Random matrices: Universality of local eigenvalue statistics

Tao, Terence; Vu, Van

doi:10.1007/s11511-011-0061-3

Cited by 391 publications

(744 citation statements)

References 46 publications

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“…Using duality between the spectrum of sparse and almost full graphs [14], we see from the Fig.4 that SD in almost complete graph fits perfectly the shifted SD of sparse matrix ensemble, meaning that our identification of clusters with cliques and separated eigenvalues is true. The striking difference between the SD of single clique and the whole network indicates that the triangle-shape SD of the whole networks occurs due to the inter-clique connections.…”

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confidence: 66%

Eigenvalue tunneling and decay of quenched random network

et al. 2016

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We consider the canonical ensemble of N -vertex Erdős-Rényi (ER) random topological graphs with quenched vertex degree, and with fugacity µ for each closed triple of bonds. We claim complete defragmentation of large-N graphs into the collection of [p −1 ] almost full subgraphs (cliques) above critical fugacity, µc, where p is the ER bond formation probability. Evolution of the spectral density, ρ(λ), of the adjacency matrix with increasing µ leads to the formation of two-zonal support for µ > µc. Eigenvalue tunnelling from one (central) zone to the other means formation of a new clique in the defragmentation process. The adjacency matrix of the ground state of a network has the block-diagonal form where number of vertices in blocks fluctuate around the mean value N p. The spectral density of the whole network in this regime has triangular shape. We interpret the phenomena from the viewpoint of the conventional random matrix model and speculate about possible physical applications.Investigation of critical and collective effects in graphs and networks has becoming a new rapidly developing interdisciplinary area, with diverse applications and variety of questions to be asked, see [1] for review. Ensembles of random Erdős-Rényi topological graphs (networks) provide an efficient laboratory for testing collective phenomena in statistical physics of complex systems, being also tightly linked to conventional random matrix theory. Besides investigating typical statistical properties of networks, like vertex degree distribution, clustering coefficients, "small world" structure etc, last two decades have been marked by rapidly growing interest in more refined graph characteristics, such as distribution of small subgraphs involving triads of vertices.Triadic interactions, being the simplest interactions beyond the free-field theory, play crucial role in the network statistics. Presence of such interactions is responsible for emergence of phase transitions in complex distributed systems. First example of a phase transition in random networks, known as Strauss clustering [2], has been treated by the Random Matrix Theory (RMT) in [3]. It was argued that, when the increasing fugacity, µ, the system develops two phases with essentially different triad concentrations. At large µ the system falls into the Strauss phase with the single clique of nodes. The condensation of triads is a non-perturbative phenomenon identified in [4] with the 1st order phase transition in the framework of mean-field cavity-like approach.Similar critical behavior was found in [5] for the vertexdegree-conserved ER graphs. It was demonstrated in the framework of the mean-field approach that the phase transition takes place in this case as well. The hysteresis for dependence of the triad concentration on the fugacity, µ also has been observed in [5]. For bi-color networks with conserved vertex degree a new phenomena of a wide plateau formation in concentration of black-white bonds as a function of the fugacity of unicolor triples of bonds has been found in...

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confidence: 66%

Eigenvalue tunneling and decay of quenched random network

et al. 2016

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“…The WDM conjecture has recently been proved in increasing generality in a series of papers [18,21,24,25] for both the real symmetric and complex hermitian symmetry classes via the Dyson Brownian motion. An alternative approach introducing the four-moment comparison theorem was presented in [49,50,52]. In this paper we only discuss universality in the bulk of the spectrum, but we remark that a similar development took place for the edge universality.…”

Section: Introductionmentioning

confidence: 91%

Stability of the matrix Dyson equation and random matrices with correlations

Ajanki

Erdős

Krüger

2018

Probab. Theory Relat. Fields

145

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We consider real symmetric or complex hermitian random matrices with correlated entries. We prove local laws for the resolvent and universality of the local eigenvalue statistics in the bulk of the spectrum. The correlations have fast decay but are otherwise of general form. The key novelty is the detailed stability analysis of the corresponding matrix valued Dyson equation whose solution is the deterministic limit of the resolvent.

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“…the distribution of eigenvalues near the points λ in which the limiting eigenvalue density ρ(λ) = 0, is studied for many ensembles of random matrices (see e.g. [7], [15,16], [21], [8]). In particular, universality for the DGUE (1.1) was proved in [10,11] for H (0) n being the Wigner matrix (i.e.…”

Section: Introductionmentioning

confidence: 99%

On Universality of Local Edge Regime for the Deformed Gaussian Unitary Ensemble

Shcherbina

2011

J Stat Phys

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We consider the deformed Gaussian ensemble H n = H (0) n +M n in which H (0) n is a hermitian matrix (possibly random) and M n is the Gaussian unitary random matrix (GUE) independent of H (0) n . Assuming that the Normalized Counting Measure ofn converges weakly (in probability if random) to a non-random measure N (0) with a bounded support and assuming some conditions on the convergence rate, we prove universality of the local eigenvalue statistics near the edge of the limiting spectrum of H n .

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Random matrices: Universality of local eigenvalue statistics

Cited by 391 publications

References 46 publications

Eigenvalue tunneling and decay of quenched random network

Eigenvalue tunneling and decay of quenched random network

Stability of the matrix Dyson equation and random matrices with correlations

On Universality of Local Edge Regime for the Deformed Gaussian Unitary Ensemble

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