The 1-vertex transfer matrix and accurate estimation of channel capacity
IEEE Transactions on Information Theory
Journal of Combinatorial Theory Series B
Delay performance in random-access grid networks
Performance Evaluation
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The even discrete torus is the graphTL,d on vertex set{0,…,L 1}d (with L even) inwhich two vertices are adjacent if they differ on exactly onecoordinate and differ by 1(modL) on that coordinate. Thehard-core measure with activity λ onTL,d is the probabilitydistribution πλ on the independent sets (setsof vertices spanning no edges) ofTL,d in which an independent setI is chosen with probability proportional toλ|I|. This distribution occurs naturallyin problems from statistical physics and the study of communicationnetworks.We study Glauber dynamics, a single-site update Markov chain onthe set of independent sets of TL,dwhose stationary distribution is πλ. Weshow that for λ = ω(d-1-4 log3-4d) and d sufficiently large theconvergence to stationarity is (essentially) exponentially slow inLd-1. This improves a result of Borgs,Chayes, Frieze, Kim, Tetali, Vigoda, and Vu (Proceedings of theIEEE FOCS (1999), 218229) [5] who had shown slow mixing of Glauberdynamics for λ growing exponentially with d.Our proof, which extends to ρ-local chains (chains whichalter the state of at most a proportion ρ of the vertices ineach step) for suitable ρ, closely follows the conductanceargument of Borgs et al., [5] adding to it some combinatorialenumeration methods that are modifications of those used by Galvinand Kahn (Combinatorics, Probability and Computing 13 (2004),137164) [12] to show that the hard-core model with parameterλ on the integer lattice ℤd exhibitsphase coexistence for λ =ω(d-1-4 log3-4d).The discrete even torus is a bipartite graph, with partitionclasses µ (consisting of those vertices the sum of whosecoordinates is even) and $$ \cal{O} $$. Our result can be expressedcombinatorially as the statement that for each sufficiently largeλ, there is a ρ(λ) 0 such that if Iis an independent set chosen according toπλ, then the probability that‖I ∩ε||I ∩$ \cal{O} $‖ isat most ρ(λ)Ld is exponentiallysmall in Ld-1. In particular, we obtainthe combinatorial result that for all ε 0 theprobability that a uniformly chosen independent set fromTL,d satisfies ‖I∩ε||I ∩$ \cal{O} $‖≤ (.25 -ε)Ld is exponentially smallin Ld-1. © 2008 Wiley Periodicals,Inc. Random Struct. Alg., 2008Author supported by National Science Foundation (DMS-0111298)while a member of Institute for Advanced Study.