Asymptotic behavior of a selfinteracting random walk

S. A. Nadtochii

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We consider a simple one-dimensional random walk with the statistical weight of each sample path given by $\pi_t(\omega)=\exp\{-\beta\sum_{0\leq i<j\le n}V(|\omega_j-\omega_i|)\}$, where $\beta$ has the meaning of negative temperature, and $V$ is a nonnegative decreasing function with finite support. We show that for $\beta>\beta_0$ the distribution of $\omega_n$ is concentrated in the area $\{|\omega_n|>c\,n\}$, where $c=c(\beta)>0$, and for $\beta<0$ every sample path becomes localized, in the sense that $\omega_n$ never leaves some fixed interval.

Keywords: potential, random walk, self-repulsive random walk, asymptotic behavior.

Received: 12.09.2005

DOI: 10.4213/tvp150

 English version: Theory of Probability and its Applications, 2007, 51:1, 182–188

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