Related papers: The growth exponent for planar loop-erased random …

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…

We study the shrinking Pearson random walk in two dimensions and greater, in which the direction of the Nth is random and its length equals lambda^{N-1}, with lambda<1. As lambda increases past a critical value lambda_c, the endpoint…

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 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…

Consider an infinite planar graph with uniform polynomial growth of degree d > 2. Many examples of such graphs exhibit similar geometric and spectral properties, and it has been conjectured that this is necessary. We present a family of…

The exponential growth rate of non polynomially growing subgroups of $GL_d$ is conjectured to admit a uniform lower bound. This is known for non-amenable subgroups, while for amenable subgroups it is known to imply the Lehmer conjecture…

We study the growth of the values of binary quadratic forms $Q$ on a binary planar tree as it was described by Conway. We show that the corresponding Lyapunov exponents $\Lambda_Q(x)$ as a function of the path determined by $x\in \mathbb…

Using the results obtained by the non commutative geometry techniques applied to the Harper equation, we derive the areas distribution of random walks of length $ N $ on a two-dimensional square lattice for large $ N $, taking into account…

We study the extremes of branching random walks under the assumption that the underlying Galton-Watson tree has infinite progeny mean. It is assumed that the displacements are either regularly varying or they have lighter tails. In the…

Looptrees have recently arisen in the study of critical percolation on the uniform infinite planar triangulation. Here we consider random infinite looptrees defined as the local limit of the looptree associated with a critical…

We investigate a lattice model of comb polymers and derive bounds on the exponential growth rate of the number of embeddings of the comb. A comb is composed of a backbone that is a self-avoiding walk and a set of $t$ teeth, also modelled as…

Consider a branching random walk, where the branching mechanism is governed by a Galton-Watson process, and the migration by a finite range symmetric irreducible random walk on the integer lattice $\mathbb{Z}^d$. Let $Z_n(z)$ be the number…

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…

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…

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…

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 $\beta \in (1, \frac{5}{3}]$ such…

We study the one-dimensional branching random walk in the case when the step size distribution has a stretched exponential tail, and, in particular, no finite exponential moments. The tail of the step size $X$ decays as $\mathbb{P}[X \geq…

We study branching random walks in random environment on the $d$-dimensional square lattice, $d \geq 1$. In this model, the environment has finite range dependence, and the population size cannot decrease. We prove limit theorems (laws of…

In this note, we compute the probability that a two-dimensional symmetric random walk visits more vertices than expected, for deviations on scales between the mean behavior and linear growth.

Let $M$ be the infinite spanning-tree-weighted random planar map, which is the local limit of finite random planar maps sampled with probability proportional to the number of spanning trees they admit. We show that a.s. the…