Dynamic Young Diagrams

Abstract

Young diagrams can be regarded as decreasing interfaces which separate two distinct phases, see Figs. 2.1 and 2.17 below. Such random interfaces appear also in zero-temperature Ising models at the corner, see Fig. 2.3. The goal of this chapter is to study the dynamics of random Young diagrams, sometimes called SOS (solid on solid) dynamics, mostly in two-dimensional (2D) case, which is naturally associated with the grand canonical and canonical ensembles both in uniform and restricted uniform statistics, introduced by Vershik (Funct Anal Appl 30:90–105, 1996). We first recall ensembles of two-dimensional Young diagrams and their scaling limits, that is, the law of large numbers, which leads to the so-called Vershik curves in the limits, their fluctuations and large deviation principle. Then, we introduce the corresponding dynamics, and establish its space-time scaling limits such as hydrodynamic limit (law of large numbers, obtained with Funaki and Sasada (Commun Math Phys 299:335–363, 2010)) and non-equilibrium fluctuations (obtained with Funaki et al. (Stoch Proc Appl 123:1229–1275, 2013)) for non-conservative case, i.e., for the dynamics associated with the grand canonical ensembles. Vershik curves can be recovered in these dynamic scaling limits. We also discuss the dynamics associated with the canonical ensembles, which has a conservation law. We finally discuss the three-dimensional (3D) case. Cerf and Kenyon (Commun Math Phys 222:147–179, 2001) derived the limit surface called Wulff shape, which is characterized by a certain variational formula under uniform statistics. We discuss the corresponding dynamic problem.