Convergence to the thermodynamic limit for random-field random surfaces. (English) Zbl 1517.82016
Summary: We study random surfaces with a uniformly convex gradient interaction in the presence of quenched disorder taking the form of a random independent external field. Previous work on the model has focused on proving existence and uniqueness of infinite-volume gradient Gibbs measures with a given tilt and on studying the fluctuations of the surface and its discrete gradient.
In this work, we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions \(d\geq 4\) while the distribution of the surface itself converges in dimensions \(d\geq 5\). Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by C. Cotar and C. Külske [Ann. Appl. Probab. 22, No. 4, 1650–1692 (2012; Zbl 1254.60095)].
In this work, we focus on the convergence of the thermodynamic limit, establishing convergence of the finite-volume distributions with Dirichlet boundary conditions to translation-covariant (gradient) Gibbs measures. Specifically, it is shown that, when the law of the random field has finite second moment and is symmetric, the distribution of the gradient of the surface converges in dimensions \(d\geq 4\) while the distribution of the surface itself converges in dimensions \(d\geq 5\). Moreover, a power-law upper bound on the rate of convergence in Wasserstein distance is obtained. The results partially answer a question discussed by C. Cotar and C. Külske [Ann. Appl. Probab. 22, No. 4, 1650–1692 (2012; Zbl 1254.60095)].
MSC:
| 82B24 | Interface problems; diffusion-limited aggregation arising in equilibrium statistical mechanics |
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
Citations:
Zbl 1254.60095References:
| [1] | ADAMS, S., KOTECKÝ, R. and MÜLLER, S. (2016). Strict convexity of the surface tension for non-convex potentials. Preprint. Available at arXiv:1606.09541. |
| [2] | AIZENMAN, M., HAREL, M. and PELED, R. (2020). Exponential decay of correlations in the 2D random field Ising model. J. Stat. Phys. 180 304-331. · Zbl 1460.60105 · doi:10.1007/s10955-019-02401-5 |
| [3] | AIZENMAN, M. and PELED, R. (2019). A power-law upper bound on the correlations in the \[2D\] random field Ising model. Comm. Math. Phys. 372 865-892. · Zbl 1447.82005 · doi:10.1007/s00220-019-03450-3 |
| [4] | AIZENMAN, M. and WEHR, J. (1989). Rounding of first-order phase transitions in systems with quenched disorder. Phys. Rev. Lett. 62 2503-2506. · doi:10.1103/PhysRevLett.62.2503 |
| [5] | AIZENMAN, M. and WEHR, J. (1990). Rounding effects of quenched randomness on first-order phase transitions. Comm. Math. Phys. 130 489-528. · Zbl 0714.60090 |
| [6] | ARMSTRONG, S. and WU, W. (2022). \[{C^2}\] regularity of the surface tension for the \[\nabla \phi\] interface model. Comm. Pure Appl. Math. 75 349-421. · Zbl 1494.82015 · doi:10.1002/cpa.22031 |
| [7] | BISKUP, M. and KOTECKÝ, R. (2007). Phase coexistence of gradient Gibbs states. Probab. Theory Related Fields 139 1-39. · Zbl 1120.82003 · doi:10.1007/s00440-006-0013-6 |
| [8] | BISKUP, M. and SPOHN, H. (2011). Scaling limit for a class of gradient fields with nonconvex potentials. Ann. Probab. 39 224-251. · Zbl 1222.60076 · doi:10.1214/10-AOP548 |
| [9] | Bovier, A. and Külske, C. (1994). A rigorous renormalization group method for interfaces in random media. Rev. Math. Phys. 6 413-496. · Zbl 0802.60098 · doi:10.1142/S0129055X94000171 |
| [10] | BOVIER, A. and KÜLSKE, C. (1996). There are no nice interfaces in \[(2+1)\]-dimensional SOS models in random media. J. Stat. Phys. 83 751-759. · Zbl 1081.82571 · doi:10.1007/BF02183747 |
| [11] | Brascamp, H. J. and Lieb, E. H. (1976). On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22 366-389. · Zbl 0334.26009 · doi:10.1016/0022-1236(76)90004-5 |
| [12] | BRASCAMP, H. J. and LIEB, E. H. (2002). Some inequalities for Gaussian measures and the long-range order of the one-dimensional plasma. In Inequalities 403-416. Springer, Berlin. |
| [13] | BRASCAMP, H. J., LIEB, E. H. and LEBOWITZ, J. L. (1975). The statistical mechanics of anharmonic lattices. Bull. Int. Stat. Inst. 46 393-404. · Zbl 0357.60051 |
| [14] | BRICMONT, J. and KUPIAINEN, A. (1988). Phase transition in the 3d random field Ising model. Comm. Math. Phys. 116 539-572. · Zbl 1086.82573 |
| [15] | BRYDGES, D. and YAU, H.-T. (1990). Grad \(ϕ\) perturbations of massless Gaussian fields. Comm. Math. Phys. 129 351-392. · Zbl 0705.60101 |
| [16] | CHATTERJEE, S. (2018). On the decay of correlations in the random field Ising model. Comm. Math. Phys. 362 253-267. · Zbl 1397.82011 · doi:10.1007/s00220-018-3085-0 |
| [17] | COTAR, C. and DEUSCHEL, J.-D. (2012). Decay of covariances, uniqueness of ergodic component and scaling limit for a class of \[\nabla \phi\] systems with non-convex potential. Ann. Inst. Henri Poincaré Probab. Stat. 48 819-853. · Zbl 1247.60133 · doi:10.1214/11-AIHP437 |
| [18] | Cotar, C., Deuschel, J.-D. and Müller, S. (2009). Strict convexity of the free energy for a class of non-convex gradient models. Comm. Math. Phys. 286 359-376. · Zbl 1173.82010 · doi:10.1007/s00220-008-0659-2 |
| [19] | Cotar, C. and Külske, C. (2012). Existence of random gradient states. Ann. Appl. Probab. 22 1650-1692. · Zbl 1254.60095 · doi:10.1214/11-AAP808 |
| [20] | COTAR, C. and KÜLSKE, C. (2015). Uniqueness of gradient Gibbs measures with disorder. Probab. Theory Related Fields 162 587-635. · Zbl 1334.60207 · doi:10.1007/s00440-014-0580-x |
| [21] | DARIO, P., HAREL, M. and PELED, R. (2021). Random-field random surfaces. Preprint. Available at arXiv:2101.01711. |
| [22] | DE GIORGI, E. (1957). Sulla differenziabilità e l’analiticità delle estremali degli integrali multipli regolari. Mem. Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. (3) 3 25-43. · Zbl 0084.31901 |
| [23] | DELMOTTE, T. (1997). Inégalité de Harnack elliptique sur les graphes. Colloq. Math. 72 19-37. · Zbl 0871.31008 · doi:10.4064/cm-72-1-19-37 |
| [24] | DELMOTTE, T. and DEUSCHEL, J.-D. (2005). On estimating the derivatives of symmetric diffusions in stationary random environment, with applications to \[\nabla \phi\] interface model. Probab. Theory Related Fields 133 358-390. · Zbl 1083.60082 · doi:10.1007/s00440-005-0430-y |
| [25] | Deuschel, J.-D., Giacomin, G. and Ioffe, D. (2000). Large deviations and concentration properties for \[\nabla \phi\] interface models. Probab. Theory Related Fields 117 49-111. · Zbl 0988.82018 · doi:10.1007/s004400050266 |
| [26] | DING, J. and WIRTH, M. (2020). Correlation length of two-dimensional random field Ising model via greedy lattice animal. Preprint. Available at arXiv:2011.08768. |
| [27] | DING, J. and XIA, J. (2021). Exponential decay of correlations in the two-dimensional random field Ising model. Invent. Math. 224 999-1045. · Zbl 1492.60272 · doi:10.1007/s00222-020-01024-y |
| [28] | FUNAKI, T. (2005). Stochastic interface models. In Lectures on Probability Theory and Statistics. Lecture Notes in Math. 1869 103-274. Springer, Berlin. · Zbl 1119.60081 · doi:10.1007/11429579_2 |
| [29] | FUNAKI, T. and SPOHN, H. (1997). Motion by mean curvature from the Ginzburg-Landau \[\nabla \phi\] interface model. Comm. Math. Phys. 185 1-36. · Zbl 0884.58098 · doi:10.1007/s002200050080 |
| [30] | GIACOMIN, G., OLLA, S. and SPOHN, H. (2001). Equilibrium fluctuations for \[\nabla \phi\] interface model. Ann. Probab. 29 1138-1172. · Zbl 1017.60100 · doi:10.1214/aop/1015345600 |
| [31] | Gilbarg, D. and Trudinger, N. S. (2001). Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin. Reprint of the 1998 edition. · Zbl 1042.35002 |
| [32] | Gloria, A. and Otto, F. (2011). An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 779-856. · Zbl 1215.35025 · doi:10.1214/10-AOP571 |
| [33] | Gloria, A. and Otto, F. (2012). An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 1-28. · Zbl 1387.35031 · doi:10.1214/10-AAP745 |
| [34] | HAN, Q. and LIN, F. (2011). Elliptic Partial Differential Equations 1. Am. Math. Soc. , Providence. · Zbl 1210.35031 |
| [35] | IMRY, Y. and MA, S.-K. (1975). Random-field instability of the ordered state of continuous symmetry. Phys. Rev. Lett. 35 1399. |
| [36] | KÜLSKE, C. and ORLANDI, E. (2006). A simple fluctuation lower bound for a disordered massless random continuous spin model in \[D=2\]. Electron. Commun. Probab. 11 200-205. · Zbl 1119.82019 · doi:10.1214/ECP.v11-1218 |
| [37] | KÜLSKE, C. and ORLANDI, E. (2008). Continuous interfaces with disorder: Even strong pinning is too weak in two dimensions. Stochastic Process. Appl. 118 1973-1981. · Zbl 1151.60348 · doi:10.1016/j.spa.2007.11.005 |
| [38] | MILLER, J. (2011). Fluctuations for the Ginzburg-Landau \[\nabla \phi\] interface model on a bounded domain. Comm. Math. Phys. 308 591-639. · Zbl 1237.82030 · doi:10.1007/s00220-011-1315-9 |
| [39] | MOSER, J. (1961). On Harnack’s theorem for elliptic differential equations. Comm. Pure Appl. Math. 14 577-591. · Zbl 0111.09302 · doi:10.1002/cpa.3160140329 |
| [40] | MOSER, J. (1964). A Harnack inequality for parabolic differential equations. Comm. Pure Appl. Math. 17 101-134. · Zbl 0149.06902 · doi:10.1002/cpa.3160170106 |
| [41] | NADDAF, A. and SPENCER, T. (1997). On homogenization and scaling limit of some gradient perturbations of a massless free field. Comm. Math. Phys. 183 55-84. · Zbl 0871.35010 · doi:10.1007/BF02509796 |
| [42] | Naddaf, A. and Spencer, T. (1998). Estimates on the variance of some homogenization problems. Unpublished preprint. · Zbl 0871.35010 |
| [43] | Nash, J. (1958). Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 931-954. · Zbl 0096.06902 · doi:10.2307/2372841 |
| [44] | Sheffield, S. (2005). Random surfaces. Astérisque 304 vi+175. · Zbl 1104.60002 |
| [45] | STROOCK, D. W. and ZHENG, W. (1997). Markov chain approximations to symmetric diffusions. Ann. Inst. Henri Poincaré Probab. Stat. 33 619-649. · Zbl 0885.60065 · doi:10.1016/S0246-0203(97)80107-0 |
| [46] | van Enter, A. C. D. and Külske, C. (2008). Nonexistence of random gradient Gibbs measures in continuous interface models in \[d=2\]. Ann. Appl. Probab. 18 109-119. · Zbl 1141.60074 · doi:10.1214/07-AAP446 |
| [47] | VAN ENTER, A. C. D. and SHLOSMAN, S. B. (2002). First-order transitions for n-vector models in two and more dimensions: Rigorous proof. Phys. Rev. Lett. 89 285702. · doi:10.1103/PhysRevLett.89.285702 |
| [48] | VAN ENTER, A. C. D. and SHLOSMAN, S. B. (2005). Provable first-order transitions for nonlinear vector and gauge models with continuous symmetries. Comm. Math. Phys. 255 21-32. · Zbl 1074.82031 · doi:10.1007/s00220-004-1286-1 |
| [49] | Velenik, Y. (2006). Localization and delocalization of random interfaces. Probab. Surv. 3 112-169. · Zbl 1189.82051 · doi:10.1214/154957806000000050 |
| [50] | VILLAIN, J. (1975). Theory of one-and two-dimensional magnets with an easy magnetization plane. II. The planar, classical, two-dimensional magnet. J. Phys. 36 581-590. |
| [51] | Villani, C. (2009). Optimal Transport: Old and New. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 338. Springer, Berlin · Zbl 1156.53003 · doi:10.1007/978-3-540-71050-9 |
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.
