The circular law conjecture for iid random matrices
Let be the random matrix whose entries are iid complex random variables with mean 0 and variance 1. The empirical spectral distribution (ESD) of is the probability measure formed by assigning mass to each eigenvalue. Circular law conjecture. The ESD of converges, in both the strong and weak senses, to the uniform distribution
on the unit disk . 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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The Circular Law conjecture for iid random matrices
Let be the random matrix whose entries are iid complex random variables with mean and variance . The Circular Law conjecture. The empirical spectral distribution of 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).
References
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).
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