Variations on Hammersley's interacting particle process

TL;DR

This paper explores generalized Hammersley-type interacting particle processes related to permutation patterns, analyzing their statistical properties, distribution structures, and asymptotic behaviors in multi-dimensional systems.

Contribution

It introduces new multi-dimensional interacting particle systems analogous to Hammersley's process and studies their statistical properties and asymptotic behaviors.

Findings

Derived estimates for mean and variance of particle counts

Analyzed distribution structures of generalized processes

Proposed three new interacting particle systems in the plane

Abstract

The longest increasing subsequence problem for permutations has been studied extensively in the last fifty years. The interpretation of the longest increasing subsequence as the longest 21-avoiding subsequence in the context of permutation patterns leads to many interesting research directions. We introduce and study the statistical properties of Hammersleytype interacting particle processes related to these generalizations and explore the finer structures of their distributions. We also propose three different interacting particle systems in the plane analogous to the Hammersley process in one dimension and obtain estimates for the asymptotic orders of the mean and variance of the number of particles in the systems.