Finite-size scaling analysis of localization transition for scalar waves in a 3D ensemble of resonant point scatterers

TL;DR

This study investigates the localization transition of scalar waves in a 3D resonant scatterer ensemble using finite-size scaling, estimating the critical exponent and confirming its universality class.

Contribution

It demonstrates that the probability density of decay rates is broad at the transition and introduces a method to estimate the critical exponent from small-$g$ scaling.

Findings

Estimated critical exponent $ u \,\simeq 1.55$

Probability density $p(g)$ is broad at transition

Transition belongs to the 3D orthogonal universality class

Abstract

We use the random Green's matrix model to study the scaling properties of the localization transition for scalar waves in a three-dimensional (3D) ensemble of resonant point scatterers. We show that the probability density $p(g)$ of normalized decay rates of quasi-modes $g$ is very broad at the transition and in the localized regime and that it does not obey a single-parameter scaling law for finite system sizes that we can access. The single-parameter scaling law holds, however, for the small-$g$ part of $p(g)$ which we exploit to estimate the critical exponent $\nu$ of the localization transition. Finite-size scaling analysis of small-$q$ percentiles $g_q$ of $p(g)$ yields an estimate $\nu \simeq 1.55 \pm 0.07$. This value is consistent with previous results for Anderson transition in the 3D orthogonal universality class and suggests that the localization transition under study belongs to the same class.