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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.

60F10 Large deviations
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