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Large deviations from the circular law. (English) Zbl 0916.60022
Let $$X^N$$ be a random $$N \times N$$ matrix whose entries are i.i.d. centered Gaussian random variables with variance $$N^{-1}$$. Denote by $$\mu^N$$ the empirical measure corresponding to the $$N$$ complex eigenvalues of $$X^N$$. The authors prove a full large deviation principle in the scale $$N^2$$ for the distributions of $$\mu^N$$, on the space of symmetric probability measures on C. (A probability measure $$\mu$$ on C is symmetric if $$\mu(A^*) = \mu(A)$$ where $$A^*$$ is the complex conjugate set of $$A$$.) The convex, good rate function is given explicitly and has a unique minimum at the circular law, i.e. at the uniform distribution on the unit disc. In particular, this yields an alternative derivation of the convergence of $$\mu^N$$ to the circular law. The corresponding result for the self-adjoint case was proved by Ben Arous and Guionnet. The authors use similar techniques and make use of precise determinant computations due to Edelman and to Lehmann and Sommers.

##### MSC:
 60F10 Large deviations
##### Keywords:
circular law; large deviations; random matrices
Full Text:
##### References:
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