Limit laws for the energy of a charged polymer
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 = 1j<kn ω j ω k 1 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 = 1j<kn 1 S j =S k
de la marche aléatoire d-dimensionnelle S k . L’étude du théorème central limite et de l’intégrabilité exponentielle de Q n (dans le cas d3) 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 = 1j<kn ω j ω k 1 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 = 1j<kn 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 d3.

DOI : https://doi.org/10.1214/07-AIHP120
Classification : 60F05,  60F10,  60F15
Mots clés : charged polymer, self-intersection local time, central limit theorem, moderate deviation, laws of the iterated logarithm
@article{AIHPB_2008__44_4_638_0,
     author = {Chen, Xia},
     title = {Limit laws for the energy of a charged polymer},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {638--672},
     publisher = {Gauthier-Villars},
     volume = {44},
     number = {4},
     year = {2008},
     doi = {10.1214/07-AIHP120},
     zbl = {1178.60024},
     mrnumber = {2446292},
     language = {en},
     url = {www.numdam.org/item/AIHPB_2008__44_4_638_0/}
}
Chen, Xia. Limit laws for the energy of a charged polymer. Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 4, pp. 638-672. doi : 10.1214/07-AIHP120. http://www.numdam.org/item/AIHPB_2008__44_4_638_0/

[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 2288063

[2] A. Asselah. Large deviation estimates for self-intersection local times for simple random walk in ℤ3. Probab. Theory Related Fields. To appear. | MR 2372964 | Zbl 1135.60340

[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 127416 | MR 2261059 | Zbl 1112.60016

[4] E. Buffet and J. V. Pulé. A model of continuous polymers with random charges. J. Math. Phys. 38 (1997) 5143-5152. | MR 1471918 | Zbl 0890.60099

[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 1857326 | Zbl 0982.60014

[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 2094445 | Zbl 1067.60071

[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 2135311 | Zbl 1066.60013

[8] X. Chen and W. Li. Large and moderate deviations for intersection local times. Probab. Theory Related Fields 128 (2004) 213-254. | MR 2031226 | Zbl 1038.60074

[9] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. | MR 1619036 | Zbl 0896.60013

[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 1295374 | Zbl 0850.82072

[12] R. Van Der Hofstad and W. König. A survey of one-dimensional random polymers. J. Statist. Phys. 103 (2001) 915-944. | MR 1851362 | Zbl 1126.82313

[13] N. C. Jain and W. E. Pruitt. The range of transient random walk. J. Anal. Math. 24 (1971) 369-393. | MR 283890 | Zbl 0249.60038

[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 478361 | Zbl 0293.60063

[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 723730 | Zbl 0538.60074

[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 1106281 | Zbl 0729.60066

[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 1385633 | Zbl 0919.60078

[19] P. Révész. Random Walks in Random and Non-Random Environments. World Scientific, London, 1990. | Zbl 0733.60091

[20] J. Rosen. Random walks and intersection local time. Ann. Probab. 18 (1990) 959-977. | MR 1062054 | Zbl 0717.60057

[21] F. Spitzer. Principles of Random Walk. Van Nostrand, Princeton, New Jersey, 1964. | MR 171290 | Zbl 0119.34304