TY - GEN
T1 - Mixing time power laws at criticality
AU - Long, Yun
AU - Nachmias, Asaf
AU - Peres, Yuval
PY - 2007
Y1 - 2007
N2 - We study the mixing time of some Markov Chains converging to critical physical models. These models are indexed by a parameter β and there exists some critical value βc where the model undergoes a phase transition. According to Physics lore, the mixing time of such Markov Chains is often of logarithmic order outside the critical regime, when β ≠ βc, and satisfies some power law at criticality, when β= βc. We prove this in the two following settings: 1. Lazy random walk on the critical percolation cluster of "mean-field" graphs, which include the complete graph and random d-regular graphs. The critical mixing time here is of order Θ(n). This answers a question of Benjamini, Kozma and Wormald[4]. 2. Swendsen-Wang dynamics [33] on the complete graph. The critical mixing time here is of order Θ(n1/4). This improves results of Cooper, Dyer, Frieze and Rue [9]. In both settings, the main tool is understanding the Markov Chain dynamics via properties of critical percolation on the underlying graph.
AB - We study the mixing time of some Markov Chains converging to critical physical models. These models are indexed by a parameter β and there exists some critical value βc where the model undergoes a phase transition. According to Physics lore, the mixing time of such Markov Chains is often of logarithmic order outside the critical regime, when β ≠ βc, and satisfies some power law at criticality, when β= βc. We prove this in the two following settings: 1. Lazy random walk on the critical percolation cluster of "mean-field" graphs, which include the complete graph and random d-regular graphs. The critical mixing time here is of order Θ(n). This answers a question of Benjamini, Kozma and Wormald[4]. 2. Swendsen-Wang dynamics [33] on the complete graph. The critical mixing time here is of order Θ(n1/4). This improves results of Cooper, Dyer, Frieze and Rue [9]. In both settings, the main tool is understanding the Markov Chain dynamics via properties of critical percolation on the underlying graph.
UR - http://www.scopus.com/inward/record.url?scp=46749123914&partnerID=8YFLogxK
U2 - 10.1109/FOCS.2007.4389493
DO - 10.1109/FOCS.2007.4389493
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AN - SCOPUS:46749123914
SN - 0769530109
SN - 9780769530109
T3 - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
SP - 205
EP - 214
BT - Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2007
T2 - 48th Annual Symposium on Foundations of Computer Science, FOCS 2007
Y2 - 20 October 2007 through 23 October 2007
ER -