Circular law for very sparse non-Hermitian random matrices

Konstantin Tikhomirov
Princeton University

Consider an n by n random matrix with i.i.d Bernoulli entries, where each entry is equal one with probability p. In the regime when p is at least logarithmic in dimension n, the circular law for the limiting spectral distribution (of appropriately rescaled matrix) as the dimension grows to infinity, was previously established in works of Tao-Vu, Gotze-Tikhomirov, Basak-Rudelson. In our work, we consider the setting allowing p to be sublogarithmic; in particular, this implies that the random matrix is singular with high probability. With some restrictions on p, we are able to establish the limiting law for the spectrum for this random model, as well as for some of its generalizations. This is joint work with Mark Rudelson.

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