Limit laws for the energy of a charged polymer

Chen, Xia

Annales de l'I.H.P. Probabilités et statistiques, Volume 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

[1] A. Asselah and F. Castell. Self-intersection local times for random walk, and random walk in random scenery in dimension d≥5. Preprint, 2005. Available at http://arxiv.org/math.PR/0509721arXiv:math.PR/0509721. | MR

[2] A. Asselah. Large deviation estimates for self-intersection local times for simple random walk in ℤ3. Probab. Theory Related Fields. To appear. | MR | Zbl

[3] R. F. Bass, X. Chen and J. Rosen. Moderate deviations and laws of the iterated logarithm for the renormalized self-intersection local times of planar random walks. Electron. J. Probab. 11 (2006) 993-1030. | EuDML | MR | Zbl

[4] E. Buffet and J. V. Pulé. A model of continuous polymers with random charges. J. Math. Phys. 38 (1997) 5143-5152. | MR | Zbl

[5] X. Chen. On the law of the iterated logarithm for local times of recurrent random walks. In High Dimensional Probability II (Seattle, WA, 1999) 249-259, 2000. | MR | Zbl

[6] X. Chen. Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks. Ann. Probab. 32 (2004) 3248-3300. | MR | Zbl

[7] X. Chen. Moderate deviations and law of the iterated logarithm for intersections of the range of random walks. Ann. Probab. 33 (2005) 1014-1059. | MR | Zbl

[8] X. Chen and W. Li. Large and moderate deviations for intersection local times. Probab. Theory Related Fields 128 (2004) 213-254. | MR | Zbl

[9] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR | Zbl

[10] B. Derrida, R. B. Griffiths and R. G. Higgs. A model of directed walks with random self interactions. Europhys. Lett. 18 (1992) 361-366.

[11] B. Derrida and P. G. Higgs. Low-temperature properties of directed walks with random self-interactions. J. Phys. A 27 (1994) 5485-5493. | MR | Zbl

[12] R. Van Der Hofstad and W. König. A survey of one-dimensional random polymers. J. Statist. Phys. 103 (2001) 915-944. | MR | Zbl

[13] N. C. Jain and W. E. Pruitt. The range of transient random walk. J. Anal. Math. 24 (1971) 369-393. | MR | Zbl

[14] N. C. Jain and W. E. Pruitt. Further limit theorem for the range of random walk. J. Anal. Math. 27 (1974) 94-117. | MR | Zbl

[15] N. C. Jain and W. E. Pruitt. Asymptotic behavior of the local time of a recurrent random walk. Ann. Probab. 11 (1984) 64-85. | MR | Zbl

[16] Y. Kantor and M. Kardar. Polymers with self-interactions. Europhys. Lett. 14 (1991) 421-426.

[17] J.-F. Le Gall and J. Rosen. The range of stable random walks. Ann. Probab. 19 (1991) 650-705. | MR | Zbl

[18] S. Martínez and D. Petritis. Thermodynamics of a Brownian bridge polymer model in a random environment. J. Phys. A 29 (1996) 1267-1279. | MR | Zbl

[19] P. Révész. Random Walks in Random and Non-Random Environments. World Scientific, London, 1990. | Zbl

[20] J. Rosen. Random walks and intersection local time. Ann. Probab. 18 (1990) 959-977. | MR | Zbl

[21] F. Spitzer. Principles of Random Walk. Van Nostrand, Princeton, New Jersey, 1964. | MR | Zbl