Limit laws for the energy of a charged polymer

TL;DR

This paper establishes limit theorems such as the central limit theorem, moderate deviations, and laws of the iterated logarithm for the energy of a charged polymer, linking it to self-intersection local times of random walks.

Contribution

It introduces a novel comparison approach between the polymer energy and self-intersection local times, extending results to higher dimensions and providing new insights into their probabilistic behavior.

Findings

Proved central limit theorems for polymer energy.

Derived moderate deviation principles.

Established laws of the iterated logarithm.

Abstract

In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy \[H_n=\sum_{1\le j<k\le n}\omega_j\omega_k1_{\{S_j=S_k\}}\] of the polymer $\{S_1,...,S_n\}$ equipped with random electrical charges $\{\omega_1,...,\omega_n\}$. Our approach is based on comparison of the moments between $H_n$ and the self-intersection local time \[Q_n=\sum_{1\le j<k\le n}1_{\{S_j=S_k\}}\] run by the $d$-dimensional random walk $\{S_k\}$. As partially needed for our main objective and partially motivated by their independent interest, the central limit theorems and exponential integrability for $Q_n$ are also investigated in the case $d\ge3$.