Scaling theory of the Anderson transition in random graphs: ergodicity and universality

TL;DR

This paper investigates the Anderson transition on random graphs with a tunable branching parameter, revealing a universal transition between localized and ergodic delocalized phases with distinct scaling laws.

Contribution

It introduces a comprehensive numerical analysis showing a universal Anderson transition with unique scaling behaviors on random graphs with variable branching.

Findings

A single transition separates localized and ergodic phases.

Critical wavefunctions are multifractal and located on few branches.

Scaling laws differ on either side of the transition, supporting universality.

Abstract

We study the Anderson transition on a generic model of random graphs with a tunable branching parameter $1<K\le 2$, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase which is ergodic at large scales but strongly non-ergodic at smaller scales. In the critical regime, multifractal wavefunctions are located on few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.