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I am looking for a reference for a proof of the following claim:

Assume that there exist constants $a>0, C$, where the independent entries of a Wigner matrix, $\{ X_N(i,j)\}_{1\le i\le j\le N}$ satisfy: $$ \sup E(e^{a\sqrt{N}|X_N(i,j)|})\le C$$ Prove that there exist a constant $c=c(C)$ such that $\lim \sup_{N\to \infty} \lambda^N_N\le c$ almost surely, and $\lim \sup E(\lambda_N^N)\le c$.

where $\lambda_N$ is the maximum eigenvalue of the Wigner matrix which is a symmetric matrix.

Edit: changed the $\lambda$ appearing in the exponent to $a$.

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    $\begingroup$ This is exercise 2.1.29 in Anderson, Guionnet, and Zeitouni's book on random matrices (though I stress this doesn't answer the question of where a proof can be found -- it's an exercise there, without a proof). Unfortunately I am only familiar with some parts of that book, but a proof of this statement where $\sqrt{N} X_N(i,j)$ are iid is given in Theorem 2.3.24 of Tao's RMT book, and I guess there should not be too many modifications to treat the not-necessarily-identically-distributed case here. Perhaps someone else will know a reference that does this exactly though, no extra work required. $\endgroup$
    – Brad Rodgers
    Commented Dec 16, 2017 at 8:30

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Have you looked into the Katz and Sarnak book? I have not checked wether or not this claim is indeed inside, however it is likely to be and it is my central reference for the subject:

Katz, N. M. and Sarnak, P., Random Matrices, Frobenius Eigenvalues, and Monodromy, American Mathematical Society, Colloquium Publications 1999 (45); 419 pp

It is available online here.

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    $\begingroup$ I tried searching for the keyword "Wigner" in this text, but didn't find something relevant to my question. $\endgroup$
    – Alan
    Commented Dec 16, 2017 at 7:39
  • $\begingroup$ @Alan It aims at a general account of random matrix theory and tries to encapsulate quite a long story, maybe they just don't call Wigner matrices like that. However, this was the only clue coming to my mind. $\endgroup$
    – Desiderius Severus
    Commented Dec 16, 2017 at 7:45
  • $\begingroup$ I edited my OP, and changed a typo. $\endgroup$
    – Alan
    Commented Dec 16, 2017 at 7:53

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