✦ LIBER ✦

On the Distribution of the Area Enclosed by a Random Walk onZ2

✍ Scribed by James A Mingo; Alexandru Nica

Publisher: Elsevier Science
Year: 1998
Tongue: English
Weight: 475 KB
Volume: 84
Category: Article
ISSN: 0097-3165
DOI: 10.1006/jcta.1998.2879

No coin nor oath required. For personal study only.

✦ Synopsis

Let 1 2n be the set of paths with 2n steps of unit length in Z 2 , which begin and end at (0, 0). For # # 1 2n , let area(#) # Z denote the oriented area enclosed by #. We show that for every positive even integer k, there exists a rational function R k with integer coefficients, such that:

We calculate explicitly the degree and leading coefficient of R k . We show how as a consequence of this (and by also using the enumeration of up-down permutations, and the exponential formula for cycles of permutations) one can derive the asymptotic distribution of the area enclosed by a random path in 1 2n . The formula for the asymptotic distribution can be stated as follows: for

Since the appropriately re-normalized random walks with n steps on Z 2 converge, as n Ä , to the 2-dimensional Brownian motion, this argument can be viewed as a combinatorial approach to the formula of Le vy for the area enclosed by a random 2-dimensional Brownian path.

📜 SIMILAR VOLUMES

On the fragmentation of a torus by rando

On the fragmentation of a torus by random walk

✍ Augusto Teixeira; David Windisch 📂 Article 📅 2011 🏛 John Wiley and Sons 🌐 English ⚖ 413 KB

## Abstract We consider a simple random walk on a discrete torus \input amssym $({\Bbb Z}/N{\Bbb Z})^d$ with dimension __d__ ≥ 3 and large side length __N__. For a fixed constant __u__ ≥ 0, we study the percolative properties of the vacant set, consisting of the set of vertices not visited by the r

A note on the last new vertex visited by

A note on the last new vertex visited by a random walk

✍ Lászlo Lovász; Peter Winkler 📂 Article 📅 1993 🏛 John Wiley and Sons 🌐 English ⚖ 200 KB

## Abstract A “cover tour” of a connected graph __G__ from a vertex __x__ is a random walk that begins at __x__, moves at each step with equal probability to any neighbor of its current vertex, and ends when it has hit every vertex of __G__. The cycle __C__~n~ is well known to have the curious prop

On the generation of N (μ, ∑)-distribute

On the generation of N (μ, ∑)-distributed random vectors by N (0,1)-distributed random numbers

✍ Bedall, F. K. ;Zimmermann, H. 📂 Article 📅 1976 🏛 Wiley (John Wiley & Sons) ⚖ 180 KB

On the distribution of rank of a random

On the distribution of rank of a random matrix over a finite field

✍ C. Cooper 📂 Article 📅 2000 🏛 John Wiley and Sons 🌐 English ⚖ 152 KB 👁 2 views

On the asymptotic distributions of subgr

On the asymptotic distributions of subgraph counts in a random tournament

✍ Pontus Andersson 📂 Article 📅 1998 🏛 John Wiley and Sons 🌐 English ⚖ 191 KB 👁 2 views

A random tournament T is obtained by independently orienting the edges of n 1 the complete graph on n vertices, with probability for each direction. We study the 2 asymptotic distribution, as n tends to infinity, of a suitable normalization of the number of subgraphs of T that are isomorphic to a gi

The Eigenvalue Distribution of a Random

The Eigenvalue Distribution of a Random Unipotent Matrix in Its Representation on Lines

✍ Jason Fulman 📂 Article 📅 2000 🏛 Elsevier Science 🌐 English ⚖ 120 KB

The eigenvalue distribution of a uniformly chosen random finite unipotent matrix in its permutation action on lines is studied. We obtain bounds for the mean number of eigenvalues lying in a fixed arc of the unit circle and offer an approach to other asymptotics. For the case of all unipotent matric