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TITRE / TITLE
Optimal rigidity and maximum of the characteristic polynomial of Wigner matrices.

R脡SUM脡 / ABSTRACT

We consider two related questions about the extremal statistics of Wigner matrices (random symmetric matrices with independent entries). First, how much can their eigenvalues fluctuate? It is known that the eigenvalues of such matrices behave as repelling particles, trapping them near deterministic locations. We provide optimal estimates for this 鈥渞igidity鈥� phenomenon. Second, what is the behavior of the maximum of the characteristic polynomial? This is motivated by a conjecture of Fyodorov鈥揌iary鈥揔eating on the maxima of logarithmically correlated fields, and we will present the first results on this question for Wigner matrices.

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