Large scale stochastic dynamics. Abstracts from the workshop held September 11–17, 2022. (English) Zbl 1520.00022
Summary: The goal of this workshop was to explore the recent advances in the mathematical understanding of the macroscopic properties which emerge on large space-time scales from interacting microscopic particle systems. The talks addressed the following topics: randomness emerging from deterministic dynamics, hydrodynamic limits, Markov chain mixing times and cut-off phenomenon, superdiffusivity in out-of-equilibrium 2-dimensional systems.
MSC:
| 00B05 | Collections of abstracts of lectures |
| 00B25 | Proceedings of conferences of miscellaneous specific interest |
| 82-06 | Proceedings, conferences, collections, etc. pertaining to statistical mechanics |
| 82C05 | Classical dynamic and nonequilibrium statistical mechanics (general) |
| 82C20 | Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics |
| 82C24 | Interface problems; diffusion-limited aggregation in time-dependent statistical mechanics |
| 82C40 | Kinetic theory of gases in time-dependent statistical mechanics |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
References:
| [1] | R. Aimino, C. Liverani, Deterministic walks in random environment. Preprint ArXiv:1804.11114. · Zbl 1456.60269 |
| [2] | M. Blank, G. Keller, C. Liverani. Ruelle-Perron-Frobenius spectrum for Anosov maps. Non-linearity 15 (2002), no. 6, 1905-1973. · Zbl 1021.37015 |
| [3] | Bunimovich, L. A.; Sinaȋ, Ya. G. Statistical properties of Lorentz gas with periodic config-uration of scatterers. Comm. Math. Phys. 78 (1980/81), no. 4, 479-497. · Zbl 0459.60099 |
| [4] | L. A. Bunimovich, Y. G. Sinaȋ, and N. I. Chernov. Statistical properties of two-dimensional hyperbolic billiards. Uspekhi Mat. Nauk, 46(4(280)):43-92, 192, 1991. · Zbl 0748.58014 |
| [5] | G. Cristadoro, M. Lenci, and M. Seri. Erratum: “Recurrence for quenched random Lorentz tubes” [Chaos 20, 023115 (2010)] [mr2741899]. · Zbl 1311.37030 |
| [6] | Chaos, 20(4):049903, 1, 2010. |
| [7] | G. Cristadoro, M. Lenci, and M. Seri. Recurrence for quenched random Lorentz tubes. Chaos, 20(2):023115, 7, 2010. · Zbl 1311.37030 |
| [8] | M.F Demers; C. Liverani, Stability of statistical properties in two-dimensional piecewise hyperbolic maps. Trans. Amer. Math. Soc.360 (2008), no. 9, 4777-4814. · Zbl 1153.37019 |
| [9] | M.F. Demers and H.-K. Zhang, Spectral analysis of the transfer operator for the Lorentz Gas, J. Modern Dnyam. 5:4 (2011), 665-709. · Zbl 1321.37034 |
| [10] | M.F. Demers and H.-K. Zhang, A functional analytic approach to perturbations of the Lorentz gas, Comm. Math. Phys. 324:3 (2013), 767-830. · Zbl 1385.37050 |
| [11] | M.F. Demers and H.-K. Zhang, Spectral analysis of hyperbolic systems with singularities, Nonlinearity 27 (2014), 379-433. · Zbl 1347.37070 |
| [12] | M. Demers, C. Liverani. Projective Cones for Sequential Dispersing Billiards. Preprint arXiv:2104.06947. |
| [13] | D. Dolgopyat, D. Szász, and T. Varjú. Recurrence properties of planar Lorentz process. Duke Math. J., 142(2):241-281, 2008. · Zbl 1136.37022 |
| [14] | D. Dolgopyat, D. Szász, and T. Varjú. Limit theorems for locally perturbed planar Lorentz processes. Duke Math. J., 148(3):459-499, 2009. · Zbl 1177.37042 |
| [15] | G. Gallavotti: Divergencies and the approach to equilibrium in the Lorentz and the windtree models. Phys. Rev. 185: 308-322 (1969) |
| [16] | S. Gouëzel, C. Liverani. Banach spaces adapted to Anosov systems. Ergodic Theory Dynam. Systems 26 (2006), no. 1, 189-217. · Zbl 1088.37010 |
| [17] | S. Gouëzel, C. Liverani. Compact locally maximal hyperbolic sets for smooth maps: fine statistical properties. J. Differential Geom. 79 (2008), no. 3, 433-477. · Zbl 1166.37010 |
| [18] | M. Lenci. Aperiodic Lorentz gas: recurrence and ergodicity. Ergodic Theory Dynam. Sys-tems, 23(3):869-883, 2003. · Zbl 1069.82011 |
| [19] | M. Lenci. Typicality of recurrence for Lorentz gases. Ergodic Theory Dynam. Systems, 26(3):799-820, 2006. · Zbl 1098.82028 |
| [20] | M. Lenci. Central limit theorem and recurrence for random walks in bistochastic random environments. Journal of Mathematical Physics, 49(12):125213, 2008. · Zbl 1159.81328 |
| [21] | M. Lenci and S. Troubetzkoy. Infinite-horizon Lorentz tubes and gases: recurrence and ergodic properties. Phys. D, 240(19):1510-1515, 2011. · Zbl 1228.37030 |
| [22] | C. Liverani. Decay of correlations. Ann. of Math. (2), 142(2):239-301, 1995. · Zbl 0871.58059 |
| [23] | C. Liverani. Decay of correlations for piecewise expanding maps. Journal of Statistical Physics, 78(3):1111-1129, 1995. · Zbl 1080.37501 |
| [24] | C. Liverani and V. Maume-Deschamps. Lasota-Yorke maps with holes: conditionally invari-ant probability measures and invariant probability measures on the survivor set. Ann. Inst. H. Poincaré Probab. Statist., 39(3):385-412, 2003. · Zbl 1021.37002 |
| [25] | Christopher Lutsko, Bálint Tóth Invariance Principle for the Random Lorentz Gas -Be-yond the Boltzmann-Grad Limit Preprint arXiv:1812.11325. |
| [26] | J. Marklof. The low-density limit of the lorentz gas: periodic, aperiodic and random. In Proceedings of the ICM 2014, Seoul, Vol. III, pages 623-646. 2004. · Zbl 1373.82065 |
| [27] | J. Marklof and A. Strömbergsson. The Boltzmann-Grad limit of the periodic Lorentz gas. Ann. of Math. (2), 174(1):225-298, 2011. · Zbl 1237.37014 |
| [28] | M. Seri, M. Lenci, M. degli Esposti, and G. Cristadoro. Recurrence and higher ergodic properties for quenched random Lorentz tubes in dimension bigger than two. J. Stat. Phys., 144(1):124-138, 2011. · Zbl 1226.82056 |
| [29] | J. van den Berg and H. Kesten, Inequalities for the Time Constant in First-Passage Per-colation, Annals of Applied Probability 3 (1993), 56-80. · Zbl 0771.60092 |
| [30] | E. Candellero and A. Stauffer, First passage percolation in hostile environment is not mono-tone, available from arXiv:2110.05821. |
| [31] | E. Candellero and A. Stauffer, Coexistence of competing first passage percolation on hy-perbolic graphs, Annales de l’Institut Henri Poincaré Probabilités et Statistiques 57 (2021), 2128-2164. · Zbl 1492.60268 |
| [32] | D. Dauvergne and A. Sly, Spread of infections in a heterogeneous moving population, avail-able from arXiv:2105.11947. · Zbl 1533.60172 |
| [33] | D. Dauvergne and A. Sly, The SIR model in a moving population: propagation of infection and herd immunity, available from arXiv:2209.06037. |
| [34] | T. Finn and A. Stauffer, Non-equilibrium multi-scale analysis and coexistence in competing first passage percolation, Journal of the European Mathematical Society, to appear. · Zbl 1543.60113 |
| [35] | O. Garet and R. Marchand, First-passage competition with different speeds: positive density for both species is impossible, Electronic Journal of Probability 13 (2008), 2118-2159. · Zbl 1191.60111 |
| [36] | V. Sidoravicius and A. Stauffer, Multi-particle diffusion limited aggregation, Inventiones Mathematicae 218 (2019), 491-571. · Zbl 1491.60183 |
| [37] | L. Bertini, N. Cancrini, The two-dimensional stochastic heat equation: renormalizing a multiplicative noise, J. Phys. A: Math. Gen. 31 (1998), 615-622. · Zbl 0976.82035 |
| [38] | F. Caravenna, R. Sun, N. Zygouras, Universality in marginally relevant disordered systems, Ann. Appl. Probab. 27 (2017), 3050-3112. · Zbl 1387.82032 |
| [39] | F. Caravenna, R. Sun, N. Zygouras, On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window, Commun. Math. Phys. 372 (2019), 385-440. · Zbl 1427.82063 |
| [40] | F. Caravenna, R. Sun, N. Zygouras, The Dickman subordinator, renewal theorems, and disordered systems, Electron. J. Probab. 24 (2019), paper no. 101, 40 pp. · Zbl 1466.60182 |
| [41] | F. Caravenna, R. Sun, N. Zygouras, The Critical 2d Stochastic Heat Flow, Preprint (2021), arXiv:2109.03766 [math.PR] |
| [42] | F. Caravenna, R. Sun, N. Zygouras, The critical 2d Stochastic Heat Flow is not a Gaussian Multiplicative Chaos, Preprint (2022), arXiv:2206.08766 [math.PR] |
| [43] | F. Comets, Directed Polymers in Random Environments, Springer (2017), Lecture Notes in Mathematics, 2175. · Zbl 1392.60002 |
| [44] | Y. Gu, J. Quastel, L.-C. Tsai, Moments of the 2D SHE at criticality, Prob. Math. Phys. 2 (2021), 179-219. · Zbl 1483.60093 |
| [45] | T. Bodineau and A. Guionnet. About the stationary states of vortex systems. Ann. Inst. H. Poincaré Probab. Statist., 35(2):205-237, 1999. · Zbl 0920.60095 |
| [46] | C. Geldhauser and M. Romito. Limit theorems and fluctuations for point vortices of gener-alized Euler equations. J. Stat. Phys., 182(3):Paper No. 60, 27, 2021. · Zbl 1464.76018 |
| [47] | F. Grotto and M. Romito. A central limit theorem for Gibbsian invariant measures of 2D Euler equations. Comm. Math. Phys., 376(3):2197-2228, 2020. · Zbl 1460.60114 |
| [48] | L. Onsager. Statistical hydrodynamics. Nuovo Cimento (9), 6(Supplemento, 2 (Convegno Internazionale di Meccanica Statistica)):279-287, 1949. References |
| [49] | A. Blanca and A. Sinclair. Random-cluster dynamics in Z 2 . Probab. Theory Related Fields, 168:821-847, 2017. Extended abstract appeared in Proc. of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2016), pp. 498-513. · Zbl 1419.82007 |
| [50] | T. Bodineau. Slab percolation for the Ising model. Probability Theory and Related Fields, 132(1):83-118, 2005. · Zbl 1071.82012 |
| [51] | C. Borgs, J. T. Chayes, and P. Tetali. Tight bounds for mixing of the Swendsen-Wang algorithm at the Potts transition point. Probab. Theory Related Fields, 152(3-4):509-557, 2012. · Zbl 1250.60034 |
| [52] | C. Borgs, J. Chayes, T. Helmuth, W. Perkins, and P. Tetali. Efficient sampling and counting algorithms for the Potts model on Z d at all temperatures. In Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2020, page 738-751, New York, NY, USA, 2020. Association for Computing Machinery. · Zbl 07298284 |
| [53] | Z. Chen, A. Galanis, L. A. Goldberg, W. Perkins, J. Stewart, and E. Vigoda. Fast algorithms at low temperatures via Markov chains. Random Structures & Algorithms, 58(2):294-321, 2021. · Zbl 07748484 |
| [54] | R. Gheissari and E. Lubetzky. Mixing times of critical two-dimensional Potts models. Comm. Pure Appl. Math, 71(5):994-1046, 2018. · Zbl 1392.82007 |
| [55] | R. Gheissari and A. Sinclair, Low-temperature Ising dynamics from random initializations, Symposium on Theory of Computing STOC (2022). |
| [56] | R. Gheissari and A. Sinclair, Spatial mixing and the random cluster dynamics on lattices, preprint (2022). |
| [57] | G. Grimmett. The Random-Cluster Model. In Probability on Discrete Structures, volume 110 of Encyclopaedia Math. Sci., pages 73-123. Springer, Berlin, 2004. · Zbl 1045.60105 |
| [58] | H. Guo and M. Jerrum. Random cluster dynamics for the Ising model is rapidly mixing. In Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2017, pages 1818-1827, 2017. · Zbl 1419.82013 |
| [59] | M. Harel and Y. Spinka. Finitary codings for the random-cluster model and other infinite-range monotone models, Electronic Journal of Probability 27: 1-32 (2022). · Zbl 1498.60387 |
| [60] | T. Helmuth, W. Perkins, and G. Regts. Algorithmic Pirogov-Sinai theory. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, page 1009-1020, New York, NY, USA, 2019. · Zbl 1437.82006 |
| [61] | D. A. Huse and D. S. Fisher. Dynamics of droplet fluctuations in pure and random Ising systems. Phys. Rev. B, 35:6841-6846, May 1987. |
| [62] | M. Jerrum and A. Sinclair. Polynomial-time approximation algorithms for the Ising model. SIAM Journal on Computing, 22:1087-1116, 1993. · Zbl 0782.05076 |
| [63] | E. Lubetzky, F. Martinelli, A. Sly, and F. L. Toninelli. Quasi-polynomial mixing of the 2D stochastic Ising model with “plus” boundary up to criticality. J. Eur. Math. Soc. (JEMS), 15(2):339-386, 2013. · Zbl 1266.60161 |
| [64] | F. Martinelli and E. Olivieri. Approach to equilibrium of Glauber dynamics in the one phase region. I. The attractive case. Communications in Mathematical Physics, 161:447-486, 1994. · Zbl 0793.60110 |
| [65] | A. Pisztora. Surface order large deviations for Ising, Potts and percolation models. Proba-bility Theory and Related Fields, 104(4):427-466, 1996. · Zbl 0842.60022 |
| [66] | R. H. Swendsen and J.-S. Wang. Nonuniversal critical dynamics in Monte Carlo simulations. Phys. Rev. Lett., 58:86-88, 1987. |
| [67] | A.-S. Sznitman. Vacant set of random interlacements and percolation. Annals of Mathemat-ics (2010): 2039-2087. · Zbl 1202.60160 |
| [68] | A. Teixeira. Interlacement percolation on transient weighted graphs. Electronic Journal of Probability 14 (2009): 1604-1627. · Zbl 1192.60108 |
| [69] | D. Windisch. Random walk on a discrete torus and random interlacements. Electronic Com-munications in Probability 13 (2008): 140-150. · Zbl 1187.60089 |
| [70] | B. Doyon, Lecture Notes on Generalised Hydrodynamics, SciPost Phys. Lecture Notes 18 (2020). |
| [71] | A. Abanov, B. Doyon, J. Dubai, A. Kamenev, and H. Spohn, eds., Hydrodynamics of low-dimensional quantum systems, Special Issue Collection, J. Phys. A (2022). |
| [72] | A. Bastianello, B. Bertini, B. Doyon, and R. Vasseur, eds., Emergent hydrodynamics in integrable many-body systems, Special Issue Collection, Journ. Stat. Mech. (2022). |
| [73] | H. Spohn, Hydrodynamic Scales of Integrable Many-Body Systems, in preparation (2023). |
| [74] | F. Calogero, Classical Many-Body Problems Amenable to Exact Treatments, Springer-Verlag, Heidelberg, 2001. · Zbl 1011.70001 |
| [75] | A. Faggionato, Hydrodynamic limit of simple exclusion processes in symmetric random en-vironments via duality and homogenization, Probab. Theory Relat. Fields (2022), to appear. Available online. · Zbl 1515.60139 |
| [76] | M. Biskup, Recent progress on the random conductance model, Probability Surveys, 6 (2011), 294-373. · Zbl 1245.60098 |
| [77] | D.J. Daley, D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer Verlag, 1988. · Zbl 0657.60069 |
| [78] | A. Faggionato, Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit, Electron. J. Probab. 13 (2008), 2217-2247. · Zbl 1189.60172 |
| [79] | A. Faggionato, Stochastic homogenization of random walks on point processes, Ann. Inst. H. Poincaré Probab. Statist. (to appear), arXiv:2009.08258. · Zbl 1533.60185 |
| [80] | A. Faggionato, C. Tagliaferri, Homogenization, simple exclusion processes and random re-sistor networks on Delaunay triangulations, in preparation. |
| [81] | P. Gonçalves, M. Jara, Scaling limit of gradient systems in random environment, J. Stat. Phys. 131 (2008), 691-716. · Zbl 1144.82043 |
| [82] | M. Jara, C. Landim, Quenched nonequilibrium central limit theorem for a tagged particle in the exclusion process with bond disorder, Ann. Inst. H. Poincaré Probab. Statist. 44 (2008), 341-361. · Zbl 1195.60124 |
| [83] | C. Kipnis, C. Landim, Scaling limits of interacting particle systems, Grundlehren der Math-ematischen Wissenschaften 320 (1999), Springer-Verlag. · Zbl 0927.60002 |
| [84] | K. Nagy, Symmetric random walk in random environment, Period. Math. Hung. 45 (2002), 101-120. · Zbl 1064.60202 |
| [85] | L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim; Macroscopic current fluctuations in stochastic lattice gases. Phys. Rev. Lett. 94, 030601, (2005). |
| [86] | L. Bertini, A. De Sole, D. Gabrielli, G. Jona-Lasinio, C. Landim; Non equilibrium current fluctuations in stochastic lattice gases. J. Stat. Phys. 123, 237-276 (2006). · Zbl 1097.82016 |
| [87] | L. Bertini, D. Gabrielli, C. Landim; Concurrent Donsker-Varadhan and hydrodynamical large deviations arXiv:2111.05892, (2021) · Zbl 1518.60035 |
| [88] | T. Bodineau, B. Derrida; Current fluctuations in nonequilibrium diffusive systems: an additivity principle. Phys. Rev. Lett. 92, 180601 (2004). |
| [89] | K. Bannai, Y. Kametani, M. Sasada, Topological Structures of Large Scale Interacting Sys-tems via Uniform Functions and Forms, arXiv:2009.04699 (2020). |
| [90] | K. Bannai, M. Sasada, Varadhan’s Decomposition of Shift-Invariant Closed L 2 -forms for Large Scale Interacting Systems on the Euclidean Lattice, arXiv:2111.08934 (2021) |
| [91] | S. R. S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions. II, Asymptotic problems in probability theory: stochastic models and diffusions on fractals, (Sanda/Kyoto 1990), vol. 1. Edited by K. D. Elworthy and N. Ikeda. Pitman Research Notes in Mathematics 283. Longman Scientific & Technical (Harlow), (1993) References · Zbl 0780.00028 |
| [92] | Riddhipratim Basu, Jonathan Hermon, and Yuval Peres. Characterization of cutoff for re-versible Markov chains. Ann. Probab., 45(3):1448-1487, 2017. · Zbl 1374.60129 |
| [93] | Pietro Caputo, Thomas M. Liggett, and Thomas Richthammer. Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc., 23(3):831-851, 2010. · Zbl 1203.60145 |
| [94] | Persi Diaconis. The cutoff phenomenon in finite Markov chains. Proc. Nat. Acad. Sci. U.S.A., 93(4):1659-1664, 1996. · Zbl 0849.60070 |
| [95] | Nina Gantert, Evita Nestoridi, and Dominik Schmid. Mixing times for the simple exclusion process with open boundaries, 2021. · Zbl 1525.60113 |
| [96] | Patrícia Gonçalves, Milton Jara, Rodrigo Marinho, and Otávio Menezes. Sharp convergence to equilibrium for the ssep with reservoirs, 2021. References |
| [97] | D. Aldous, and D. Lanoue A lecture on the averaging process, Probab. Surv. 9 (2012), 90-102. · Zbl 1245.60088 |
| [98] | S. Chatterjee, P. Diaconis, A. Sly and L. Zang A phase transition for repeated averages, Ann. Probab. 50(1) (2022), 1-17. · Zbl 1485.60069 |
| [99] | R. Movassagh, M. Szegedy and G. Wang Repeated averages on graph, arXiv:2205.04535 (2022). |
| [100] | M. Quattropani and F. Sau Mixing time of the Averaging process and its discrete dual in finite dimensional geometries, Ann. Appl. Probab., to appear. · Zbl 1521.60018 |
| [101] | C. Bezuidenhout and L. Gray, Critical attractive spin systems, Ann. Probab. 22 (1994), no. 3, 1160-1194. MR 1303641 · Zbl 0819.60094 |
| [102] | C. Bezuidenhout and G. Grimmett, Exponential decay for subcritical contact and percolation processes, Ann. Probab. 19 (1991), no. 3, 984-1009. MR 1112404 · Zbl 0743.60107 |
| [103] | H. Duminil-Copin, A. Raoufi, and V. Tassion, Sharp phase transition for the random-cluster and Potts models via decision trees, Ann. of Math. (2) 189 (2019), no. 1, 75-99. MR 3898174 · Zbl 1482.82009 |
| [104] | L. F. Gray, Toom’s stability theorem in continuous time, Perplexing problems in probability, Progr. Probab., vol. 44, Birkhäuser, Boston, MA, 1999, pp. 331-353. MR 1703139 · Zbl 0948.60093 |
| [105] | I. Hartarsky, U -bootstrap percolation: critical probability, exponential decay and applica-tions, Ann. Inst. Henri Poincaré Probab. Stat. 57 (2021), no. 3, 1255-1280. MR 4291442 · Zbl 1480.60292 |
| [106] | , Bootstrap percolation, probabilistic cellular automata and sharpness, J. Stat. Phys. 187 (2022), no. 3, Article No. 21, 17. MR 4408459 · Zbl 1492.60276 |
| [107] | I. Hartarsky and R. Szabó, Generalised oriented site percolation, Markov Process. Related Fields 28 (2022), no. 2. · Zbl 1539.60127 |
| [108] | T. M. Liggett, Stochastic interacting systems: contact, voter and exclusion processes, Grundlehren der mathematischen Wissenschaften, vol. 324, Springer, Berlin, Heidelberg, 1999. MR 1717346 · Zbl 0949.60006 |
| [109] | , Interacting particle systems, Classics in mathematics, Springer, Berlin, Heidelberg, 2005, Originally published by Springer, New York (1985). MR 2108619 · Zbl 0559.60078 |
| [110] | J. M. Swart, R. Szabó, and C. Toninelli, Peierls bounds from Toom contours, arXiv e-prints (2022). |
| [111] | A. L. Toom, Stable and attractive trajectories in multicomponent systems, Multicomponent random systems, Adv. Probab. Related topics, vol. 6, Dekker, New York, 1980, pp. 549-575. MR 599548 · Zbl 0441.68053 |
| [112] | O. Busani, Diffusive scaling limit of the Busemann process in Last Passage Percolation, arXiv:2110.03808. |
| [113] | O. Busani, T. Seppäläinen, E. Sorensen, The stationary horizon and semi-infinite geodesics in the directed landscape, arXiv:2203.13242. |
| [114] | Boldrighini, C., Dobrushin, R.L. and Suhov, Yu.M.: Hydrodynamical limit for a degenerate model of classical statistical mechanics. Uspekhi Matem. Nauk [Russian], 35 no 4, 152 (1980) |
| [115] | Boldrighini, C., Dobrushin, R.L. and Suhov, Yu.M.: One dimensional hard rod caricature of hydrodynamics. J. Stat. Phys. 31, 577-616 (1983) |
| [116] | C. Boldrighini, W. David Wick Fluctuations in a One-Dimensional Mechanical System. I. The Euler Limit, JSP, Vol. 52, Nos. 3/4, 1988 · Zbl 1084.82505 |
| [117] | Boldrighini, C., Dobrushin, R.L. and Suhov, Yu.M.: One-Dimensional Hard-Rod Caricature of Hydrodynamics: Navier-Stokes Correction,1990 |
| [118] | C. Boldrighini, Yu.M. Suhov, One-Dimensional Hard-Rod Caricature of Hydrodynamics: “Navier-Stokes Correction” for Local Equilibrium Initial States, Commun. Math. Phys. 189, 577-590, 1997. · Zbl 0895.76002 |
| [119] | Croydon, D.A., Sasada, M., Generalized Hydrodynamic Limit for the Box-Ball System. Commun. Math. Phys. 383, 427-463 (2021). https://doi.org/10.1007/s00220-020-03914-x · Zbl 1479.82044 · doi:10.1007/s00220-020-03914-x |
| [120] | B Doyon, and H Spohn, Dynamics of hard rods with initial domain wall state, J. Stat. Mech. (2017) 073210 · Zbl 1457.82247 |
| [121] | P.A.Ferrari, C. Franceschini, D.G.E. Grevino, H.Spohn, Generalized hydrodynamics for size inhomogeneous hard rods, in preparation. |
| [122] | P. A. Ferrari and D. Gabrielli, BBS invariant measures with independent soliton components, Electron. J. Probab. 25 1 -26, 2020. https://doi.org/10.1214/20-EJP475 · Zbl 1451.37018 · doi:10.1214/20-EJP475 |
| [123] | JL Lebowitz, JK Percus, J Sykes, Time evolution of the Total Distribution Function of a One-Dimensional System of Hard Rods, Physical review, 171, 1, 1968 |
| [124] | E. Presutti, W D Wick, Macroscopic Stochastic Fluctuations in a One-Dimensional Me-chanical System, Journal of Statistical Physics, Vol. 51, Nos. 5/6, 1988 · Zbl 1083.82502 |
| [125] | H. Spohn, Hydrodynamical Theory for Equilibrium Time Correlation Functions of Hard Rods, Annals of Physics, 141,353-364 (1982) |
| [126] | H. Spohn, Large Scale Dynamics of Interacting Particles Systems, Springer 1991. · Zbl 0742.76002 |
| [127] | H. Spohn, Hydrodynamic Equations for the Toda Lattice, 2021, https://doi.org/10.48550/arXiv.2101.06528 · doi:10.48550/arXiv.2101.06528 |
| [128] | Bayati, M., and Montanari, A.: The dynamics of message passing on dense graphs, with applications to compressed sensing. IEEE Trans. Inform. Theory, 57, 764-785, 2011. · Zbl 1366.94079 |
| [129] | Bolthausen, E.: An iterative construction of solutions of the TAP equations for the Sherrington-Kirkpatrick model. Comm. Math. Phys., 325, 333-366, 2014. · Zbl 1288.82038 |
| [130] | Bolthausen, E.: A Morita type proof of the replica symmetric formula for SK. In “Statistical mechanics of classical and disordered systems” Springer Proc. Math. Stat., 293, 63-93, 2019. · Zbl 1446.82083 |
| [131] | Bolthausen, E., Nakajima, Sh., Sun, N., and Xu, Ch.: Gardner’s formula for Ising perceptron models at small density. Proceedings of Machine Learning Research, 1-126, 2022. |
| [132] | Brennecke, Ch, and Yau, H.T. : A note on the replica symmetric formula for the SK model. arXiv:2109.07354, 2021. |
| [133] | Ding, J., Sun, N.: Capacity lower bound for the Ising perceptron. arXiv:1809.07742, 2018. |
| [134] | Mézard, M.: The space of interactions in neural networks: Gardner’s computation with the cavity method. J. Phys. A, 22, 2181, 1989. |
| [135] | Talagrand, M.: Mean field models for spin glasses. Volume II, Chap 5. Ergebnisse der Math-ematik und ihrer Grenzgebiete. 3. Folge, Vol 55. Springer, Heidelberg, 2011. References · Zbl 1214.82002 |
| [136] | Eric A. Carlen, Shigeo Kusuoka and Daniel W. Stroock, Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. 23:2 suppl. (1987), 245-287. · Zbl 0634.60066 |
| [137] | Terry Lyons, Instability of the Liouville property for quasi-isometric Riemannian manifolds and reversible Markov chains. J. Differential Geom. 26 (1987), no. 1, 33-66. · Zbl 0599.60011 |
| [138] | Itai Benjamini, Instability of the Liouville property for quasi-isometric graphs and manifolds of polynomial volume growth. J. Theoret. Probab. 4 (1991), no. 3, 631-637. · Zbl 0725.60087 |
| [139] | Jian Ding and Yuval Peres, Sensitivity of mixing times. Electron. Commun. Probab. 18 (2013), no. 88, 6 pp. · Zbl 1307.60051 |
| [140] | Jonathan Hermon and Yuval Peres, On sensitivity of mixing times and cutoff. Electron. J. Probab. 23 (2018), Paper No. 25, 34 pp. · Zbl 1387.60112 |
| [141] | Jonathan Hermon and Gady Kozma, Sensitivity of mixing times of Cayley graphs. Available at: arXiv:2008.07517 |
| [142] | R. Bauerschmidt, N. Crawford, and T. Helmuth. Percolation transition for random forests in d ≥ 3. arXiv preprint arXiv:2107.01878, 2021. |
| [143] | R. Bauerschmidt, N. Crawford, T. Helmuth, and A. Swan. Random spanning forests and hyperbolic symmetry. Communications in Mathematical Physics, 381(3):1223-1261, 2021. · Zbl 1470.60270 |
| [144] | R. Bauerschmidt and T. Helmuth. Spin systems with hyperbolic symmetry: a survey. arXiv preprint arXiv:2109.02566, 2021. |
| [145] | D. Brydges and G. Slade. A renormalisation group method. I. Gaussian integration and normed algebras, II. Approximation by local polynomials. J. Stat. Phys., 159(3), 2015. · Zbl 1317.82013 |
| [146] | S. Caracciolo, J. Jacobsen, H. Saleur, A. Sokal, and A. Sportiello. Fermionic field theory for trees and forests. Phys. Rev. Lett., aracc(8):080601, 4, 2004. |
| [147] | J. Chayes, L. Chayes, G. Grimmett, H. Kesten, and R. H. Schonmann. The correlation length for the high-density phase of bernoulli percolation. The Annals of Probability, pages 1277-1302, 1989. · Zbl 0696.60094 |
| [148] | P. Easo. The wired arboreal gas on regular trees. Electronic Communications in Probability, 27:1-10, 2022. · Zbl 1491.60174 |
| [149] | G. Grimmett. The random-cluster model, volume 333 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, Berlin, 2006. · Zbl 0858.60093 |
| [150] | T. Lubensky and J. Isaacson. Field theory for the statistics of branched polymers, gelation, and vulcanization. Physical Review Letters, 41(12):829, 1978. |
| [151] | T. Luczak and B. Pittel. Components of random forests. Combin. Probab. Comput., 1(1):35-52, 1992. · Zbl 0793.05109 |
| [152] | J. Martin and D. Yeo. Critical random forests. ALEA Lat. Am. J. Probab. Math. Stat., 15(2):913-960, 2018. · Zbl 1393.05241 |
| [153] | G. Ray and B. Xiao. Forests on wired regular trees. ALEA, Lat. Am. J. Probab. Math. Stat., 19:1035-1043, 2022. · Zbl 1502.60162 |
| [154] | J.B. Walsh An introduction to stochastic partial differential equations BT -École d’Été de Probabilités de Saint Flour XIV -1984École d’été de probablités de Saint-Flour XIV (1984), Lecture Notes in Mathematics 1080, Springer. |
| [155] | Q. Berger C. Chongand H. Lacoin The stochastic heat equation with multiplicative Lévy noise: Existence, moments, and intermittency, preprint, arXiv:2111.07988. |
| [156] | Q. Berger and H. Lacoin The Scaling Limit of the Directed Polymer with Power-Law Tail Disorder, Communications in Mathematical Physics 386 (2021) 1051-1105. · Zbl 1470.60271 |
| [157] | Q. Berger and H. Lacoin The continuum directed polymer in Lévy Noise, J.École Polytech-nique (in press). · Zbl 1482.82038 |
| [158] | J. Ding, J. Song and R. Sun, A New Correlation Inequality for Ising Models with External Fields, arXiv:2107.09243, 2021. |
| [159] | D. Bakry and M.Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 177-206, 1985. |
| [160] | R. Bauerschmidt and T. Bodineau, Log-Sobolev inequality for the continuum sine-Gordon model, Commun. Pure Appl. Math. 74 2064-2113, 2021. References · Zbl 1475.60144 |
| [161] | M. Balázs, J. Quastel, T. Seppäläinen. Fluctuation exponent of the KPZ/stochastic Burgers equation, Jour. American Math. Society 24, no. 3, (2011), 683-708. · Zbl 1227.60083 |
| [162] | F. Comets, C. Cosco, C. Mukherjee. Renormalizing the Kardar-Parisi-Zhang equation in d ≥ 3 in weak disorder, Jour. Stat. Phys. 179, (2020), 713-728. · Zbl 1434.60278 |
| [163] | S. Chatterjee, A. Dunlap. Constructing a solution of the (2 + 1)-dimensional KPZ equation, Ann. Probab. 48, no. 2, (2020), 1014-1055. · Zbl 1434.60148 |
| [164] | G. Cannizzaro, D. Erhard, P. Schönbauer. 2D anisotropic KPZ at stationarity: scaling, tightness and nontriviality, Ann. Probab. 49, no. 1, (2021), 122-156. · Zbl 1457.60112 |
| [165] | G. Cannizzaro, D. Erhard, F. Toninelli. The stationary AKPZ equation: logarithmic su-perdiffusivity, Comm. Pure Applied Math. (to appear). arXiv:2007.12203. |
| [166] | G. Cannizzaro, D. Erhard, F. Toninelli. Weak coupling limit of the Anisotropic KPZ equa-tion, Duke Math. Jour. (to appear). arXiv:2108.09046 |
| [167] | F. Caravenna, R. Sun, N. Zygouras. The two-dimensional KPZ equation in the entire sub-critical regime, Ann. Probab. 48, no. 3, (2020), 1086-1127. · Zbl 1444.60061 |
| [168] | C. Cosco, S. Nakajima, M. Nakashima. Law of large numbers and fluctuations in the sub-critical and L 2 regions for SHE and KPZ equation in dimension d ≥ 3. Stochastic Process. Appl. 151 (2022), 127-173. · Zbl 1493.60149 |
| [169] | M. Gubinelli, P. Imkeller, N. Perkowski. Paracontrolled distributions and singular PDEs, Forum Math. Pi 3, (2015), e6, 75. · Zbl 1333.60149 |
| [170] | M. Gubinelli, N. Perkowski. The infinitesimal generator of the stochastic Burgers equation, Probab. Theory Rel. Fields 178, (2020), 1067-1124. · Zbl 1470.60192 |
| [171] | Y. Gu. Gaussian fluctuations from the 2D KPZ equation. Stoch. Partial Differ. Equ. Anal. Comput. 8, no. 1, (2020), 150-185. · Zbl 1431.35257 |
| [172] | M. Hairer. A theory of regularity structures, Invent. Math. 198, no. 2, (2014), 269-504. · Zbl 1332.60093 |
| [173] | M. Kardar, G. Parisi, Y. Zhang. Dynamic scaling of growing interfaces, Phys. Rev. Lett. 56, no. 9, (1986), 889-892. · Zbl 1101.82329 |
| [174] | D. Lygkonis, N. Zygouras. Edwards-Wilkinson fluctuations for the directed polymer in the full L 2 -regime for dimensions d ≥ 3. Ann. Inst. H. Poincaré (B): Prob. Stat. 58(1), (2022), 65-104. · Zbl 1484.82076 |
| [175] | J. Magnen, J. Unterberger. The scaling limit of the KPZ equation in space dimension 3 and higher, J. Stat. Phys. 171, no. 4, (2018), 543-598. · Zbl 1394.35508 |
| [176] | J. Quastel, S. Sarkar. Convergence of exclusion processes and KPZ equation to the KPZ fixed point, Journal of the American Mathematical Society (to appear). arXiv:2008.06584. References · Zbl 1520.60063 |
| [177] | B. Alder, T. Wainwright (1967). Velocity autocorellations for hard speres. Physical review letters 18 988. |
| [178] | G. Cannizzaro, F. Toninelli, L. Haunschmid-Sibitz (2022) √ log t -superdiffusivity for a Brownian particle in the curl of the 2d GFF, accepted at Annals of Probability · Zbl 1502.82015 |
| [179] | B., Toth, B. Valko (2012)Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d = 2. Journal of Statistical Physics, 147(1), 113-131. · Zbl 1245.82092 |
| [180] | H.-T. Yau (2004). (log t) 2 3 law of the two dimensional asymmetric simple exclusion process. |
| [181] | Ann. of Math. (2) 159 377-405 |
| [182] | B. J. Alder, T. E. Wainwright, Velocity Autocorrelations for Hard Spheres, Phys. Rev. Lett. 18 (1967), no. 23, 998-990. |
| [183] | G. Cannizzaro, L. Haunschmid-Sibitz, F. Toninelli, √ log t-superdiffusivity for a Brownian particle in the curl of the 2d GFF, ArXiv e-prints, arXiv:2106.06264 (2021). |
| [184] | G. L. Feltes, H. Weber, Brownian particle in the curl of 2-d stochastic heat equations, In preparation. · Zbl 1532.60163 |
| [185] | B. Tóth, B. Valkó Superdiffusive bounds on self-repellent Brownian polymers and diffusion in the curl of the Gaussian free field in d=2, Journal of Statistical Physics 147 (2012), no. 1, 113-131. · Zbl 1245.82092 |
| [186] | H.-T. Yau, (logt)2/3 law of the two dimensional asymmetric simple exclusion process, Ann. of Math. 159 (2004), no. 2, 377-405. · Zbl 1060.60099 |
| [187] | A.V. Bobylev,F.A. Maaøo, A. Hansen and E.H. Hauge.two-dimensional magnetotransport according to the classical Lorentz model, Phys. Rev. Lett. 75 (1995) 2. |
| [188] | G. Gallavotti, Grad-Boltzmann limit and Lorentz’s gas. Statistical mechanics: a short trea-tise, appendix 1.A2. Springer, Berlin (1999). · Zbl 0932.82002 |
| [189] | S. Chatteerjee and P. Diaconis, Speeding up Markov chains with deterministic jumps, Prob-ability Theory and Related Fields 178(3) (2020), 1193-1214. · Zbl 1468.60088 |
| [190] | F.R. Chung, P. Diaconis and R.L. Graham, Random walks arising in random number gen-eration, The Annals of Probability 15(3) (1987), 1148-1165. · Zbl 0622.60016 |
| [191] | S. Eberhard and P. Varjú, Mixing time of the Chung-Diaconis-Graham random process, Probability Theory and Related Fields 179(1) (2021), 317-344. · Zbl 1478.60201 |
| [192] | A. Ben-Hamou and Y. Peres, Cutoff for permuted Markov chains, Annales de l’IHP (B), to appear, arXiv preprint arXiv:2104.03568, 2021. · Zbl 1508.60073 |
| [193] | E. Lubetzky and A. Sly, Cutoff phenomena for random walks on random regular graphs, Duke Mathematical Journal 153(3) (2010), 475-510. · Zbl 1202.60012 |
| [194] | N. Berestycki, E. Lubetzky, Y. Peres and A. Sly, Random walks on the random graph, The Annals of Probability 46(1) (2018), 456-490. · Zbl 1393.60077 |
| [195] | A. Ben-Hamou and J. Salez, Cutoff for non-backtracking random walks on sparse random graphs, The Annals of Probability 45(3) (2017), 1752-1770. · Zbl 1372.60101 |
| [196] | C. Bordenave, P. Caputo and J. Salez, Random walk on sparse random digraphs, Probability Theory and Related Fields 170(3-4) (2018), 933-960. · Zbl 1383.05294 |
| [197] | C. Bordenave, P. Caputo and J. Salez, Cutoff at the “entropic time” for sparse Markov chains, Probability Theory and Related Fields 173(1) (2019), 261-292. · Zbl 1480.60202 |
| [198] | J. Hermon and S. Olesker-Taylor, Cutoff for Almost All Random Walks on Abelian Groups, arXiv preprint arXiv:2102.02809, 2021. |
| [199] | J. Hermon, A. Sly and P. Sousi, Universality of cutoff for graphs with an added random matching, The Annals of Probability 50(1) (2022), 203-240. · Zbl 1486.05278 |
| [200] | C. Bordenave and H. Lacoin, Cutoff at the entropic time for random walks on covered expander graphs, Journal of the Institute of Mathematics of Jussieu (2021), 1-46. |
| [201] | J. He, H.T. Pham and M.W. Xu, Mixing time of fractional random walk on finite fields, arXiv preprint arXiv:2102.02781 (2021). |
| [202] | H. van Beijeren, O. E. Lanford, III, J. L. Lebowitz, and H. Spohn, Equilibrium time corre-lation functions in the low-density limit, J. Statist. Phys. 22:2 (1980), 237-257. · Zbl 0508.60089 |
| [203] | T. Bodineau, I. Gallagher, and L. Saint-Raymond, The Brownian motion as the limit of a deterministic system of hard-spheres, Invent. Math. 203:2 (2016), 493-553. · Zbl 1337.35107 |
| [204] | T. Bodineau, I. Gallagher, and L. Saint-Raymond, From hard sphere dynamics to the Stokes-Fourier equations: An L 2 analysis of the Boltzmann-Grad limit, Annals of PDE 3(1):2 (2017). · Zbl 1403.35194 |
| [205] | T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella, Statistical dynamics of a hard sphere gas: fluctuating Boltzmann equation and large deviations, arXiv:2008.10403. |
| [206] | T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella, Long-time correlations for a hard-sphere gas at equilibrium, Comm. Pure Appl. Math., to appear. · Zbl 1542.76056 |
| [207] | T. Bodineau, I. Gallagher, L. Saint-Raymond, and S. Simonella, Long-time derivation at equilibrium of the fluctuating Boltzmann equation, arXiv:2201.04514. |
| [208] | D. Burago, S. Ferleger, and A. Kononenko, Uniform estimates on the number of collisions in semi-dispersing billiards, Ann. of Math. 147:3 (1998), 695-708. · Zbl 0995.37025 |
| [209] | R. Denlinger, The propagation of chaos for a rarefied gas of hard spheres in the whole space, Arch. Ration. Mech. Anal. 229:2 (2018) 885-952. · Zbl 1397.35164 |
| [210] | H. Grad, Principles of the kinetic theory of gases, Springer-Verlag, Berlin-Göttingen-Heidelberg (1958), 205-294. |
| [211] | R. Illner, and M. Pulvirenti, Global validity of the Boltzmann equation for a two-and three-dimensional rare gas in vacuum: erratum and improved result, Commun. Math. Phys. 121 (1989), 143-146. · Zbl 0850.76600 |
| [212] | O. E. Lanford III, Time evolution of large classical systems, Lecture Notes in Phys. 38 (1975), 1-111. · Zbl 0329.70011 |
| [213] | F. Rezakhanlou. Equilibrium fluctuations for the discrete Boltzmann equation, Duke Math. J. 93(2) (1998), 257-288. · Zbl 0976.82039 |
| [214] | H. Spohn, Fluctuations around the Boltzmann equation, J. Statist. Phys. 26:2 (1981), 285-305. |
| [215] | H. Spohn, Fluctuation theory for the Boltzmann equation, in: Nonequilibrium phenomena I 10 of Stud. Statist. Mech. (1983), 225-251. |
| [216] | H. Spohn, Large scale dynamics of interacting particles, Springer Berlin, Heidelberg (1991). References · Zbl 0742.76002 |
| [217] | F. Caravenna, R. Sun and N. Zygouras, Universality in marginally relevant disordered sys-tems. Ann. Appl. Probab. 27(5):30503112, (2017) · Zbl 1387.82032 |
| [218] | F. Caravenna, R. Sun and N. Zygouras, The two-dimensional KPZ equa-tion in the entire subcritical regime. Annals Probab. 48:10861127, (2020) |
| [219] | C. Cosco, O. Zeitouni, Moments of partition functions of 2D Gaussian polymers in the weak disorder regime -I. arXiv:2112.03767 (2021) |
| [220] | P. Lammers, S. Ott, Delocalisation and absolute-value-FKG in the solid-on-solid model, arXiv:2101.05139. |
| [221] | S. Sheffield, Random Surfaces, Asterisque (2005). · Zbl 1104.60002 |
| [222] | P. Lammers, Height function delocalisation on cubic planar graphs, Probability Theory and Related Fields (2021). |
| [223] | M. Aizenman, M. Harel, R. Peled, J. Shapiro, Depinning in integer-restricted Gaussian Fields and BKT phases of two-component spin models, arXiv:2110.09498. |
| [224] | D. van Engelenburg, M. Lis, An elementary proof of phase transition in the planar XY model, arXiv:2110.09465. · Zbl 07672993 |
| [225] | Márton Balázs, Ofer Busani, and Timo Seppäläinen. Local stationarity of exponential last passage percolation. Probability Theory and Related Fields, 180:113-162, 2021. · Zbl 1483.60144 |
| [226] | Márton Balázs, Eric Cator, and Timo Seppäläinen. Cube root fluctuations for the corner growth model associated to the exclusion process. Electr. J. Prob., 11:no. 42, 1094-1132 (electronic), 2006. · Zbl 1139.60046 |
| [227] | Wai-Tong Louis Fan and Timo Seppäläinen. Joint distribution of busemann functions in the exactly solvable corner growth model. Probability and Mathematical Physics, 1(1):55 -100, 2020. · Zbl 1487.60180 |
| [228] | P.A. Ferrari and J. Martin. Stationary distributions of multi-type totally asymmetric exclu-sion processes. Ann. Probab., 35(3):807-832, 2007. · Zbl 1117.60089 |
| [229] | Nicos Georgiou, Firas Rassoul-Agha, and Timo Seppäläinen. Stationary cocycles and Buse-mann functions for the corner growth model. Probab. Theory Related Fields, 169(1-2):177-222, 2017. Reporter: Ivailo Hartarsky · Zbl 1407.60122 |
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.
