# Random Walk: A Modern Introduction

@inproceedings{Lawler2010RandomWA, title={Random Walk: A Modern Introduction}, author={Gregory F. Lawler and Vlada Limic}, year={2010} }

Random walks are stochastic processes formed by successive summation of independent, identically distributed random variables and are one of the most studied topics in probability theory. This contemporary introduction evolved from courses taught at Cornell University and the University of Chicago by the first author, who is one of the most highly regarded researchers in the field of stochastic processes. This text meets the need for a modern reference to the detailed properties of an important…

## Topics from this paper

## 646 Citations

Random walk on random walks

- Mathematics
- 2014

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson…

Analysis of random walks on a hexagonal lattice

- Mathematics
- 2016

We consider a discrete-time random walk on the nodes of an unbounded hexagonal lattice. We determine the probability generating functions, the transition probabilities and the relevant moments. The…

Long-time Behavior of Random Walks in Random Environment

- Mathematics
- 2013

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…

Some case studies of random walks in dynamic random environments

- Mathematics
- 2012

This thesis is dedicated to the study of random walks in dynamic random environments.
These are models for the motion of a tracer particle in a disordered medium, which is called
a static random…

Notes on stochastic processes Paul Keeler

- 2018

A stochastic process is a type of mathematical object studied in mathematics, particularly in probability theory, which can be used to represent some type of random evolution or change of a system.…

APPROXIMATING THE RANDOM WALK USING THE CENTRAL LIMIT THEOREM

- 2011

This paper will define the random walk on an integer lattice and will approximate the probability that the random walk is at a certain point after a certain number of steps by using a modified…

Capacity of the range of branching random walks in low dimensions

- Mathematics
- 2021

where Px denotes the law of a (discrete) random walk (Sn) with jump distribution η started at x, and τ A := inf{n ≥ 1 : Sn ∈ A} is (Sn)’s first returning time to A. Let μ be a probability…

On The Range Of a Random Walk In A Torus

- Mathematics
- 2010

Let a simple random walk run inside a torus of dimension three or higher for a number of steps which is a constant proportion of the volume. We examine geometric properties of the range, the random…

The potential function and ladder heights of a recurrent random walk on Z with infinite variance

- 2020

We consider a recurrent random walk of i.i.d. increments on the one-dimensional integer lattice and obtain a formula relating the hitting distribution of a half-line with the potential function,…

Random walk on random walks: higher dimensions

- MathematicsElectronic Journal of Probability
- 2019

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to…

## References

SHOWING 1-10 OF 22 REFERENCES

Principles Of Random Walk

- Mathematics
- 1964

This book is devoted exclusively to a very special class of random processes, namely to random walk on the lattice points of ordinary Euclidean space. The author considered this high degree of…

Intersections of random walks

- Mathematics
- 1991

A more accurate title for this book would be "Problems dealing with the non-intersection of paths of random walks. " These include: harmonic measure, which can be considered as a problem of…

Values of Brownian intersection exponents, II: Plane exponents

- Mathematics, Physics
- 2001

This paper is the follow-up of the paper [27], in which we derived the exact value of intersection exponents between Brownian motions in a half-plane. In the present paper, we will derive the value…

The Intersection Exponent for Simple Random Walk

- Mathematics, Computer ScienceCombinatorics, Probability and Computing
- 2000

It is shown that the intersection exponent for random walks is the same as that for Brownian motion and that the probability of nonintersection up to distance n is comparable (equal up to multiplicative constants) to n−ξ.

Walks on the slit plane

- Mathematics
- 2000

Abstract. In the first part of this paper, we enumerate exactly walks on the square lattice that start from the origin, but otherwise avoid the half-line . We call them walks on the slit plane. We…

The growth exponent for planar loop-erased random walk

- Mathematics
- 2008

We give a new proof of a result of Kenyon that the growth exponent for loop-erased random walks in two dimensions is 5/4. The proof uses the convergence of LERW to Schramm-Loewner evolution with…

Random walk loop soup

- Mathematics
- 2004

The Brownian loop soup introduced by Lawler and Werner (2004) is a Poissonian realization from a σ-finite measure on unrooted loops. This measure satisfies both conformal invariance and a restriction…

Potential kernel for two-dimensional random walk

- Mathematics
- 1996

It is proved that the potential kernel of a recurrent, aperiodic random walk on the integer lattice Z 2 admits an asymptotic expansion of the form (2π√|Q| -1 ln Q(x 2 , -x 1) + const + |x|U 1 (ω x )…

Values of Brownian intersection exponents, I: Half-plane exponents

- Mathematics
- 1999

Theoretical physics predicts that conformal invariance plays a crucial role in the macroscopic behavior of a wide class of two-dimensional models in statistical physics (see, e.g., [4], [6]). For…

Exact partition functions and correlation functions of multiple Hamiltonian walks on the Manhattan lattice

- Physics
- 1988

This is a general and exact study of multiple Hamiltonian walks (HAW) filling the two-dimensional (2D) Manhattan lattice. We generalize the original exact solution for a single HAW by Kasteleyn to a…