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Limit shapes for random square Young tableaux. (English) Zbl 1122.60009
Summary: Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random $$n\times$$n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems studied by B. F. Logan and L. A. Shepp [Adv. Math. 26, 206–222 (1977; Zbl 0363.62068)], A. M. Vershik and S. V. Kerov [Sov. Math., Dokl. 18, 527–531 (1977); translation from Dokl. Akad. Nauk SSSR 233, 1024–1027 (1977; Zbl 0406.05008)], and H. Cohn, M. Larsen and J. Propp [New York J. Math. 4, 137–165, electronic only (1998; Zbl 0908.60083)]. We solve this problem by developing a general technique for solving variational problems of this kind. An extension to rectangular Young tableaux is also given. We also apply the main result to show that the location of a particular entry in the tableau is in the limit governed by a semicircle distribution, and to the study of extremal Erdős-Szekeres permutations, namely, permutations of the numbers $$1,2,\dots ,n^{2}$$ whose longest monotone subsequence is of length $$n$$.

##### MSC:
 60C05 Combinatorial probability 05E10 Combinatorial aspects of representation theory 60F10 Large deviations
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