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

TL;DR

This paper derives precise asymptotic probabilities for the (2+1)-dimensional SOS interface remaining positive over large regions at low temperature, revealing a unique exponential decay involving surface tension and logarithmic factors.

Contribution

It provides the first sharp asymptotics for the probability of positive SOS interfaces in large regions at low temperature, highlighting a distinct behavior from continuous models.

Findings

Probability decays as exp(-tau_beta(0) L log L)

Behavior differs from continuous height models

Results apply under both infinite volume and zero boundary conditions

Abstract

We obtain 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 $\Lambda$, both under the infinite volume measure and under the measure with zero boundary conditions around $\Lambda$, this probability turns out to behave like $\exp(-\tau_\beta(0) L \log L )$, with $\tau_\beta(0)$ 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.