Rate of convergence for polymers in a weak disorder.(English)Zbl 1373.60164
Summary: We consider directed polymers in random environment on the lattice \(\mathbb{Z}^d\) at small inverse temperature and dimension \(d \geq 3\). Then, the normalized partition function \(W_n\) is a regular martingale with limit \(W\). We prove that \(n^{(d - 2) / 4}(W_n - W) / W_n\) converges in distribution to a Gaussian law. Both the polynomial rate of convergence and the scaling with the martingale \(W_n\) are different from those for polymers on trees.
MSC:
| 60K37 | Processes in random environments |
| 60F05 | Central limit and other weak theorems |
| 82D60 | Statistical mechanics of polymers |
Keywords:
directed polymers;random environment;weak disorder;rate of convergence;central limit theorem for martingales;stable and mixing convergenceCite
References:
| [1] | Alberts, T.; Khanin, K.; Quastel, J., The intermediate disorder regime for directed polymers in dimension \(1 + 1\), Ann. Probab., 42, 1212-1256 (2014) ·Zbl 1292.82014 |
| [2] | Alberts, T.; Ortgiese, M., The near-critical scaling window for directed polymers on disordered trees, Electron. J. Probab., 18, 19 (2013) ·Zbl 1279.82008 |
| [3] | Aldous, D.; Eagleson, G., On mixing and stability of limit theorems, Ann. Probab., 6, 325-331 (1978) ·Zbl 0376.60026 |
| [4] | Birkner, M., A condition for weak disorder for directed polymers in random environment, Electron. Commun. Probab., 9, 22-25 (2004) ·Zbl 1067.82030 |
| [5] | Bolthausen, E., A note on diffusion of directed polymers in a random environment, Comm. Math. Phys., 123, 529-534 (1989) ·Zbl 0684.60013 |
| [6] | Camanes, A.; Carmona, P., The critical temperature of a directed polymer in a random environment, Markov Process. Related Fields, 15, 105-116 (2009) ·Zbl 1203.60148 |
| [7] | Caravenna, F.; Sun, R.; Zygouras, N., Polynomial chaos and scaling limits of disordered systems, J. Eur. Math. Soc., 19, 1, 1-65 (2017) ·Zbl 1364.82026 |
| [8] | Comets, F.; Popov, S.; Vachkovskaia, M., The number of open paths in an oriented \(ρ\)-percolation model, J. Stat. Phys., 131, 357-379 (2008) ·Zbl 1144.82031 |
| [9] | Comets, F.; Vargas, V., Majorizing multiplicative cascades for directed polymers in random media, ALEA Lat. Am. J. Probab. Math. Stat., 2, 267-277 (2006) ·Zbl 1105.60074 |
| [10] | Comets, F.; Yoshida, N., Directed polymers in random environment are diffusive at weak disorder, Ann. Probab., 34, 1746-1770 (2006) ·Zbl 1104.60061 |
| [11] | Dean, D.; Majumdar, S., Extreme-value statistics of hierarchically correlated variables deviation from Gumbel statistics and anomalous persistence, Phys. Rev. E, 64, Article 046121 pp. (2001) |
| [12] | Derrida, B.; Spohn, H., Polymers on disordered trees, spin glasses, and traveling waves, J. Stat. Phys., 51, 817-840 (1988) ·Zbl 1036.82522 |
| [13] | Giacomin, G., Random Polymer Models (2007), Imperial College Press ·Zbl 1125.82001 |
| [14] | Hall, P.; Heyde, C. C., Martingale Limit Theory and Its Application (1980), Academic Press ·Zbl 0462.60045 |
| [15] | Halpin-Healy, T.; Takeuchi, K., A KPZ cocktail - shaken, not stirred …Toasting 30 years of kinetically roughened surfaces, J. Stat. Phys., 160, 794-814 (2015) ·Zbl 1327.82065 |
| [16] | Heyde, C., A rate of convergence result for the super-critical Galton-Watson process, J. Appl. Probab., 7, 451-454 (1970) ·Zbl 0198.22501 |
| [17] | Heyde, C., Some central limit analogues for supercritical Galton-Watson processes, J. Appl. Probab., 8, 52-59 (1971) ·Zbl 0222.60054 |
| [18] | Hu, Y.; Shao, Q.-M., A note on directed polymers in Gaussian environments, Electron. Commun. Probab., 14, 518-528 (2009) ·Zbl 1191.60123 |
| [19] | Huang, C.; Liu, Q., Convergence rates for a supercritical branching process in a random environment, Markov Process. Related Fields, 20, 265-285 (2014) ·Zbl 1318.60087 |
| [20] | Iksanov, O., On the rate of convergence of a regular martingale related to a branching random walk, Ukrainian Math. J., 58, 368-387 (2006) ·Zbl 1097.60013 |
| [21] | Iksanov, A.; Kabluchko, Z., A central limit theorem and a law of the iterated logarithm for the Biggins martingale of the supercritical branching random walk, J. Appl. Probab., 53, 4, 1178-1192 (2016) ·Zbl 1356.60055 |
| [22] | Imbrie, J. Z.; Spencer, T., Diffusion of directed polymer in a random environment, J. Stat. Phys., 52, 609-626 (1988) ·Zbl 1084.82595 |
| [23] | Lacoin, H., New bounds for the free energy of directed polymers in dimension \(1 + 1\) and \(1 + 2\), Comm. Math. Phys., 294, 471-503 (2010) ·Zbl 1227.82098 |
| [24] | Lawler, G., Intersections of Random Walks (1991), Birkhäuser: Birkhäuser Boston ·Zbl 1228.60004 |
| [25] | Lawler, G.; Limic, V., Random Walk: A Modern Introduction (2010), Cambridge University Press ·Zbl 1210.60002 |
| [26] | Liu, Q.; Watbled, F., Exponential inequalities for martingales and asymptotic properties of the free energy of directed polymers in random environment, Stochastic Process. Appl., 119, 3101-3132 (2009) ·Zbl 1177.60043 |
| [28] | Rényi, A., On stable sequences of events, Sankhya, Ser. A, 25, 293-302 (1963) ·Zbl 0141.16401 |
| [29] | Sinai, Y., A remark concerning random walks with random potentials, Fund. Math., 147, 173-180 (1995) ·Zbl 0835.60062 |
| [30] | Vargas, V., A local limit theorem for directed polymers in random media: the continuous and the discrete case, Ann. Inst. Henri Poincaré Probab. Stat., 42, 521-534 (2006) ·Zbl 1104.60067 |
| [31] | Wang, H.; Gao, Z.; Liu, Q., Central limit theorems for a supercritical branching process in a random environment, Statist. Probab. Lett., 81, 539-547 (2011) ·Zbl 1210.60095 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
