On the probability of staying above a wall for the ( 2 + 1 )-dimensional SOS model at low temperature

We obtains a sharp asymptotics for the probability that the (2+1)-dimensional discrete SOS interface at low temperature is positive in a large region. For a square region , both under the infinite volume measure and under the measure with zero boundary conditions around , this probability turns out to behave like exp(−τβ(0)L log L), with τβ(0) the surface tension at zero tilt, also called step free energy, and L the box side. This behavior is qualitatively different from the one found for continuous height massless gradient interface models (Bolthausen et al., Commun Math Phys 170(2):417–443, 1995; Deuschel et al., Stochastic Process Appl 89(2):333– 354, 2000)