Related papers: Topics in loop measures and the loop-erased walk

Here I share a few notes I used in various course lectures, talks, etc. Some may be just calculations that in the textbooks are more complicated, scattered, or less specific; others may be simple observations I found useful or curious.

In this note, we consider random walks in the quarter plane with arbitrary big jumps. We announce the extension to that class of models of the analytic approach of [G. Fayolle, R. Iasnogorodski, and V. Malyshev, Random walks in the quarter…

We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a L\'evy flight on a line. After a brief survey of the theory of records for independent and…

We have studied the probability distribution of the perimeter and the area of the k-th 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…

Random walk has wide applications in many fields, such as machine learning, biology, physics, and chemistry. Random walk can be discrete or continuous in time and space. Asymmetric random walk could be described by drift-diffusion equation.…

These lecture notes are based on a set of six lectures that I gave in Edinburgh in 2008/2009 and they cover some topics in the interface between Geometry and Physics. They involve some unsolved problems and conjectures and I hope they may…

In this note, we try to analyze and clarify the intriguing interplay between some counting problems related to specific thermalized weighted graphs and random walks consistent with such graphs.

Human-in-the-loop topic modelling incorporates users' knowledge into the modelling process, enabling them to refine the model iteratively. Recent research has demonstrated the value of user feedback, but there are still issues to consider,…

The fourfold research proposal regards in particular: critical oriented percolation; random walk limit laws; neural networks with long-range connections; the ant in a labyrinth.

This paper is a collection of recent results on discrete-time and continuous-time branching random walks. Some results are new and others are known. Many aspects of this theory are considered: local, global and strong local survival, the…

These notes deal with metric spaces, Hausdorff measures and dimensions, Lipschitz mappings, and related topics. The reader is assumed to have some familiarity with basic analysis, which is also reviewed.

Random walks are ubiquitous in the sciences, and they are interesting from both theoretical and practical perspectives. They are one of the most fundamental types of stochastic processes; can be used to model numerous phenomena, including…

The main topic of these notes are Markov loops, studied in the context of continuous time Markov chains on discrete state spaces. We refer to [1] and [2] for the short "history" of the subject. In contrast with these references, symmetry is…

We prove strong theorems for the local time at infinity of a nearest neighbor transient random walk. First, laws of the iterated logarithm are given for the large values of the local time. Then we investigate the length of intervals over…

Quantum walks can reconstruct quantum algorithms for quantum computation, where the precise controls of quantum state transfers between arbitrary distant sites are required. Here, we investigate quantum walks using a periodically…

In this paper, we make a few random explorations that relate directly to the items mentioned in the title. We define transient chains and recurrent chains with "killing", the Green's function, the Laplacian operator, and harmonic functions.…

We study behavior in space and time of random walks in an i.i.d. random environment on Z^d, d>=3. It is assumed that the measure governing the environment is isotropic and concentrated on environments that are small perturbations of the…

The characterization of record events is considered for a discrete-time random walk model with long-term memory arising from correlations between successive steps. An important feature is that the correlations are strong enough to give rise…

Random walks provide a simple conventional model to describe various transport processes, for example propagation of heat or diffusion of matter through a medium. However, in many practical cases the medium is highly irregular due to…

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichm\"uller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the…