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Sorry if the wording of this question is sloppy, I have a weak background in probability theory (hence the quotation marks throughout).

Is there some "ergodicity-type" result for Wigner's semicircle law?

By that I mean whether it is true that the "average" eigenvalue distribution for real $n\times n$ symmetric matrices with entries distributed "such that Wigner's semicircle law holds" (e.g. as in the link) is also "close to a" semicircle? The size of the matrices should be considered as sufficiently large BUT finite.

Something like that is mentioned at the end of the Mathworld article linked, but I would be grateful if I could be pointed to a more precise statement or to a reference.

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    $\begingroup$ If I understand your question correctly, I believe that the answer is positive and you can find more details in Tao's book Topics in Random Matrix Theory, , especially Theorem 2.4.2. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 27, 2013 at 14:31
  • $\begingroup$ Thanks. I guess with the statement that the ESD's converge in expectation to a semicircle one can formulate quantitative statements that for $n$ "sufficiently large", the expectation of the ESD's are "sufficiently close"to a semicircle? $\endgroup$
    – Camilo Sarmiento
    Commented Nov 27, 2013 at 15:21
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    $\begingroup$ Yes, you are correct. For special types of distributions you can even prove large deviation estimates, which show that the "odds of $ESD$ being far from the semicircle are small". For precise statements, see the book of Anderson-Guionnet-Zeitouni on random matrices, especially section 2.6. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 27, 2013 at 15:31
  • $\begingroup$ Thanks! (now i have to write some stuff to post the comment) $\endgroup$
    – Camilo Sarmiento
    Commented Nov 27, 2013 at 15:34

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You can take a look at http://terrytao.wordpress.com/2010/02/02/254a-notes-4-the-semi-circular-law/. I remember to have read this some time ago, it could help you.

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