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The circular law conjecture for iid random matrices

Let MnM_n be the n×nn \times n random matrix whose entries are iid complex random variables with mean 0 and variance 1. The empirical spectral distribution (ESD) of 1nMn\frac{1}{\sqrt n}M_n is the probability measure formed by assigning mass 1/n1/n to each eigenvalue. Circular law conjecture. The ESD of 1nMn\frac{1}{\sqrt n}M_n converges, in both the strong and weak senses, to the uniform distribution

μ:=1π1∣z∣≤1 dz\mu:= \frac{1}{\pi}1_{|z|\leq 1}\,dz

on the unit disk {z∈C:∣z∣≤1}\{z\in\mathbf{C}:|z|\leq 1\}. This is the non-Hermitian counterpart of the semi-circular law and had remained a central question since the 1950s; the source presents it as an open conjecture, although later work established the claim in this stated generality.

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  1. The Circular Law conjecture for iid random matrices

    Let AnA_n be the n×nn \times n random matrix whose entries are iid complex random variables with mean 00 and variance 11. The Circular Law conjecture. The empirical spectral distribution of 1nAn\frac{1}{\sqrt n}A_n converges, both in probability and almost surely, to the uniform distribution on the unit disk. This conjecture asserts universality of the limiting empirical spectral distribution for non-Hermitian random matrices. The real Gaussian case was confirmed by Edelman, and the conjecture has since been resolved in the stated iid setting.

    source: Terence Tao, Van Vu and Manjunath Krishnapur, “Random matrices: Universality of ESDs and the circular law”, arXiv:0807.4898 (2009).

Sources & referencesView supporting material

Primary source

Terence Tao and Van Vu, “From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices”, arXiv:0810.2994 (2009).

circular-lawrandom-matricesspectral-distributionuniversality

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