Maximum height of low-temperature 3D Ising interfaces

Stochastics Seminar
Thursday, October 10, 2019 - 3:05pm for 1 hour (actually 50 minutes)
Skiles 006
Reza Gheissari – University of California, Berkeley –
Michael Damron

Consider the random surface given by the interface separating the plus and minus phases in a low-temperature Ising model in dimensions $d\geq 3$. Dobrushin (1972) famously showed that in cubes of side-length $n$ the horizontal interface is rigid, exhibiting order one height fluctuations above a fixed point. 

We study the large deviations of this interface and obtain a shape theorem for its pillar, conditionally on it reaching an atypically large height. We use this to analyze the law of the maximum height $M_n$ of the interface: we prove that for every $\beta$ large, $M_n/\log n \to c_\beta$, and $(M_n - \mathbb E[M_n])_n$ forms a tight sequence. Moreover, even though this centered sequence does not converge, all its sub-sequential limits satisfy uniform Gumbel tail bounds. Based on joint work with Eyal Lubetzky.