with stops (MERWS). In contrast with the standard elephant random walk, the elephant is allowed to stay on his own position. We…

Avalanche frontiers in Abelian Sandpile Model (ASM) are random simple curves whose continuum limit is known to be a Schramm-Loewner Evolution (SLE) with diffusivity parameter $\kappa = 2$. In this paper we consider the dissipative ASM and…

We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively…

Schramm-Loewner Evolutions (SLEs) have proved an efficient way to describe a single continuous random conformally invariant interface in a simply-connected planar domain; the admissible probability distributions are parameterized by a…

Standard Schramm-Loewner evolution (SLE) is driven by a continuous Brownian motion which then produces a trace, a continuous fractal curve connecting the singular points of the motion. If jumps are added to the driving function, the trace…

In this work, we discuss the smoothly amnesia-reinforced multidimensional elephant random walk (MARW). The scaling limit of the MARW is shown to exist in the diffusive, critical and superdiffusive regimes. We also establish the almost sure…

In the simple random walk the steps are independent, whereas in the Elephant Random Walk (ERW), which was introduced by Sch\"utz and Trimper in 2004, the next step always depends on the whole path so far. In an earlier paper we investigated…

Random walk in random environment (RWRE) is a fundamental model of statistical mechanics, describing the movement of a particle in a highly disordered and inhomogeneous medium as a random walk with random jump probabilities. It has been…

Existence of Loewner trace is revisited. We identify finite energy paths (the "skeleton of Wiener measure") as natural class of regular drivers for which we find simple and natural estimates in terms of their (Cameron--Martin) norm.…