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 Zd at inverse temperature β. When the bond dilution density p is below the percolation threshold, we prove that, for any ε>0, any particle density and any β, 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+ε.