×

Universality of cutoff for exclusion with reservoirs. (English) Zbl 1519.60072

In this paper, the author studies exclusion processes with reservoirs on arbitrary networks. The author characterizes the spectral gap, mixing time, and mixing window of the processes in terms of certain simple spectral statistics of the underlying network. Using these results, the author establishes a nonconservative analogue of Aldous’s spectral gap conjecture and proves that cutoff occurs if and only if a product condition is satisfied. The author also establishes cutoff phenomena in relative entropy, Hilbert norm, separation distance and supremum norm.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C22 Interacting particle systems in time-dependent statistical mechanics

References:

[1] ALDOUS, D. and FILL, J. A. (2002). Reversible Markov Chains and Random Walks on Graphs. Unfinished monograph, recompiled 2014, available at http://www.stat.berkeley.edu/texttildelowaldous/RWG/book.html.
[2] ALON, G. and KOZMA, G. (2020). Comparing with octopi. Ann. Inst. Henri Poincaré Probab. Stat. 56 2672-2685. · Zbl 1471.60063 · doi:10.1214/20-AIHP1054
[3] BASU, R., HERMON, J. and PERES, Y. (2017). Characterization of cutoff for reversible Markov chains. Ann. Probab. 45 1448-1487. · Zbl 1374.60129 · doi:10.1214/16-AOP1090
[4] BERTINI, L., DE SOLE, A., GABRIELLI, D., JONA-LASINIO, G. and LANDIM, C. (2003). Large deviations for the boundary driven symmetric simple exclusion process. Math. Phys. Anal. Geom. 6 231-267. · Zbl 1031.82039 · doi:10.1023/A:1024967818899
[5] BORCEA, J., BRÄNDÉN, P. and LIGGETT, T. M. (2009). Negative dependence and the geometry of polynomials. J. Amer. Math. Soc. 22 521-567. · Zbl 1206.62096 · doi:10.1090/S0894-0347-08-00618-8
[6] BRISTIEL, A. and CAPUTO, P. (2021). Entropy inequalities for random walks and permutations.
[7] CAPUTO, P. (2008). On the spectral gap of the Kac walk and other binary collision processes. ALEA Lat. Am. J. Probab. Math. Stat. 4 205-222. · Zbl 1176.60081
[8] CAPUTO, P., LABBÉ, C. and LACOIN, H. (2020). Mixing time of the adjacent walk on the simplex. Ann. Probab. 48 2449-2493. · Zbl 1456.60185 · doi:10.1214/20-AOP1428
[9] CAPUTO, P., LABBÉ, C. and LACOIN, H. (2022). Spectral gap and cutoff phenomenon for the Gibbs sampler of \[\nabla \varphi\] interfaces with convex potential. Ann. Inst. Henri Poincaré Probab. Stat. 58 794-826. · Zbl 1502.37032 · doi:10.1214/21-aihp1174
[10] Caputo, P., Liggett, T. M. and Richthammer, T. (2010). Proof of Aldous’ spectral gap conjecture. J. Amer. Math. Soc. 23 831-851. · Zbl 1203.60145 · doi:10.1090/S0894-0347-10-00659-4
[11] CHEN, G.-Y. and SALOFF-COSTE, L. (2008). The cutoff phenomenon for ergodic Markov processes. Electron. J. Probab. 13 26-78. · Zbl 1190.60007 · doi:10.1214/EJP.v13-474
[12] Diaconis, P. (1996). The cutoff phenomenon in finite Markov chains. Proc. Natl. Acad. Sci. USA 93 1659-1664. · Zbl 0849.60070 · doi:10.1073/pnas.93.4.1659
[13] DING, J., LUBETZKY, E. and PERES, Y. (2010). Total variation cutoff in birth-and-death chains. Probab. Theory Related Fields 146 61-85. · Zbl 1190.60005 · doi:10.1007/s00440-008-0185-3
[14] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York. · Zbl 0592.60049 · doi:10.1002/9780470316658
[15] GANTERT, N., NESTORIDI, E. and SCHMID, D. (2021). Mixing times for the simple exclusion process with open boundaries.
[16] GONÇALVES, P., JARA, M., MARINHO, R. and MENEZES, O. (2021). Sharp Convergence to Equilibrium for the SSEP with Reservoirs.
[17] HERMON, J. and PYMAR, R. (2020). The exclusion process mixes (almost) faster than independent particles. Ann. Probab. 48 3077-3123. · Zbl 1476.60133 · doi:10.1214/20-AOP1455
[18] Hermon, J. and Salez, J. (2019). A version of Aldous’ spectral-gap conjecture for the zero range process. Ann. Appl. Probab. 29 2217-2229. · Zbl 1466.60208 · doi:10.1214/18-AAP1449
[19] Lacoin, H. (2016). Mixing time and cutoff for the adjacent transposition shuffle and the simple exclusion. Ann. Probab. 44 1426-1487. · Zbl 1408.60061 · doi:10.1214/15-AOP1004
[20] LACOIN, H. (2016). The cutoff profile for the simple exclusion process on the circle. Ann. Probab. 44 3399-3430. · Zbl 1410.37008 · doi:10.1214/15-AOP1053
[21] LACOIN, H. (2017). The simple exclusion process on the circle has a diffusive cutoff window. Ann. Inst. Henri Poincaré Probab. Stat. 53 1402-1437. · Zbl 1379.82023 · doi:10.1214/16-AIHP759
[22] LANDIM, C., MILANÉS, A. and OLLA, S. (2008). Stationary and nonequilibrium fluctuations in boundary driven exclusion processes. Markov Process. Related Fields 14 165-184. · Zbl 1157.60087
[23] Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. Second edition of [ MR2466937], With contributions by Elizabeth L. Wilmer, With a chapter on “Coupling from the past” by James G. Propp and David B. Wilson. · Zbl 1390.60001 · doi:10.1090/mbk/107
[24] Liggett, T. M. (2005). Interacting Particle Systems. Classics in Mathematics. Springer, Berlin. · doi:10.1007/b138374
[25] LUBETZKY, E. and SLY, A. (2013). Cutoff for the Ising model on the lattice. Invent. Math. 191 719-755. · Zbl 1273.82014 · doi:10.1007/s00222-012-0404-5
[26] LUBETZKY, E. and SLY, A. (2014). Cutoff for general spin systems with arbitrary boundary conditions. Comm. Pure Appl. Math. 67 982-1027. · Zbl 1292.82025 · doi:10.1002/cpa.21489
[27] LUBETZKY, E. and SLY, A. (2015). An exposition to information percolation for the Ising model. Ann. Fac. Sci. Toulouse Math. (6) 24 745-761. · Zbl 1333.60207 · doi:10.5802/afst.1462
[28] LUBETZKY, E. and SLY, A. (2016). Information percolation and cutoff for the stochastic Ising model. J. Amer. Math. Soc. 29 729-774. · Zbl 1342.60173 · doi:10.1090/jams/841
[29] Montenegro, R. and Tetali, P. (2006). Mathematical aspects of mixing times in Markov chains. Found. Trends Theor. Comput. Sci. 1 x+121. · Zbl 1193.68138 · doi:10.1561/0400000003
[30] MORRIS, B. (2006). The mixing time for simple exclusion. Ann. Appl. Probab. 16 615-635. · Zbl 1133.60037 · doi:10.1214/105051605000000728
[31] Oliveira, R. I. (2013). Mixing of the symmetric exclusion processes in terms of the corresponding single-particle random walk. Ann. Probab. 41 871-913. · Zbl 1274.60242 · doi:10.1214/11-AOP714
[32] PERES, Y. American Institute of Mathematics (AIM) research workshop “Sharp Thresholds for Mixing Times” (Palo Alto, December 2004). Summary available at http://www.aimath.org/WWN/mixingtimes.
[33] QUATTROPANI, M. and SAU, F. (2021). Mixing of the Averaging process and its discrete dual on finite-dimensional geometries.
[34] SALEZ, J. (2021). Cutoff for non-negatively curved Markov chains.
[35] SPITZER, F. (1970). Interaction of Markov processes. Adv. Math. 5 246-290 (1970). · Zbl 0312.60060 · doi:10.1016/0001-8708(70)90034-4
[36] Wilson, D. B. (2004). Mixing times of Lozenge tiling and card shuffling Markov chains. Ann. Appl. Probab. 14 274-325. · Zbl 1040.60063 · doi:10.1214/aoap/1075828054
[37] YAU, H.-T. (1991). Relative entropy and hydrodynamics of Ginzburg-Landau models. Lett. Math. Phys. 22 63-80 · Zbl 0725.60120 · doi:10.1007/BF00400379
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.