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 \[H_n=\sum_{1\le j<k\le n}\omega_j\omega_k1_{\{S_j=S_k\}}\] 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 $d\ge3$.
- Publication:
-
Annales de L'Institut Henri Poincare Section (B) Probability and Statistics
- Pub Date:
- August 2008
- DOI:
- 10.1214/07-AIHP120
- arXiv:
- arXiv:0808.3037
- Bibcode:
- 2008AIHPB..44..638C
- Keywords:
-
- Mathematics - Probability
- E-Print:
- Published in at http://dx.doi.org/10.1214/07-AIHP120 the Annales de l'Institut Henri Poincar\'e - Probabilit\'es et Statistiques (http://www.imstat.org/aihp/) by the Institute of Mathematical Statistics (http://www.imstat.org)