L.I.F.E.: and Halloween Law

These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.Comment: Added: one reference and few comment

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“…See figure 2. 8 Girko's writing style is also quite original, see for instance the recent paper [61]. 9 .…”

Section: Quarter Circular and Circular Lawsmentioning

confidence: 99%

Around the circular law

Bordenave¹,

Chafäı²

2012

Probab. Surveys

191

252

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“…, n we have similar formulas). Despite that, even in this case, the structure of this matrix is quite complicated, we prove the following assertion [7].…”

Section: The ∼ Life ∼-Operatormentioning

confidence: 66%

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Girko¹

2011

Acta Phys. Pol. B

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Devoted to the memory of Marian Ritter von Smolan Smoluchowski which after several years spent at other universities (Paris, Glasgow, and Berlin) moved to Lvov in 1899, where he took a position at the University of Lvov, before he moved to Kraków in 1913, to take over the chair in Experimental Physics Department. In 1906, independently of Albert Einstein, he described Brownian motion. Smoluchowski presented an equation which became an important basis of the theory of stochastic processes.

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L.I.F.E.: and Halloween Law

Cited by 2 publications

References 5 publications

Around the circular law

Around the circular law

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