Directed polymers in random environment are diffusive at weak disorder. (English) Zbl 1104.60061

Summary: We consider directed polymers in random environment with discrete space and time. For transverse dimension at least equal to 3, we prove that diffusivity holds for the path in the full weak disorder region, that is, where the partition function differs from its annealed value only by a nonvanishing factor. Deep inside this region, we also show that the quenched averaged energy has fluctuations of order 1. In complete generality (arbitrary dimension and temperature), we prove monotonicity of the phase diagram in the temperature.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G42 Martingales with discrete parameter
82D30 Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses)
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[1] Albeverio, S. and Zhou, X. (1996). A martingale approach to directed polymers in a random environment. J. Theoret. Probab. 9 171–189. · Zbl 0837.60069
[2] Atlagh, M. and Weber, M. (2000). Le théorème central limite presque sûr. Expo. Math. 18 97–126. · Zbl 0959.60028
[3] Baik, J., Deift, P. and Johansson, K. (1999). On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 1119–1178. JSTOR: · Zbl 0932.05001
[4] Birkner, M. (2004). A condition for weak disorder for directed polymers in random environment. Electron. Comm. Probab. 9 22–25. · Zbl 1067.82030
[5] Bolthausen, E. (1989). A note on diffusion of directed polymers in a random environment. Comm. Math. Phys. 123 529–534. · Zbl 0684.60013
[6] Carmona, P. and Hu, Y. (2002). On the partition function of a directed polymer in a random environment. Probab. Theory Related Fields 124 431–457. · Zbl 1015.60100
[7] Carmona, P. and Hu, Y. (2004). Fluctuation exponents and large deviations for directed polymers in a random environment. Stochastic Process. Appl. 112 285–308. · Zbl 1072.60091
[8] Comets, F. and Neveu, J. (1995). The Sherrington–Kirkpatrick model of spin glasses and stochastic calculus: The high temperature case. Comm. Math. Phys. 166 549–564. · Zbl 0811.60098
[9] Comets, F., Shiga, T. and Yoshida, N. (2003). Directed polymers in random environment: Path localization and strong disorder. Bernoulli 9 705–723. · Zbl 1042.60069
[10] Comets, F., Shiga, T. and Yoshida, N. (2004). Probabilistic analysis of directed polymers in a random environment: A review. In Stochastic Analysis on Large Scale Interacting Systems (T. Funaki and H. Osada, eds.) 115–142. Math. Soc. Japan, Tokyo. · Zbl 1114.82017
[11] Dudley, R. (1989). Real Analysis and Probability . Wadsworth, Pacific Grove, CA. · Zbl 0686.60001
[12] Durrett, R. (1988). Lecture Notes on Particle Systems and Percolation . Wadsworth, Pacific Grove, CA. · Zbl 0659.60129
[13] Durrett, R. (1995). Probability Theory and Examples , 2nd ed. Duxbury Press, Belmont, CA. · Zbl 1202.60001
[14] Fisher, D. S. and Huse, D. A. (1991). Directed paths in random potential. Phys. Rev. B 43 10728–10742.
[15] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89–103. · Zbl 0346.06011
[16] Grimmett, G. (1999). Percolation , 2nd ed. Springer, Berlin. · Zbl 0926.60004
[17] Grimmett, G. and Hiemer, P. (2002). Directed percolation and random walk. In In and Out of Equilibrium 273–297. Birkhäuser, Boston. · Zbl 1010.60087
[18] Imbrie, J. Z. and Spencer, T. (1988). Diffusion of directed polymer in a random environment. J. Statist. Phys. 52 609–626. · Zbl 1084.82595
[19] Johansson, K. (2000). Shape fluctuations and random matrices. Comm. Math. Phys. 209 437–476. · Zbl 0969.15008
[20] Johansson, K. (2000). Transversal fluctuations for increasing subsequences on the plane. Probab. Theory Related Fields 116 445–456. · Zbl 0960.60097
[21] Kahane, J. P. and Peyriere, J. (1976). Sur certaines martingales de Benoit Mandelbrot. Adv. in Math. 22 131–145. · Zbl 0349.60051
[22] Kardar, M., Parisi, G. and Zhang, Y.-C. (1986). Dynamical scaling of growing interfaces. Phys. Rev. Lett. 56 889–892. · Zbl 1101.82329
[23] Krug, H. and Spohn, H. (1991). Kinetic roughenning of growing surfaces. In Solids Far from Equilibrium (C. Godrèche, ed.) 412–525. Cambridge Univ. Press.
[24] Licea, C., Newman, C. and Piza, M. (1996). Superdiffusivity in first-passage percolation. Probab. Theory Related Fields 106 559–591. · Zbl 0870.60096
[25] Liggett, T. (1985). Interacting Particle Systems . Springer, New York. · Zbl 0559.60078
[26] Mejane, O. (2004). Upper bound of a volume exponent for directed polymers in a random environment. Ann. Inst. H. Poincaré Probab. Statist. 40 299–308. · Zbl 1041.60079
[27] Menshikov, M. V. (1986). Coincidence of critical points in percolation problems. Dokl. Akad. Nauk SSSR 288 1308–1311. (In Russian.) · Zbl 0615.60096
[28] Rudin, W. (1987). Real and Complex Analysis , 3rd ed. McGraw–Hill, New York. · Zbl 0925.00005
[29] Piza, M. S. T. (1997). Directed polymers in a random environment: Some results on fluctuations. J. Statist. Phys. 89 581–603. · Zbl 0945.82527
[30] Sinai, Y. (1995). A remark concerning random walks with random potentials. Fund. Math. 147 173–180. · Zbl 0835.60062
[31] Song, R. and Zhou, X. Y. (1996). A remark on diffusion on directed polymers in random environment. J. Statist. Phys. 85 277–289. · Zbl 0924.60053
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