Cutoff for the East process. (English) Zbl 1408.82007
Summary: The East process is a 1D kinetically constrained interacting particle system, introduced in the physics literature in the early 1990s to model liquid-glass transitions. Spectral gap estimates of D. Aldous and P. Diaconis [J. Stat. Phys. 107, No. 5–6, 945–975 (2002; Zbl 1006.60095)] imply that its mixing time on \(L\) sites has order \(L\). We complement that result and show cutoff with an \(O(\sqrt{L})\)-window.
The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an \(O(\sqrt{L})\)-window. The law of the process behind the front plays a crucial role: in [Stochastic Processes Appl. 123, No. 9, 3430–3465 (2013; Zbl 1291.60199)], O. Blondel showed that it converges to an invariant measure \(\nu\), on which very little is known. Here, we obtain quantitative bounds on the speed of convergence to \(\nu\), finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of E. Bolthausen [Ann. Probab. 10, 1047–1050 (1982; Zbl 0496.60020)] implies a CLT for the location of the front, yielding the cutoff result.
Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an \(O(1)\)-window.
The main ingredient is an analysis of the front of the process (its rightmost zero in the setup where zeros facilitate updates to their right). One expects the front to advance as a biased random walk, whose normal fluctuations would imply cutoff with an \(O(\sqrt{L})\)-window. The law of the process behind the front plays a crucial role: in [Stochastic Processes Appl. 123, No. 9, 3430–3465 (2013; Zbl 1291.60199)], O. Blondel showed that it converges to an invariant measure \(\nu\), on which very little is known. Here, we obtain quantitative bounds on the speed of convergence to \(\nu\), finding that it is exponentially fast. We then derive that the increments of the front behave as a stationary mixing sequence of random variables, and a Stein-method based argument of E. Bolthausen [Ann. Probab. 10, 1047–1050 (1982; Zbl 0496.60020)] implies a CLT for the location of the front, yielding the cutoff result.
Finally, we supplement these results by a study of analogous kinetically constrained models on trees, again establishing cutoff, yet this time with an \(O(1)\)-window.
MSC:
| 82C41 | Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics |
| 82C26 | Dynamic and nonequilibrium phase transitions (general) in statistical mechanics |
References:
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