Limit shapes for random square Young tableaux

An NxN square Young tableau can be thought of as a set of instructions for building an NxN brick wall by laying bricks sequentially in such a way that the heights of the brick columns, read from left to right, form a monotone (weakly) decreasing sequence at any point during the construction. I will talk about the problem of the "typical" limit shape, or growth profile, of such a wall - namely, if a square Young tableau is chosen at random uniformly from all tableaux of this size, what can one say about the growth profile of the wall at various times during its construction when N is large? The analysis leads to a problem in the calculus of variations which has a similar structure to the problem solved by Vershik-Kerov and Logan-Shepp in 1977 in their solution of the famous Ulam problem on the length of the longest increasing subsequences of a random permutation. At the end of the talk, if time permits I will tell about some recent applications of the limit shape result. The talk is based on joint work with Boris Pittel.