Limit Shapes and Fluctuations of Bounded Random Partitions
Random partitions of integers, bounded both in the number of summands and the size of each summand are considered, subject to the probability measure which assigns a probability proportional to some fixed positive number to the power of the number being partitioned. This corresponds to considering Young diagrams confined to a rectangle. When the rectangle grows, and diagrams are rescaled, the probability measure degenerates to a delta measure on a continuous curve, the limit shape. In the intermediate scaling, the fluctuations around the limit shape turn out to be governed by an Ornstein-Uhlenbeck process. Similar behaviour occurs in the related models bounded only on one side or not at all, which were studied by Vershik and others.
Thesis advisor: Nicolai Reshetikhin and Henning Haahr Andersen