# Universality Classes and Fluctuations in Disordered Systems

J. B. Pendry, A. MacKinnon, P. J. Roberts

## Abstract

Waves transmitted through disordered media show increasing fluctuations with thickness of material so that averages of different properties of the wavefield have very different scaling with thickness traversed. We have been able to classify these properties according to a scheme that is independent of the nature of the medium, such that members of a class have a universal scaling independent of the nature of the medium. We apply this result to trace (T$_{L}$ T$_{L}^{\dagger}$)$^{M}$, where T$_{L}$ is the amplitude transmission matrix. The eigenfunctions of T$_{L}$ T$_{L}^{\dagger}$ define a set of channels through which the current flows, and the eigenvalues are the corresponding transmission coefficients. We prove that these coefficients are either $\approx$ 0 or $\approx$ 1. As L increases more channels are shut down. This is the maximal fluctuation theorem: fluctuations cannot be greater than this. We expect that our classification scheme will prove of further value in proving theorems about limiting distributions. We show by numerical simulations that our theorem holds good for a wide variety of systems, in one, two and three dimensions.