Distribution of extremes in the fluctuations of two-dimensional equilibrium interfaces

TL;DR

This paper analyzes the statistical distribution of the maximum fluctuations in two-dimensional Gaussian interfaces, revealing a connection to entropic repulsion and showing that the distribution aligns with a Gumbel distribution, with implications for understanding correlated extreme events.

Contribution

It derives the average maximal fluctuation and the distribution form for 2D Gaussian interfaces, highlighting the role of correlations in extreme value statistics.

Findings

Average maximal fluctuation scales as rac{1}{\u221a{ ext{K}}} \, ext{ln} N

Distribution matches Gumbel's first asymptote with a non-integer parameter

Standardized distribution is independent of interface size and tension

Abstract

We investigate the statistics of the maximal fluctuation of two-dimensional Gaussian interfaces. Its relation to the entropic repulsion between rigid walls and a confined interface is used to derive the average maximal fluctuation $<m> \sim \sqrt{2/(\pi K)} \ln N$ and the asymptotic behavior of the whole distribution $P(m) \sim N^2 e^{-{\rm (const)} N^2 e^{-\sqrt{2\pi K} m} - \sqrt{2\pi K} m}$ for $m$ finite with $N^2$ and $K$ the interface size and tension, respectively. The standardized form of $P(m)$ does not depend on $N$ or $K$, but shows a good agreement with Gumbel's first asymptote distribution with a particular non-integer parameter. The effects of the correlations among individual fluctuations on the extreme value statistics are discussed in our findings.