LIMIT SHAPES AND THEIR ANALYTIC PARAMETERIZATIONS

Abstract

A “limit shape” is a form of the law of large numbers, and happens when a large random system, typically consisting of many interacting particles, can be described, after an appropriate normalization, by a certain nonrandom object. Limit shapes occur in, for example, random integer partitions or in random interface models such as the dimer model. Typically limit shapes can be described by some variational formula based on a large deviations estimate. We discuss limit shapes for certain 2-dimensional interface models, and explain how their underlying analytic structure is related to a (conjectural in some cases) conformal invariance property for the models. 1 Limit shapes: integer partitions We illustrate the notion of limit shape with a fundamental example. Given a uniform random integer partition of n for n large, a theorem of Vershik and Kerov [1981] asserts that, when both axes are scaled by p n, the graph of (that is, the Young diagram associated to ) converges with probability tending to 1 to a nonrandom curve, given by the equation e cx + e cy = 1, with c = p 2/6, see Figure 1. This is an example (in fact, one of the first examples) of a limit shape theorem: in the limit of large system size, the typical random object will, when appropriately scaled, concentrate on a fixed nonrandom shape. One way to make a more precise formulation of this statement is say that for each n, the random partition of n defines a certain probability measure n (on the space of nonincreasing functions f : [0; 1) ! [0; 1) of integral 1) and as n ! 1 this sequence of measures converges in probability1 to a point mass on the Vershik-Kerov curve. MSC2010: 82B20. 1 The topology of convergence for the sequence of random functions can be taken to be uniform convergence on compact subsets of (0; 1).