Probability to be positive for the membrane model in dimensions 2 and 3

We consider the membrane model on a box V-N subset of Z(n) of size (2N + 1)(n) with zero boundary condition in the subcritical dimensions n = 2 and n = 3. We show optimal estimates for the probability that the field is positive in a subset D-N of V-N. In particular we obtain for D-N = V-N that the probability to be positive on the entire domain is exponentially small and the rate is of the order of the surface area N-n-(1).