Limit laws for the energy of a charged polymer

Chen, Xia

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 638-672

In this paper we obtain the central limit theorems, moderate deviations and the laws of the iterated logarithm for the energy

$$H_n=\sum _{1\le j<k\le n}{\omega }_j{\omega }_k{1}_{\left\{S_j=S_k\right\}}$$

of the polymer $\left\{S_1,...,S_n\right\}$ equipped with random electrical charges $\left\{{\omega }_1,...,{\omega }_n\right\}$. Our approach is based on comparison of the moments between $H_n$ and the self-intersection local time

$$Q_n=\sum _{1\le j<k\le n}{1}_{\left\{S_j=S_k\right\}}$$

run by the $d$-dimensional random walk $\left\{S_k\right\}$. 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 $d\ge 3$.

Cet article est consacré à l’étude du théorème central limite, des déviations modérées et des lois du logarithme itéré pour l’énergie

$$H_n=\sum _{1\le j<k\le n}{\omega }_j{\omega }_k{1}_{\left\{S_j=S_k\right\}}$$

du polymère $\left\{S_1,...,S_n\right\}$ doté de charges électriques $\left\{{\omega }_1,...,{\omega }_n\right\}$. Notre approche se base sur la comparaison des moments de $H_n$ et du temps local de recoupements

$$Q_n=\sum _{1\le j<k\le n}{1}_{\left\{S_j=S_k\right\}}$$

de la marche aléatoire $d$-dimensionnelle $\left\{S_k\right\}$. L’étude du théorème central limite et de l’intégrabilité exponentielle de $Q_n$ (dans le cas $d\ge 3$) est également menée, tant pour comme outil pour notre principal objectif que pour son intérêt intrinsèque.

Classification : 60F05, 60F10, 60F15
Keywords: charged polymer, self-intersection local time, central limit theorem, moderate deviation, laws of the iterated logarithm

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