Diffusive scaling of the spectral gap for the dilute Ising lattice gas dynamics below the percolation threshold

We consider a conservative stochastic lattice-gas dynamics reversible with respect to the canonical Gibbs measure of the bond dilute Ising model on Z(d) at inverse temperature beta. When the bond dilution density p is below the percolation threshold we prove that for any particle density and any beta, with probability one, the spectral gap of the generator of the dynamics in a box of side L centered at the origin scales like L-2. Such an estimate is then used to prove a decay to equilibrium for local functions of the form 1/t(alpha-epsilon) where epsilon is positive and arbitrarily small and alpha = 1/2 for d = 1, alpha = I for d greater than or equal to 2. In particular our result shows that, contrary to what happens for the Glauber dynamics, there is no dynamical phase transition when beta crosses the critical value beta (c) of the pure system.