The even discrete torus is the graph T_{L,d} on vertex set {0,...,L-1}^d (L
even) 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 on
the independent sets (sets of vertices spanning no edges) of T_{L,d} in which a
set I is chosen with probability proportional to x^|I|. This distribution
occurs in problems from statistical physics and communication networks.

We study Glauber dynamics, a single-site update Markov chain on the set of
independent sets of T_{L,d} whose stationary distribution is pi_x. We show that
for x > cd^{-1/4}log^{3/4}d (and d large) the convergence to stationarity is
exponentially slow in L^{d-1}. This improves a result of Borgs et al., who had
shown slow mixing for x > c^d.

Our proof, which extends to r-local chains (chains which alter the state of
at most a proportion r of the vertices in each step) for suitable r, follows
the conductance argument of Borgs et al., adding to it some combinatorial
enumeration methods that are modifications of those used by Galvin and Kahn to
show that the hard-core model with parameter x on the integer lattice Z^d
exhibits phase coexistence for x > cd^{-1/4}log^{3/4}d.

The graph T_{L,d} is bipartite, with partition classes E (the vertices the
sum of whose coordinates is even) and O. Our result can be expressed
combinatorially as the statement that for each sufficiently large x, there is
an r(x)>0 such that if I is an independent set chosen according to pi_x, then
the probability that ||I \cap E|-|I \cap O|| is at most r(x)L^d is
exponentially small in L^{d-1}. In particular, for all eps>0 the probability
that 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}.