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The even discrete torus is the graph T_{L,d} on vertex set {0,...,L-1}^d (Leven) with two vertices adjacent if they differ by 1 (mod L) on one coordinate.The hard-core measure with activity x on T_{L,d} is the distribution pi_x onthe independent sets (sets of vertices spanning no edges) of T_{L,d} in which aset I is chosen with probability proportional to x^|I|. This distributionoccurs in problems from statistical physics and communication networks.  We study Glauber dynamics, a single-site update Markov chain on the set ofindependent sets of T_{L,d} whose stationary distribution is pi_x. We show thatfor x > cd^{-1/4}log^{3/4}d (and d large) the convergence to stationarity isexponentially slow in L^{d-1}. This improves a result of Borgs et al., who hadshown slow mixing for x > c^d.  Our proof, which extends to r-local chains (chains which alter the state ofat most a proportion r of the vertices in each step) for suitable r, followsthe conductance argument of Borgs et al., adding to it some combinatorialenumeration methods that are modifications of those used by Galvin and Kahn toshow that the hard-core model with parameter x on the integer lattice Z^dexhibits phase coexistence for x > cd^{-1/4}log^{3/4}d.  The graph T_{L,d} is bipartite, with partition classes E (the vertices thesum of whose coordinates is even) and O. Our result can be expressedcombinatorially as the statement that for each sufficiently large x, there isan r(x)>0 such that if I is an independent set chosen according to pi_x, thenthe probability that ||I \cap E|-|I \cap O|| is at most r(x)L^d isexponentially small in L^{d-1}. In particular, for all eps>0 the probabilitythat a uniformly chosen independent set from T_{L,d} satisfies ||I \cap E|-|I\cap O|| \leq (.25 - eps)L^d is exponentially small in L^{d-1}.

Sampling independent sets in the discrete torus