Aproxima\c{c}\~ao do Equil\'ibrio e Tempos Exponenciais para o Passeio Aleat\'orio no Hipercubo

TL;DR

This paper analyzes the behavior of a random walk in high-dimensional hypercubes, focusing on stopping times and convergence to exponential distributions as the dimension grows large.

Contribution

It provides new results on the asymptotic distribution of various stopping times for random walks in hypercubes, including coupling times and return times, as the dimension tends to infinity.

Findings

Coupling times converge to exponential distribution

Return times to points and sets also converge to exponential law

Provides bounds on convergence velocity to equilibrium

Abstract

We study a random walk in a N dimensional hypercube and exhibit results about stopping times when N diverges. The first theorem discusses the time in which two coupling processes spend to meet. A corollary provides a majorant for the velocity of convergence to equilibrium. Other three theorems treat, respectively, the time of first return to a point, the time of first return to a fixed set and the time of first arrival in a random set. We prove that these times, under a suitable rescaling, converge in law to a mean one exponential random time.