## Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

### Limit laws for the energy of a charged polymer

Xia Chen

#### Abstract

In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy $$H_n=\sum_{1≤j<k≤n}ω_jω_k1_{\{S_j=S_k\}}$$ 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 $$Q_n=\sum_{1≤j<k≤n}1_{\{S_j=S_k\}}$$ 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 $d≥3$.

#### Résumé

Cet article est consacré à l’étude du théorème central limite, des déviations modérées et des lois du logarithme itéré pour l’énergie $$H_n=\sum_{1≤j<k≤n}ω_jω_k1_{\{S_j=S_k\}}$$ du polymère $\{S_1, …, S_n\}$ doté de charges électriques $\{ω_1, …, ω_n\}$. Notre approche se base sur la comparaison des moments de $H_n$ et du temps local de recoupements $$Q_n=\sum_{1≤j<k≤n}1_{\{S_j=S_k\}}$$ de la marche aléatoire $d$-dimensionelle $\{S_k\}$. L’étude du théorème central limite et de l’intégrabilité exponentielle de $Q_n$ (dans le cas $d≥3$) est également menée, tant pour comme outil pour notre principal objectif que pour son intérêt intrinsèque.

#### Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 44, Number 4 (2008), 638-672.

Dates
First available in Project Euclid: 5 August 2008

https://projecteuclid.org/euclid.aihp/1217964114

Digital Object Identifier
doi:10.1214/07-AIHP120

Mathematical Reviews number (MathSciNet)
MR2446292

Zentralblatt MATH identifier
1178.60024

#### Citation

Chen, Xia. Limit laws for the energy of a charged polymer. Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), no. 4, 638--672. doi:10.1214/07-AIHP120. https://projecteuclid.org/euclid.aihp/1217964114

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