Asymptotic geometry of discrete interlaced patterns: Part I
Abstract
A discrete Gelfand–Tsetlin pattern is a configuration of particles in ℤ2. The particles are arranged in a finite number of consecutive rows, numbered from the bottom. There is one particle on the first row, two particles on the second row, three particles on the third row, etc., and particles on adjacent rows satisfy an interlacing constraint. We consider the uniform probability measure on the set of all discrete Gelfand–Tsetlin patterns of a fixed size where the particles on the top row are in deterministic positions. This measure arises naturally as an equivalent description of the uniform probability measure on the set of all tilings of certain polygons with lozenges. We prove a determinantal structure, and calculate the correlation kernel. We consider the asymptotic behavior of the system as the size increases under the assumption that the empirical distribution of the deterministic particles on the top row converges weakly. We consider the asymptotic "shape" of such systems. We provide parameterizations of the asymptotic boundaries and investigate the local geometric properties of the resulting curves. We show that the boundary can be partitioned into natural sections which are determined by the behavior of the roots of a function related to the correlation kernel. This paper should be regarded as a companion piece to the paper [E. Duse and A. Metcalfe, Asymptotic geometry of discrete interlaced patterns: Part II, in preparation], in which we resolve some of the remaining issues. Both of these papers serve as background material for the papers [E. Duse and A. Metcalfe, Universal edge fluctuations of discrete interlaced particle systems, in preparation; E. Duse and K. Johansson and A. Metcalfe, Cusp Airy process of discrete interlaced particle systems, in preparation], in which we examine the edge asymptotic behavior.
