Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models

TL;DR

This paper demonstrates that the distribution of the maximum relative height in a broad class of one-dimensional solid-on-solid models follows a universal Airy distribution in the large size limit, supported by analytical and numerical evidence.

Contribution

It establishes the universality of the maximal relative height distribution and derives it analytically for a specific model, supported by numerical simulations.

Findings

The distribution converges to the Airy distribution in the large size limit.

The subleading scaling function is also universal, given by the derivative of the Airy distribution.

Exact results are obtained for the anisotropic Ising model using transfer matrix techniques.

Abstract

We study the probability density function $P(h_m,L)$ of the maximum relative height $h_m$ in a wide class of one-dimensional solid-on-solid models of finite size $L$. For all these lattice models, in the large $L$ limit, a central limit argument shows that, for periodic boundary conditions, $P(h_m,L)$ takes a universal scaling form $P(h_m,L) \sim (\sqrt{12}w_L)^{-1}f(h_m/(\sqrt{12} w_L))$, with $w_L$ the width of the fluctuating interface and $f(x)$ the Airy distribution function. For one instance of these models, corresponding to the extremely anisotropic Ising model in two dimensions, this result is obtained by an exact computation using transfer matrix technique, valid for any $L>0$. These arguments and exact analytical calculations are supported by numerical simulations, which show in addition that the subleading scaling function is also universal, up to a non universal amplitude, and simply given by the derivative of the Airy distribution function $f'(x)$.