Hitting times for special patterns in the symmetric exclusion process on Z^d

We consider the symmetric exclusion process {\eta_t,t>0} on {0,1}^{Z^d}. We fix a pattern A:={\eta:\sum_{\Lambda}\eta(i)\ge k}, where \Lambda is a finite subset of Z^d and k is an integer, and we consider the problem of establishing sharp estimates for \tau, the hitting time of A. We present a novel argument based on monotonicity which helps in some cases to obtain sharp tail asymptotics for \tau in a simple way. Also, we characterize the trajectories {\eta_s,s\le t} conditioned on {\tau>t}.