Scaling limits for the random walk penalized by its range in dimension one

TL;DR

This paper investigates the asymptotic behavior of a one-dimensional penalized random walk model, revealing phase transitions and precise trajectory descriptions through exact partition function asymptotics.

Contribution

It provides the first detailed analysis of the scaling limits and phase transitions for a one-dimensional range-penalized random walk model.

Findings

Exact asymptotics for the partition function.

Identification of a phase transition at h_n ≈ n^{1/4}.

Description of trajectory scaling limits.

Abstract

In this article we study a one dimensional model for a polymer in a poor solvent: the random walk on $\mathbb{Z}$ penalized by its range. More precisely, we consider a Gibbs transformation of the law of the simple symmmetric random walk by a weight $\exp(-h_n|R_n|)$, with $|R_n|$ the number of visited sites and $h_n$ a size-dependent positive parameter. We use gambler's ruin estimates to obtain exact asymptotics for the partition function, that enables us to obtain a precise description of trajectories, in particular scaling limits for the center and the amplitude of the range. A phase transition for the fluctuations around an optimal amplitude is identified at $h_n \approx n^{1/4}$ , inherent to the underlying lattice structure.