Random plane partitions and corner distributions

Damir Yeliussizov

doi:10.5802/alco.171

Abstract

We explore some probabilistic applications arising in connections with $K$-theoretic symmetric functions. For instance, we determine certain corner distributions of random lozenge tilings and plane partitions. We also introduce some distributions that are naturally related to the corner growth model. Our main tools are dual symmetric Grothendieck polynomials and normalized Schur functions.

Highlights

Combinatorics arising in connection with K-theoretic Schubert calculus is quite rich
While the subject is intensively studied from combinatorial, algebraic and geometric aspects, see [20, 7, 8, 34, 19, 13, 35] and many references therein, much less is known about probabilistic connections
We mostly focus on one deformation of Schur functions, the dual Grothendieck polynomials, whose associated combinatorics is fairly neat

Summary

Introduction

Combinatorics arising in connection with K-theoretic Schubert calculus is quite rich. (The levels in Theorem 1.3 can first be viewed as fixed, and using exchangeability in Theorem 1.2 we can let be arbitrary and grow.) As it will be clear later, the arctic phenomenon passes to the first object of N-matrices with bounded last passage time, so that upper frozen boundary is recorded in the last column of a large matDrAixMaIRndYtEhLeIUmSaStIZriOxVitself has frozen regions in top left and bottom right quadrants, see Fig. 2. Consider the probability distribution Pg,b,c, which we call g-measure or gdistribution, on the (infinite) set PP(∞, b, c) of plane partitions with at most b rows and maximal entry at most c, defined as follows: Pg,b,c(π). This map is different from the well-known Robinson–Schensted–Knuth (RSK) correspondence; its description is somewhat simpler but has analogous properties Another key tool is in dual Grothendieck polynomials whose combinatorics is tied to plane partitions. Combined with this interesting coincidence (see Lemma 3.4), we use known asymptotics for normalized Schur polynomials from [5, 12]

A bijection between plane partitions and N-matrices

Dual Grothendieck polynomials

Normalized Schur polynomials

Multivariate generating function for corner distributions

Corner growth model