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The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model

Yuhan Jiang

Abstract

The multispecies asymmetric simple exclusion process (mASEP) is a Markov chain in which particles of different species hop along a one-dimensional lattice. This paper studies the doubly asymmetric simple exclusion process $\mathrm{DASEP}(n,p,q)$ in which $q$ particles with species $1, \dots, p$ hop along a circular lattice with $n$ sites, but also the particles are allowed to spontaneously change from one species to another. In this paper, we introduce two related Markov chains called the colored Boolean process and the restricted random growth model, and we show that the DASEP lumps to the colored Boolean process, and the colored Boolean process lumps to the restricted random growth model. This allows us to generalize a theorem of David Ash on the relations between sums of steady state probabilities. We also give explicit formulas for the stationary distribution of $\mathrm{DASEP}(n,2,2)$.

The doubly asymmetric simple exclusion process, the colored Boolean process, and the restricted random growth model

Abstract

The multispecies asymmetric simple exclusion process (mASEP) is a Markov chain in which particles of different species hop along a one-dimensional lattice. This paper studies the doubly asymmetric simple exclusion process in which particles with species hop along a circular lattice with sites, but also the particles are allowed to spontaneously change from one species to another. In this paper, we introduce two related Markov chains called the colored Boolean process and the restricted random growth model, and we show that the DASEP lumps to the colored Boolean process, and the colored Boolean process lumps to the restricted random growth model. This allows us to generalize a theorem of David Ash on the relations between sums of steady state probabilities. We also give explicit formulas for the stationary distribution of .
Paper Structure (5 sections, 12 theorems, 31 equations, 5 figures, 1 table)

This paper contains 5 sections, 12 theorems, 31 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. The DASEP lumps to the colored Boolean process
  3. The colored Boolean process lumps to the restricted random growth model
  4. The stationary distribution of DASEP(n,2,2)
  5. Homomesy

Key Result

Theorem 1.4

Consider $\mathop{\mathrm{DASEP}}\nolimits(n,p,q)$ for any positive integers $n,p,q$ with $n >q$. In other words, the sum of steady state probabilities of states within one equivalence class is proportional to each other and the ratio only depends on the sum of all parts and the multiplicities in the partition; all steady state probabilities with respect to binary words are equal.

Figures (5)

  • Figure 1: The state diagram of $\mathop{\mathrm{DASEP}}\nolimits(2,2,1)$ and $\mathop{\mathrm{DASEP}}\nolimits(3,2,2)$. We omit loops at each state. Bold edges denote changes in species, while regular edges denote exchanges of particles of different species or between particles and holes.
  • Figure 2: The state diagram of $\Omega^{2,2}_3$, as a lumping of $\mathop{\mathrm{DASEP}}\nolimits(3,2,2)$ as in \ref{['fig:dasep221322']}. The bold edges denote the changes of species, while the regular edges denote the exchanges between particles of different species or between particles and holes.
  • Figure 3: The state diagram of the restricted random growth model on $\chi^{2,2}$. The Markov chain on the left can be viewed as a lumping of the Markov chain on the right in which we do not rearrange the parts in weakly decreasing order.
  • Figure 4: $a_1 = u+3t+4$
  • Figure 5: $b_1 = u+2t+3$

Theorems & Definitions (35)

  • Definition 1.1
  • Remark 1.2
  • Example 1.3
  • Theorem 1.4
  • Remark 1.5
  • Example 1.6
  • Corollary 1.7
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • ...and 25 more