The subcritical phase for a homopolymer model

TL;DR

This paper investigates the long-term behavior of a homopolymer model, a random walk penalized by its occupation time at the origin, revealing different scaling limits depending on the dimension and the sign of the penalty parameter.

Contribution

It provides a scaling limit for the homopolymer model in the subcritical regime, extending previous work to the case of negative penalization, especially in recurrent dimensions.

Findings

In one dimension, the homopolymer converges to a penalized Brownian motion.

In two dimensions, the model scales to standard Brownian motion.

The approach uses potential analysis and martingale techniques.

Abstract

We study a model of continuous-time nearest-neighbor random walk on $\mathbb{Z}^d$ penalized by its occupation time at the origin, also known as a homopolymer. For a fixed real parameter $\beta$ and time $t>0$, we consider the probability measure on paths of the random walk starting from the origin whose Radon-Nikodym derivative is proportional to the exponent of the product $\beta$ times the occupation time at the origin up to time $t$. The case $\beta>0$ was studied previously by Cranston and Molchanov arXiv:1508.06915. We consider the case $\beta<0$, which is intrinsically different only when the underlying walk is recurrent, that is $d=1,2$. Our main result is a scaling limit for the distribution of the homopolymer on the time interval $[0,t]$, as $t\to\infty$, a result that coincides with the scaling limit for penalized Brownian motion due to Roynette and Yor. In two dimensions, the penalizing effect is asymptotically diminished, and the homopolymer scales to standard Brownian motion. Our approach is based on potential analytic and martingale approximation for the model. We also apply our main result to recover a scaling limit for a wetting model. We study the model through analysis of resolvents.