Limit shapes for random square Young tableaux

Our main result is a limit shape theorem for the two-dimensional surface defined by a uniform random n×n square Young tableau. The analysis leads to a calculus of variations minimization problem that resembles the minimization problems studied by Logan–Shepp, Vershik–Kerov, and Cohn–Larsen–Propp. 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 Erdös–Szekeres permutations, namely permutations of the numbers 1,2,…,n2 whose longest monotone subsequence is of length n.