Ordered spectral statistics in 1D disordered supersymmetric quantum mechanics and Sinai diffusion with dilute absorbers

TL;DR

This paper reviews eigenvalue statistics in 1D disordered supersymmetric quantum mechanics and applies these results to analyze Sinai diffusion with dilute absorbers, revealing a transition from logarithmic to power-law decay in return probability.

Contribution

It introduces a novel connection between eigenvalue statistics in supersymmetric quantum mechanics and classical diffusion with absorbers, providing new insights into decay behaviors.

Findings

Eigenvalue correlations lead to nontrivial ordered statistics.

The return probability transitions from logarithmic to power-law decay.

Power-law decay exponent depends on absorber density and force strength.

Abstract

Some results on the ordered statistics of eigenvalues for one-dimensional random Schr\"odinger Hamiltonians are reviewed. In the case of supersymmetric quantum mechanics with disorder, the existence of low energy delocalized states induces eigenvalue correlations and makes the ordered statistics problem nontrivial. The resulting distributions are used to analyze the problem of classical diffusion in a random force field (Sinai problem) in the presence of weakly concentrated absorbers. It is shown that the slowly decaying averaged return probability of the Sinai problem, $\mean{P(x,t|x,0)}\sim \ln^{-2}t$, is converted into a power law decay, $\mean{P(x,t|x,0)}\sim t^{-\sqrt{2\rho/g}}$, where $g$ is the strength of the random force field and $\rho$ the density of absorbers.