Eigenfunction statistics of Laguerre Brownian ensemble

TL;DR

This paper investigates the fluctuation behavior of eigenfunctions in a Laguerre Brownian ensemble, revealing diffusive dynamics and energy-dependent differences from Gaussian ensembles, with implications for understanding spectral statistics.

Contribution

It provides a theoretical analysis of eigenfunction fluctuations in Laguerre Brownian ensembles, highlighting their diffusive behavior and energy dependence, extending prior Gaussian ensemble results.

Findings

Eigenfunction fluctuations follow diffusive dynamics with drift.

Energy dependence of fluctuations differs from Gaussian ensembles.

Mathematical expressions for fluctuations are similar locally across ensembles.

Abstract

We theoretically analyze the eigenfunction fluctuation measures for a Hermitian ensemble which appears as an intermediate state of the perturbation of a stationary ensemble by another stationary ensemble of Laguerre type. Similar to the perturbation by a Gaussian stationary ensemble, the measures undergo a diffusive dynamics in terms of the pertubation parameter along with a drift due to level-repulsion. The energy-dependences of the fluctuations for the Laguerre case is in general different from the Gaussian case but locally they can be expressed in a same mathematical form.