Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix

Bai, Zhidong; Yin, Yanqing

doi:10.1214/aop/1176991594

Cited by 163 publications

(282 citation statements)

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“…Conditions (ii) of Theorem 2 for the Wigner Ensembles can also be easily checked, because equation (1.6) for f is quadratic. The result [4] yields (iii).…”

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confidence: 76%

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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“…Conditions (ii) of Theorem 2 for the Wigner Ensembles can also be easily checked, because equation (1.6) for f is quadratic. The result [4] yields (iii).…”

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confidence: 76%

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

Shcherbina

2011

J Stat Phys

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“…Some extensions of the Füredi-Komlós and SinaiSoshnikov techniques can also be found in [Kho01]. Finally, conditions for the almost sure convergence of the maximal eigenvalue of Wigner matrices appear in [BaY88].…”

Section: Bibliographical Notesmentioning

confidence: 99%

An Introduction to Random Matrices

2009

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The theory of random matrices plays an important role in many areas of pure mathematics and employs a variety of sophisticated mathematical tools (analytical, probabilistic and combinatorial). This diverse array of tools, while attesting to the vitality of the field, presents several formidable obstacles to the newcomer, and even the expert probabilist. This rigorous introduction to the basic theory is sufficiently self-contained to be accessible to graduate students in mathematics or related sciences, who have mastered probability theory at the graduate level, but have not necessarily been exposed to advanced notions of functional analysis, algebra or geometry. Useful background material is collected in the appendices and exercises are also included throughout to test the reader's understanding. Enumerative techniques, stochastic analysis, large deviations, concentration inequalities, disintegration and Lie algebras all are introduced in the text, which will enable readers to approach the research literature with confidence.

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“…Proof We follow Wigner's trace method [37] as outlined in [3,33,36]. For any integer k > 0 and any λ > 0, the fact that A 2k 2 ≤ trace(A 2k ) and Markov's inequality combine to show…”

Section: Lemma 15 (Courant-fischer Max-min Formula)mentioning

confidence: 99%

Swarming on Random Graphs

et al. 2013

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We consider a compromise model in one dimension in which pairs of agents interact through first-order dynamics that involve both attraction and repulsion. In the case of all-to-all coupling of agents, this system has a lowest energy state in which half of the agents agree upon one value and the other half agree upon a different value. The purpose of this paper is to study the behavior of this compromise model when the interaction between the N agents occurs according to an Erdős-Rényi random graph G (N, p). We study the effect of changing p on the stability of the compromised state, and derive both rigorous and asymptotic results suggesting that the stability is preserved for probabilities greater than p c = O( log N N ). In other words, relatively few interactions are needed to preserve stability of the state. The results rely on basic probability arguments and the theory of eigenvalues of random matrices.

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Necessary and Sufficient Conditions for Almost Sure Convergence of the Largest Eigenvalue of a Wigner Matrix

Cited by 163 publications

References 3 publications

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

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

An Introduction to Random Matrices

Swarming on Random Graphs

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