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Factorised stationary states for a long range misanthrope process

Arvind Ayyer, Saham Sil

TL;DR

This paper develops a comprehensive framework for factorised stationary states in the long-range misanthrope family, introducing PALRMP on a finite ring with site-dependent rates and its specializations TALRMP and SLRMP. Through master-equation analysis and systematic reductions to smaller configurations, it derives necessary and sufficient conditions for product-form stationary measures across PALRMP, TALRMP, and homogeneous variants, establishing equivalences between the long-range and short-range (or symmetric) cases for $q\neq1$. The authors provide explicit one-point weight forms and constructive criteria, including a detailed treatment of the discrete HAD process as a motivating example. The results unify and extend cocozza-1985-type conditions to long-range dynamics, offering broad families of factorised models and shedding light on how long-range hopping interacts with local occupation numbers to yield product measures. The HAD discretisation further demonstrates the applicability of these conditions to classical interacting particle systems in discrete settings.

Abstract

The misanthrope process is an interacting particle system where particles move between neighbouring sites with hop rates depending only on the number of particles at the departure and arrival sites. Motivated by a discretised version of the Hammersley--Aldous--Diaconis process, we introduce a partially asymmetric long range misanthrope process (PALRMP) on a finite one-dimensional lattice with periodic boundary conditions where particles can move between sites that are not necessarily neighbours, as long as there are no particles in between the departure and arrival sites. In this model, each site $\ell$ has an inhomogeneous rate parameter $x_\ell$ associated to it, and the hop rate of a particle moving from site $k$ to site $\ell$ depends upon the parameter associated to the target site $x_\ell$, the direction the particle moves, and the number of particles at sites $k$ and $\ell$. We also consider the homogeneous PALRMP, where all the $x_\ell$'s are 1. We find necessary and sufficient conditions on the hop rates under which the stationary distribution is of factorised form for both the PALRMP and the homogeneous PALRMP, as well as the extreme variants, namely the ones where the particle motion is totally asymmetric (TALRMP) and symmetric (SLRMP). As an illustrative example, we study in detail the discrete Hammersley--Aldous--Diaconis process.

Factorised stationary states for a long range misanthrope process

TL;DR

This paper develops a comprehensive framework for factorised stationary states in the long-range misanthrope family, introducing PALRMP on a finite ring with site-dependent rates and its specializations TALRMP and SLRMP. Through master-equation analysis and systematic reductions to smaller configurations, it derives necessary and sufficient conditions for product-form stationary measures across PALRMP, TALRMP, and homogeneous variants, establishing equivalences between the long-range and short-range (or symmetric) cases for . The authors provide explicit one-point weight forms and constructive criteria, including a detailed treatment of the discrete HAD process as a motivating example. The results unify and extend cocozza-1985-type conditions to long-range dynamics, offering broad families of factorised models and shedding light on how long-range hopping interacts with local occupation numbers to yield product measures. The HAD discretisation further demonstrates the applicability of these conditions to classical interacting particle systems in discrete settings.

Abstract

The misanthrope process is an interacting particle system where particles move between neighbouring sites with hop rates depending only on the number of particles at the departure and arrival sites. Motivated by a discretised version of the Hammersley--Aldous--Diaconis process, we introduce a partially asymmetric long range misanthrope process (PALRMP) on a finite one-dimensional lattice with periodic boundary conditions where particles can move between sites that are not necessarily neighbours, as long as there are no particles in between the departure and arrival sites. In this model, each site has an inhomogeneous rate parameter associated to it, and the hop rate of a particle moving from site to site depends upon the parameter associated to the target site , the direction the particle moves, and the number of particles at sites and . We also consider the homogeneous PALRMP, where all the 's are 1. We find necessary and sufficient conditions on the hop rates under which the stationary distribution is of factorised form for both the PALRMP and the homogeneous PALRMP, as well as the extreme variants, namely the ones where the particle motion is totally asymmetric (TALRMP) and symmetric (SLRMP). As an illustrative example, we study in detail the discrete Hammersley--Aldous--Diaconis process.
Paper Structure (11 sections, 29 theorems, 141 equations, 3 figures)

This paper contains 11 sections, 29 theorems, 141 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. A partially asymmetric long range misanthrope process
  4. Master equation for the partially asymmetric long range misanthrope process
  5. Totally asymmetric long range misanthrope process
  6. Partially asymmetric long range misanthrope process
  7. Symmetric long range misanthrope process
  8. Homogeneous totally asymmetric long range misanthrope process
  9. Homogeneous partially asymmetric long range misanthrope process
  10. Homogeneous symmetric long range misanthrope process
  11. The discrete Hammersley-Aldous-Diaconis process

Key Result

Proposition 2.2

If an interacting particle system on $\Omega_{L,N}$ has factorised stationary distribution determined by one-point functions $(f_\ell)_{\ell=1}^L$, then it is also factorised with respect to the one-point functions $(g_\ell)_{\ell=1}^L$, where $g_\ell(m) = a b^m f_\ell(m)$ for some $a,b \in \mathbb{

Figures (3)

  • Figure 1.1: All the models in this work. We use the prefix H to stand for homogeneous.
  • Figure 3.1: Some transitions, along with their rates, in the PALRMP from the configuration $(0,3,1,0,2) \in \Omega_{5,6}$.
  • Figure 3.2: All transitions in the TALRMP on $\Omega_{3,2}$.

Theorems & Definitions (70)

  • Definition 2.1: Factorised probability distribution
  • Proposition 2.2
  • Proposition 3.1
  • proof
  • Proposition 3.2: Translation covariance
  • proof
  • Corollary 3.3
  • Remark 3.4
  • Theorem 3.5
  • Theorem 3.6
  • ...and 60 more