×

zbMATH — the first resource for mathematics

Polynomial chaos and scaling limits of disordered systems. (English) Zbl 1364.82026
The paper deals with statistical models defined on lattices, in which disorder is taken into account in the form of external random fields. One considers more especially the disordered partition function, and it is shown that, under some conditions, it admits a non-trivial limit in distribution, which does not depend on the parameters of the initial model (say a universal limit). The main purpose of this approach is to explain the issue of disorder relevance, in accordance of which the addition of a small amount of disorder modifies the structure of the phase transition of the underlying homogeneous model. Following the introduction, the authors discuss polynomial chaos and Wiener chaos via Lindeberg principle, and then deal with the scaling limit of disordered systems.

MSC:
82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
82D60 Statistical mechanical studies of polymers
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Aizenman, M., Wehr, J.: Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130, 489-528 (1990)Zbl 0714.60090 MR 1060388 · Zbl 0714.60090
[2] 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 MR 3189070 · Zbl 1292.82014
[3] Alberts, T., Khanin, K., Quastel, J.: The continuum directed random polymer. J. Statist. Phys. 154, 305-326 (2014)Zbl 1291.82143 MR 3162542 · Zbl 1291.82143
[4] Alexander, K. S.: The effect of disorder on polymer depinning transitions. Comm. Math. Phys. 279, 117-146 (2008)Zbl 1175.82034 MR 2377630 · Zbl 1175.82034
[5] Alexander, K. S., Zygouras, N.: Quenched and annealed critical points in polymer pinning models. Comm. Math. Phys. 291, 659-689 (2009)Zbl 1188.82154 MR 2534789 · Zbl 1188.82154
[6] Amir, G., Corwin, I., Quastel, J.: Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions. Comm. Pure Appl. Math. 64, 466-537 (2011)Zbl 1222.82070 MR 2796514 [BC+14]Berger, Q., Caravenna, F., Poisat, J., Sun, R., Zygouras, N.: The critical curve of the random pinning and copolymer models at weak coupling. Comm. Math. Phys. 326, 507-530 (2014)MR 3165465
[7] Bingham, N. H., Goldie, C. M., Teugels, J. L.: Regular Variation. Encyclopedia Math. Appl. 27, Cambridge Univ. Press, Cambridge (1987)Zbl 0667.26003 MR 1015093 · Zbl 0617.26001
[8] Bolthausen, E., den Hollander, F.: Localization transition for a polymer near an interface. Ann. Probab. 25, 1334-1366 (1997)Zbl 0885.60022 MR 1457622 · Zbl 0885.60022
[9] Bourgain, J., Kahn, J., Kalai, G., Katznelson, Y., Linial, N.: The influence of variables in product spaces. Israel J. Math. 77, 55-64 (1992)Zbl 0771.60002 MR 1194785 · Zbl 0771.60002
[10] Bovier, A.: Statistical Mechanics of Disordered Systems: a Mathematical Perspective. Cambridge Univ. Press (2006)Zbl 1108.82002 MR 2252929 · Zbl 1108.82002
[11] Bricmont, J., Kupiainen, A.: Phase transition in the 3d random field Ising model. Comm. Math. Phys. 116, 539-572 (1988)MR 0943702 · Zbl 1086.82573
[12] Camia, F., Garban, C., Newman, C. M.: Planar Ising magnetization field I. Uniqueness of the critical scaling limit. Ann. Probab. 43, 528-571 (2015)Zbl 1332.82012 MR 3305999 · Zbl 1332.82012
[13] Camia, F., Garban, C., Newman, C. M.: The Ising magnetization exponent on Z2 is 1/15. Probab. Theory Related Fields 160, 175-187 (2014)Zbl 1307.82004 MR 3256812 · Zbl 1307.82004
[14] Camia, F., Garban, C., Newman, C. M.: Planar Ising magnetization field II. Properties of the critical and near-critical scaling limits. Ann. Inst. H. Poincar´e Probab. Statist. 52, 146-161 (2016)Zbl 1338.82009 MR 3449298 64Francesco Caravenna et al. · Zbl 1338.82009
[15] Caravenna, F., Giacomin, G.: The weak coupling limit of disordered copolymer models. Ann. Probab. 38, 2322-2378 (2010)Zbl 1242.82022 MR 2683632 · Zbl 1242.82022
[16] Caravenna, F., Sun, R., Zygouras, N.: The continuum disordered pinning model. Probab. Theory Related Fields 164, 17-59 (2016)Zbl 1341.82040 MR 3449385 · Zbl 1341.82040
[17] Cardy, J. (ed.): Finite-Size Scaling. North-Holland (1988)MR 1103125
[18] Chatterjee, S.: A generalization of the Lindeberg principle. Ann. Probab. 34, 2061- 2076 (2006)Zbl 1117.60034 MR 2294976 · Zbl 1117.60034
[19] Chatterjee, S., Dey, P. S.: Multiple phase transitions in long-range first-passage percolation on square lattices. Comm. Pure Appl. Math. 69, 203-256 (2016)Zbl 1332.60133 MR 3434612 · Zbl 1332.60133
[20] Chayes, J. T., Chayes, L., Fischer, D. S., Spencer, T.: Finite-size scaling and correlation lengths for disordered systems. Phys. Rev. Lett. 57, 2999-3002 (1986)MR 0925751
[21] Chayes, J. T., Chayes, L., Fischer, D. S., Spencer, T.: Correlation length bounds for disordered Ising ferromagnets. Comm. Math. Phys. 120, 501-523 (1989) Zbl 0658.60137 MR 0981216 · Zbl 0658.60137
[22] Chelkak, D., Hongler, C., Izyurov, C.: Conformal invariance of spin correlations in the planar Ising model. Ann. of Math. 181, 1087-1138 (2015)Zbl 1318.82006 MR 3296821 · Zbl 1318.82006
[23] Comets, F., Shiga, T., Yoshida, N.: Probabilistic analysis of directed polymers in a random environment: a review. In: Stochastic Analysis on Large Scale Interacting Systems, Adv. Stud. Pure Math. 39, Math. Soc. Japan, 115-142 (2004)Zbl 1114.82017 MR 2073332 · Zbl 1114.82017
[24] Conus, D., Joseph, M., Khoshnevisan, D., Shiu, S-Y.: Initial measures for the stochastic heat equation. Ann. Inst. H. Poincar´e Probab. Statist. 50, 136-153 (2014) Zbl 1288.60077 MR 3161526 · Zbl 1288.60077
[25] Corwin, I.: The Kardar-Parisi-Zhang equation and universality class. Random Matrices Theory Appl. 1, 1130001, 76 pp. (2012)Zbl 1247.82040 MR 2930377 · Zbl 1247.82040
[26] Dey, P., Zygouras, N.: High temperature limits of directed polymers with heavy tail disorder. Ann. Probab. (to appear) · Zbl 1359.60117
[27] Doney, R. A.: One-sided local large deviation and renewal theorems in the case of infinite mean. Probab. Theory Related Fields 107, 451-465 (1997)Zbl 0883.60022 MR 1440141 · Zbl 0883.60022
[28] Ellis, R.: Entropy, Large Deviations, and Statistical Mechanics. Springer, Berlin (2006) Zbl 1102.60087 MR 2189669 · Zbl 1102.60087
[29] Giacomin, G.: Random Polymer Models. Imperial College Press, London (2007) Zbl 1125.82001 MR 2380992 · Zbl 1125.82001
[30] Giacomin, G.: Disorder and Critical Phenomena Through Basic Probability Models. Lecture Notes in Math. 2025, Springer (2011)Zbl 1230.82004 MR 2816225 · Zbl 1230.82004
[31] Grimmett, G.: The Random-Cluster Model. Grundlehren Math. Wiss. 333, Springer, Berlin (2006)Zbl 1122.60087 MR 2243761 · Zbl 1122.60087
[32] Harris, A. B.: Effect of random defects on the critical behaviour of Ising models. J. Phys. C 7, 1671-1692 (1974)
[33] Huse, D. A., Henley, C. L.: Pinning and roughening of domain wall in Ising systems due to random impurities. Phys. Rev. Lett. 54, 270-2711 (1985)
[34] Janson, S.: Gaussian Hilbert Spaces. Cambridge Tracts in Math. 129, Cambridge Univ. Press, Cambridge (1997)Zbl 0887.60009 MR 1474726
[35] Kahn, J., Kalai, G., Linial, N.: The influence of variables on boolean functions. In: Proceedings of the 29th Annual Symposium on Foundations of Computer Science, 68- 80 (1988) Polynomial chaos and scaling limits of disordered systems65
[36] Lacoin, H.: New bounds for the free energy of directed polymer in dimension 1+1 and 1+2. Comm. Math. Phys. 294, 471-503 (2010)Zbl 1227.82098 MR 2579463 · Zbl 1227.82098
[37] Lieb, E. H., Loss, M.: Analysis. 2nd ed., Grad. Stud. Math. 14, Amer. Math. Soc., Providence, RI (2001)Zbl 0966.26002 MR 1817225 · Zbl 0966.26002
[38] Liggett, T. L.: An invariance principle for conditioned sums of independent random variables. J. Math. Mech. 18, 559-570 (1968)Zbl 0181.20502 MR 0238373 · Zbl 0181.20502
[39] Mossel, E., O’Donnell, R., Oleszkiewicz, K.: Noise stability of functions with low influences: Invariance and optimality. Ann. of Math. 171, 295-341 (2010) Zbl 1201.60031 MR 2630040 · Zbl 1201.60031
[40] Peccati, G., Taqqu, M. S.: Wiener Chaos: Moments, Cumulants and Diagrams. Springer (2010)Zbl 1231.60003 MR 2791919 · Zbl 1231.60003
[41] Sasamoto, T., Spohn, H.: Exact height distributions for the KPZ equation with narrow wedge initial condition. Nuclear Phys. B 834, 523-542 (2010)Zbl 1204.35137 MR 2628936 · Zbl 1204.35137
[42] Sohier, J.: Finite size scaling for homogeneous pinning models. Alea 6, 163-177 (2009) Zbl 1172.60336 MR 2506863 · Zbl 1172.60336
[43] Tao, T., Vu, V.: Random matrices: Universality of the local eigenvalue statistics. Acta Math. 206, 127-204 (2011)Zbl 1217.15043 MR 2784665 · Zbl 1217.15043
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.