Large-deviation theory for diluted Wishart random matrices

Abstract

Wishart random matrices with a sparse or diluted structure are ubiquitous in the processing of large datasets, with applications in physics, biology, and economy. In this work, we develop a theory for the eigenvalue fluctuations of diluted Wishart random matrices based on the replica approach of disordered systems. We derive an analytical expression for the cumulant generating function of the number of eigenvalues I N (x) smaller than x ∈ R + , from which all cumulants of I N (x) and the rate function x (k) controlling its large-deviation probability Prob[I N (x) = kN] e −N x (k) follow. Explicit results for the mean value and the variance of I N (x), its rate function, and its third cumulant are discussed and thoroughly compared to numerical diagonalization, showing very good agreement. The present work establishes the theoretical framework put forward in a recent letter [Phys. Rev. Lett. 117, 104101 (2016)] as an exact and compelling approach to deal with eigenvalue fluctuations of sparse random matrices.

Condensation of degrees emerging through a first-order phase transition in classical random graphs

“…In a similar context, it would be also interesting to study whether large deviations in the degree sequence can trigger the decay of the metastable states found in coupled ER random graphs [33]. Finally, in light of recent advances in the study of large deviations on diluted random matrices [34][35][36], we should consider the spectral properties corresponding to the constrained graph ensemble studied here. Some of these questions are currently under consideration.…”

Spectral theory of sparse non-Hermitian random matrices

“…The behaviour of various other spectral observables of non-Hermitian sparse random matrices have to our knowledge barely been studied, including the twopoint eigenvalue correlation function, the distribution of real eigenvalues, the localisation of eigenvectors [92,105], the study of the statistics of eigenvectors, the identification of contributions from the giant component and from finite clusters, the study of finite size corrections to the spectrum, and the computation of large deviation functions of spectral properties. In this regard, it would be very interesting to develop other methods for sparse non-Hermitian matrices such as the supersymmetric [106,107,108,109] or the replica method [110,111], which have been very successful to study properties of symmetric random matrices [112,113,114,115,116,117].…”

“…the fraction of nodes lim n→∞ |V wc |/n that contribute to the giant weakly connected component [98,104,100]. Equations (111), (115) and (116) imply that…”

Phase transitions in atypical systems induced by a condensation transition on graphs

“…Finally, we point out that condensation of degrees is driven by weak correlations between the degrees of ER random graphs. Since the eigenvalues of a sparse random matrix are weakly correlated random variables [47][48][49], it would be interesting to study whether these eigenvalues undergo a similar condensation transition.…”

“…Building on previous works on the large deviation theory of observables defined on graphs [47][48][49], here we investigate how condensation of degrees influences two different problems: the thermodynamic phase transitions of the Ising model on an ER random graph and the eigenvalue distribution of the adjacency matrix of the graph. In the first case, large deviations in the graph structure, leading to condensation of degrees, induce different thermodynamic phase transitions, which are otherwise absent if one is limited to small, typical graph fluctuations.…”