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Do you know a reference book on the law of large numbers for random Plancherel Young diagrams ? I know the book of Kerov, but actually, it is only a compilation of his articles, and i need something more detailed.

Kerov-Vershik and Logan-Shepp independently proved in 1977 a law of large numbers for random Young diagrams ditributed according to the Plancherel measure. I am interested in the asymptotics and working on the article of Vershik-Kerov : "Asymptotic of the largest and the typical dimensions of irreducible representations of a symmetric group" 1985.

In particular, I need for my master's thesis to control the $\| . \|_{\infty}$ norm of the difference between the limit shape and the normalized shape of a Young diagram with n boxes. There is an interesting result proved in the paper that I mentioned (Theorem 3), but I'm not convinced by the proof, because the authors assume that the shape has a support in $[-a, a]$ where $a$ is a constant. But in my opinion, the length of the support can be of order $O(\sqrt{n})$.

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2 Answers 2

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The length of the first row of a Plancherel-random Young diagram (with $n$ boxes) has the same distribution as the longest increasing subsequence of a random permutation $\pi$ in the symmetric group $S_n$ (hint: use Robinson-Schensted correspondence).

By Markov's inequality, the probability that $\pi$ contains a decreasing sequence of length at least $p>3\sqrt{n}$ is at most the expected number of such subsequences which is $$\binom{n}{p} \frac{1}{p!} < \left( \frac{e^2 n}{p^2} \right)^p \leq \left( \frac{e^2}{3^2} \right)^{3 \sqrt{n}} \leq e^{-d \sqrt{n}}, $$for some constant $d>0$, where we used Stirling's approximation $p!> p^p e^{-p}$. This shows that with high probability, your normalized Young diagram has support contained in the interval $[-3,3]$.

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A book on this subject: "The Surprising Mathematics of Longest Increasing Subsequences", A book in progress by Dan Romik. You are interested in Section 1.17 (version 1.1 of the draft).

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