×

Cutoff for the square plaquette model on a critical length scale. (English) Zbl 1476.60132

Summary: Plaquette models are short range ferromagnetic spin models that play a key role in the dynamic facilitation approach to the liquid glass transition. In this paper we study the dynamics of the square plaquette model at the smallest of the three critical length scales discovered in [the first author et al., J. Stat. Phys. 169, No. 3, 441–471 (2017; Zbl 1382.82009)]. Our main result is that the plaquette model with periodic boundary conditions, on this length scale, exhibits a sharp transition in the convergence to equilibrium, known as cutoff. This substantially refines our coarse understanding of mixing from previous work (Chleboun and Smith (2018)). The basic approach is to reduce the problem to an analysis of the trace process on certain “metastable” states, which may be useful in proving cutoff in other situations.

MSC:

60J27 Continuous-time Markov processes on discrete state spaces
60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)

Citations:

Zbl 1382.82009
PDFBibTeX XMLCite
Full Text: DOI arXiv Link

References:

[1] Basu, R., Hermon, J. and Peres, Y. (2015). Characterization of cutoff for reversible Markov chains. In Proceedings of the Twenty-Sixth Annual ACM-SIAM Symposium on Discrete Algorithms 1774-1791. SIAM, Philadelphia, PA. · Zbl 1371.60124 · doi:10.1137/1.9781611973730.119
[2] Berthier, L. and Biroli, G. (2011). Theoretical perspective on the glass transition and amorphous materials. Rev. Modern Phys. 83 587-645. · doi:10.1103/RevModPhys.83.587
[3] Blondel, O., Cancrini, N., Martinelli, F., Roberto, C. and Toninelli, C. (2013). Fredrickson-Andersen one spin facilitated model out of equilibrium. Markov Process. Related Fields 19 383-406. · Zbl 1321.82025
[4] Chleboun, P., Faggionato, A. and Martinelli, F. (2015). Mixing time and local exponential ergodicity of the East-like process in \[{\mathbb{Z}^d} \]. Ann. Fac. Sci. Toulouse Math. (6) 24 717-743. · Zbl 1333.60199 · doi:10.5802/afst.1461
[5] Chleboun, P., Faggionato, A. and Martinelli, F. (2016). Relaxation to equilibrium of generalized East processes on \[{\mathbb{Z}^d} \]: Renormalization group analysis and energy-entropy competition. Ann. Probab. 44 1817-1863. · Zbl 1343.60135 · doi:10.1214/15-AOP1011
[6] Chleboun, P., Faggionato, A., Martinelli, F. and Toninelli, C. (2017). Mixing length scales of low temperature spin plaquettes models. J. Stat. Phys. 169 441-471. · Zbl 1382.82009 · doi:10.1007/s10955-017-1880-1
[7] Chleboun, P. and Smith, A. (2020). Mixing of the square plaquette model on a critical length scale. Electron. J. Probab. 25 89. · Zbl 1472.60126 · doi:10.1214/20-EJP487
[8] 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
[9] Durrett, R. (2019). Probability—Theory and Examples. Cambridge Series in Statistical and Probabilistic Mathematics 49. Cambridge Univ. Press, Cambridge. · Zbl 1440.60001 · doi:10.1017/9781108591034
[10] Faggionato, A., Martinelli, F., Roberto, C. and Toninelli, C. (2013). The East model: Recent results and new progresses. Markov Process. Related Fields 19 407-452. · Zbl 1321.60208
[11] Gabrielov, A., Newman, W. I. and Turcotte, D. L. (1999). Exactly soluble hierarchical clustering model: Inverse cascades, self-similarity, and scaling. Phys. Rev. E (3) 60 5293-5300. · doi:10.1103/PhysRevE.60.5293
[12] Ganguly, S., Lubetzky, E. and Martinelli, F. (2015). Cutoff for the East process. Comm. Math. Phys. 335 1287-1322. · Zbl 1408.82007 · doi:10.1007/s00220-015-2316-x
[13] Garrahan, J. P. (2002). Glassiness through the emergence of effective dynamical constraints in interacting systems. J. Phys., Condens. Matter 14 1571-1579. · doi:10.1088/0953-8984/14/7/314
[14] Garrahan, J. P., Sollich, P. and Toninelli, C. (2011). Kinetically constrained models. In Dynamical Heterogeneities in Glasses, Colloids, and Granular Media Oxford Univ. Press, London. · doi:10.1093/acprof:oso/9780199691470.003.0010
[15] Goel, S., Montenegro, R. and Tetali, P. (2006). Mixing time bounds via the spectral profile. Electron. J. Probab. 11 1-26. · Zbl 1109.60061 · doi:10.1214/EJP.v11-300
[16] Jack, R. L., Berthier, L. and Garrahan, J. P. (2005). Static and dynamic length scales in a simple glassy plaquette model. Phys. Rev. E 72 016103. · doi:10.1103/PhysRevE.72.016103
[17] Kozma, G. (2007). On the precision of the spectral profile. ALEA Lat. Am. J. Probab. Math. Stat. 3 321-329. · Zbl 1162.60335
[18] Landim, C. (2019). Metastable Markov chains. Probab. Surv. 16 143-227. · Zbl 1491.60131 · doi:10.1214/18-PS310
[19] Levin, D. A. and Peres, Y. (2017). Markov Chains and Mixing Times. Amer. Math. Soc., Providence, RI. · Zbl 1390.60001
[20] Lubetzky, E. and Sly, A. (2017). Universality of cutoff for the Ising model. Ann. Probab. 45 3664-3696. · Zbl 1405.60148 · doi:10.1214/16-AOP1146
[21] Martinelli, F., Morris, R. and Toninelli, C. (2019). Universality results for kinetically constrained spin models in two dimensions. Comm. Math. Phys. 369 761-809. · Zbl 1419.82037 · doi:10.1007/s00220-018-3280-z
[22] Martinelli, F. and Toninelli, C. (2019). Towards a universality picture for the relaxation to equilibrium of kinetically constrained models. Ann. Probab. 47 324-361. · Zbl 1466.60210 · doi:10.1214/18-AOP1262
[23] Oliveira, R. I. (2012). Mixing and hitting times for finite Markov chains. Electron. J. Probab. 17 no. 70, 12. · Zbl 1251.60059 · doi:10.1214/EJP.v17-2274
[24] Peres, Y. and Sousi, P. (2015). Mixing times are hitting times of large sets. J. Theoret. Probab. 28 488-519. · Zbl 1323.60094 · doi:10.1007/s10959-013-0497-9
[25] Pillai, N. S. and Smith, A. (2017). Mixing times for a constrained Ising process on the torus at low density. Ann. Probab. 45 1003-1070. · Zbl 1381.60105 · doi:10.1214/15-AOP1080
[26] Pillai, N. S. and Smith, A. (2019). Mixing times for a constrained Ising process on the two-dimensional torus at low density. Ann. Inst. Henri Poincaré Probab. Stat. 55 1649-1678. · Zbl 1448.60156 · doi:10.1214/18-aihp930
[27] Saloff-Coste, L. (1997). Lectures on finite Markov chains. In Lectures on Probability Theory and Statistics (Saint-Flour, 1996). Lecture Notes in Math. 1665 301-413. Springer, Berlin. · Zbl 0885.60061 · doi:10.1007/BFb0092621
[28] Sinclair, A. (1992). Improved bounds for mixing rates of Markov chains and multicommodity flow. Combin. Probab. Comput. 1 351-370 · Zbl 0801.90039 · doi:10.1017/S0963548300000390
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.