2018/10/17 by Giulio Biroli, Biroli, Giulio, Marco Tarzia +1 · 43 citations
Mathematics · Physics and Astronomy · #Combinatorics #Continuous-time random walk #Delocalized electron #Disordered Systems and Neural Networks (cond-mat.dis-nn) #Eigenfunction #Eigenvalues and eigenvectors #Ergodic theory #Ergodicity #FOS: Physical sciences #Mathematical analysis #Mathematics #Observable #Physics #Quantum mechanics #Random Matrices and Applications #Random walk #Statistical Mechanics (cond-mat.stat-mech) #Statistical physics #Statistics #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #Thermodynamic limit #cond-mat.dis-nn #cond-mat.stat-mech
paper · pdf · doi:10.48550/arxiv.1810.07545
published in arXiv (Cornell University) (Cornell University) · This article will be submitted to the special issue of J. Phys. A 'Random Matrices: the first 90 years'. A new section will be added before submission
arxiv created 2018/10/17 · openalex publication_date 2018/10/17 · arxiv updated 2018/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We review the state of the art on the delocalized non-ergodic regime of the Anderson model on Bethe lattices. We also present new results using Belief Propagation, which consists in solving the self-consistent recursion relations for the Green's functions directly on a given sample. This allows us to numerically study very large system sizes and to directly access observables related to the eigenfunctions and energy level statistics. In agreement with recent works, we establish the existence of a delocalized non-ergodic phase on Cayley trees. On random regular graphs instead our results indicate that ergodicity is recovered when the system size is larger than a cross-over scale Nc (W), which diverges exponentially fast approaching the localization transition. This scale corresponds to the size at which the mean-level spacing becomes smaller than the Thouless energy ETh (W). Such energy scale, which vanishes exponentially fast approaching the localization transition, is the one below which ergodicity in the level statistics is restored in the thermodynamic limit. Remarkably, the behavior of random regular graphs below Nc (W) coincides with the one found close to the root of loop-less infinite Cayley trees, \it i.e. only above Nc (W) the effects of loops emerge and random regular graphs behave differently from Cayley trees. Our results indicate that ergodicity is recovered in the thermodynamic limit on random regular graph. However, all observables probing volumes smaller than Nc(W) and times smaller than ℏ/ETh (W) are expected to behave as if there were an intermediate phase. Given the very fast divergence of Nc(W) and ℏ/ETh (W) these non-ergodic effects are very pronounced in a large region preceding the localization transition, and they can be related to the intermediate phase present on Cayley trees.