# 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…

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## 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
- 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

- MathematicsCombinatorics, 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

- Physics
- 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…