Logarithmic Sobolev constant 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 epsilon > 0, any particle density and any beta, with probability one, the logarithmic Sobolev constant of the generator of the dynamics in a box of side L centered at the origin cannot grow faster that L2+epsilon.