A law of large numbers for local patterns in Schur measures and a Schur process

TL;DR

This paper proves a law of large numbers for local patterns in certain discrete point processes, including Schur measures and random plane partitions, showing convergence of pattern statistics to deterministic integrals.

Contribution

It establishes a new law of large numbers for local patterns in Schur measures and plane partition models, connecting pattern appearance to deterministic limits.

Findings

Pattern statistics converge to deterministic integrals

Results apply to Schur measures and plane partitions

Provides a unified law of large numbers for local patterns

Abstract

The aim of this note is to prove a law of large numbers for local patterns in discrete point processes. We investigate two different situations: a class of point processes on the one dimensional lattice including certain Schur measures, and a model of random plane partitions, introduced by Okounkov and Reshetikhin. The results state in both cases that the linear statistic of a function, weighted by the appearance of a fixed pattern in the random configuration and conveniently normalized, converges to the deterministic integral of that function weighted by the expectation with respect to the limit process of the appearance of the pattern.