Theory for the conditioned spectral density of noninvariant random matrices

Abstract

We develop a theoretical approach to compute the conditioned spectral density of N×N noninvariant random matrices in the limit N→∞. This large deviation observable, defined as the eigenvalue distribution conditioned to have a fixed fraction k of eigenvalues smaller than x∈R, provides the spectrum of random matrix samples that deviate atypically from the average behavior. We apply our theory to sparse random matrices and unveil strikingly different and generic properties, namely, (i) their conditioned spectral density has compact support, (ii) it does not experience any abrupt transition for k around its typical value, and (iii) its eigenvalues do not accumulate at x. Moreover, our work points towards other types of transitions in the conditioned spectral density for values of k away from its typical value. These properties follow from the weak or absent eigenvalue repulsion in sparse ensembles and they are in sharp contrast to those displayed by classic or rotationally invariant random matrices. The exactness of our theoretical findings are confirmed through numerical diagonalization of finite random matrices.

Imaginary replica analysis of loopy regular random graphs

“…The above results are similar in form to the ones derived for weighted graphs in [25], the main difference is that in [25] a second set of replicas with the traditional limit n → 0 is introduced to get the spectrum. It is interesting to see that with the functional formalism (5) both the observable dµ ̺(µ)̺(µ) and the spectrum ̺(µ) itself are calculated at the same time.…”

“…In its above form, (9) appeared first in [19,28], but similar formulae involving limits of replica dimensions to non-zero values have been introduced in different contexts, in particular when counting the number of eigenvalues in certain intervals for random matrix ensembles, see e.g. [20][21][22][23][24][25]29]. We can combine the integrals in (9) as follows:…”

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].…”

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.…”