Return probability and scaling exponents in the critical random matrix ensemble

TL;DR

This paper investigates the asymptotic behavior of return probability in critical random matrix ensembles under strong multifractality, confirming scaling laws and deriving analytical expressions for key fractal and dynamical exponents.

Contribution

It provides analytical expressions for the fractal dimension and dynamical scaling exponent, validating the Chalker's ansatz in the context of critical random matrices.

Findings

Confirmed critical scaling law for return probability

Derived analytical expressions for $d_2$ and $rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac{rac in the context of multifractality

Validated Chalker's ansatz for dynamical scaling in the studied regime

Abstract

We study an asymptotic behavior of the return probability for the critical random matrix ensemble in the regime of strong multifractality. The return probability is expected to show critical scaling in the limit of large time or large system size. Using the supersymmetric virial expansion we confirm the scaling law and find analytical expressions for the fractal dimension of the wave functions $d_2$ and the dynamical scaling exponent $\mu$. By comparing them we verify the validity of the Chalker's ansatz for dynamical scaling.