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.

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.

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 $d\ge 3$.

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