Random Growth and Random Matrices

We give a survey of some of the recent results on certain two-dimensional random growth models and their relation to random matrix theory, in particular to the Tracy-Widom distribution for the largest eigenvalue. The problems are related to that of finding the length of the longest increasing subsequence in a random permutation. We also give a new approach to certain results for the Schur measure introduced by Okounkov. 1. Random Growth Models in the Plane 1.1. Eden-Richardson growth During the last twenty years there has been alot of interest in models where an object grow by some rule involving randomness. Basically, there are two types of models, non-local models like difusion-limited aggregation (DLA) and local models, which is our concern here. There are many types of local random growth models in the plane, see [23] for a review and more background. As an example consider the Eden-Richardson growth model, [12, 29], which is defined as follows. The shape Ωt at time t of the growing object is a connected set, which is the union of unit squares centered at points in Z. Let ∂Ωt denote the set of all unit squares, centered at integer points, which are adjacent to Ωt. In the continuous time version each square in ∂Ωt is added to Ωt independently of each other and with exponential waiting times; i.e. as soon as a square joins ∂Ωt it’s clock starts to tick and the square is added to Ωt after a random time T with the exponental distribution, P [T > s] = e−s. In the discrete time version, at each time t ∈ Z, each square in ∂Ωt−1, is added to Ωt−1 with probability p = 1 − q independently of each other, and the resulting set is Ωt. At time t = 0 we take Ω0 = [−1/2, 1/2]. In both cases the object grows linearly in time, [20], Ωt/t → A, the asymptotic shape, as t→∞. We are interested in the roughness of Ωt, the fluctuations of Ωt around tA. This growth model is equivalent with a certain first-passage site percolation model. With each site (i, j) ∈ Z we associate a random variable τ(i, j), which we think of as a random time. Variables associated with different sites are independent. A path π from (0, 0) to (M,N) is a sequence {pr}r=0 ⊆ Z with |pr − pr−1| = 1, p0 = (0, 0) and pR = (M,N). The first-passage time from (0, 0) to (M,N) is T (M,N) = min π ∑ pr∈π τ(pr) . (1)