Limit laws for the energy of a charged polymer

Abstract

In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy \[H_n=\sum_{1\le j<k\le n}ω_jω_k1_{\{S_j=S_k\}}\] of the polymer $\{S_1,...,S_n\}$ equipped with random electrical charges $\{ω_1,...,ω_n\}$. 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 $d\ge3$.