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I am reading the paper "random matrices and random permutations" by Andrei Okounkov, which is very beautifully written. I just have several technical questions about some of the computations: 1. in section 2.1.2, why is it clear that only the eigenvalues near the edges of the Wigner's semicircle contribute to the asymptotics of (2.1)? In the same spirit, on page 8, section 2.1.3, the author claims that it is easy to see that by taking a suitable linear combination we can single out hte contribution of only the maximal eigenvalues in formula (2.1). I don't quite see how a winner takes all argument can be applied.

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In the expression below equation 2.1 the as n goes to infinity the expression tends towards an exponent of $y_i$ times a positive expression. The $y_i$'s from expression 1.3 are negative except in the range of 2n^.5 where they are close to zero. If they are substantially less than this then the entire expression of 1.3 is roughly -n^1/3 which which as n goes to infinity will make the exponential expression relatively small.

As for the contribution of maximimum eignenvalues in 2.1 there is more than one representation of 2.1 and a combination of these representations allows the elimination of the contributions of all but the maximum eigenvalues.

These are the lecture notes from a course on random matrices:

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