Analytic Form of a Two-Dimensional Critical Distribution

TL;DR

This paper derives an exact analytic form for the distribution of fluctuations in a two-dimensional critical spin wave and interface model, revealing connections to Gumbel distributions and temperature dependence.

Contribution

It introduces an exact Gamma function quotient expression for the distribution and a convergent Charlier series using Gumbel convolutions, advancing understanding of 2D critical phenomena.

Findings

Exact Gamma function quotient form for the distribution

Charlier series with Gumbel convolution converges to the exact distribution

Distribution's temperature dependence provides insight into Gumbel-like behavior

Abstract

This paper explores the possibility of establishing an analytic form of the distribution of the order parameter fluctuations in a two-dimensional critical spin wave model, or width fluctuations of a two dimensional Edwards-Wilkinson interface. It is shown that the characteristic function of the distribution can be expressed exactly as a Gamma function quotient, while a Charlier series, using the convolution of two Gumbel distributions as the kernel, converges to the exact result over a restricted domain. These results can also be extended to calculate the temperature dependence of the distribution and give an insight into the origin of Gumbel-like distributions in steady-state and equilibrium quantities that are not extreme values.