Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Second Class Particle Behaviour in Blocking ASEP

Daniel Adams, Márton Balázs, Jessica Jay

TL;DR

This work analyzes second-class particle behavior in blocking ASEP via a basic coupling of two ASEPs, yielding explicit distributions for second-class positions and site occupancies under blocking measures. The authors develop and exploit half-infinite and finite-volume counting results, using these to provide purely probabilistic proofs of classical partition identities such as the Durfee rectangles identity, Euler's identity, and the $q$-Binomial Theorem, while illuminating connections to the Jacobi triple product and Mallows measures. For the two-species (first and second class) ASEP, they establish a reversible stationary framework for the label process and derive exact joint distributions for any number of second-class particles under both blocking and ergodic measures, including a discrete logistic law for a single second-class particle. The findings link interacting particle systems with partition theory and random permutation models, offering new probabilistic insight into classic combinatorial identities and suggesting avenues for generalization to multi-species and multi-lane systems with potential applications in combinatorics and statistical mechanics.

Abstract

We consider any fixed $d\in\mathbb{Z}_{>0}$ number of second class particles in the asymmetric simple exclusion process (ASEP), constructed via a basic coupling of two ASEPs. We give the joint distribution of the positions of the second class particles and also the probability of there being a second class particle at a given site, under the natural blocking measure for ASEP. In order to find these distributions we use results about the number of particles in half-infinite and finite site ranges of ASEP. Our investigations also lead to probabilistic proofs of well-known combinatorial identities; the Durfee rectangles identity, Euler's identity, and the $q$-Binomial Theorem.

Second Class Particle Behaviour in Blocking ASEP

TL;DR

This work analyzes second-class particle behavior in blocking ASEP via a basic coupling of two ASEPs, yielding explicit distributions for second-class positions and site occupancies under blocking measures. The authors develop and exploit half-infinite and finite-volume counting results, using these to provide purely probabilistic proofs of classical partition identities such as the Durfee rectangles identity, Euler's identity, and the -Binomial Theorem, while illuminating connections to the Jacobi triple product and Mallows measures. For the two-species (first and second class) ASEP, they establish a reversible stationary framework for the label process and derive exact joint distributions for any number of second-class particles under both blocking and ergodic measures, including a discrete logistic law for a single second-class particle. The findings link interacting particle systems with partition theory and random permutation models, offering new probabilistic insight into classic combinatorial identities and suggesting avenues for generalization to multi-species and multi-lane systems with potential applications in combinatorics and statistical mechanics.

Abstract

We consider any fixed number of second class particles in the asymmetric simple exclusion process (ASEP), constructed via a basic coupling of two ASEPs. We give the joint distribution of the positions of the second class particles and also the probability of there being a second class particle at a given site, under the natural blocking measure for ASEP. In order to find these distributions we use results about the number of particles in half-infinite and finite site ranges of ASEP. Our investigations also lead to probabilistic proofs of well-known combinatorial identities; the Durfee rectangles identity, Euler's identity, and the -Binomial Theorem.
Paper Structure (23 sections, 49 theorems, 222 equations, 4 figures)

This paper contains 23 sections, 49 theorems, 222 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Results
  4. Distributional results for single species ASEP
  5. Combinatorial Identities
  6. Distributional results for two species ASEP
  7. The asymmetric simple exclusion process (ASEP)
  8. Symmetry of ASEP
  9. Distributional Results
  10. Combinatorial Identities coming from ASEP
  11. The asymmetric simple exclusion process with second class particles
  12. Constructing a Basic Coupling of Two ASEPs
  13. ASEP on $\mathbb{Z}$
  14. ASEP on $\mathbb{Z}_{\geq 0}$ with exactly $d$ particles (Label process)
  15. Basic Coupling of Two ASEPs
  16. ...and 8 more sections

Key Result

Theorem 1.1

For any $m \in \mathbb{Z}$, $k\in\mathbb{Z}_{\geq 0}$ and $c \in \mathbb{R}$,

Figures (4)

  • Figure 1: An example of the pair $(\underline{\xi}(t),\underline{x}(t))$ when $d=5$. The red arrows correspond to a possible second class particle label jump.
  • Figure 2: An example of constructing $\underline{\eta}(t)$ from the pair $(\underline{\xi}(t),\underline{x}(t))$ (when $d=5$).
  • Figure 3: An example of the Durfee square for an integer partition, $\lambda$, of $30$.
  • Figure 4: An example of the Durfee rectangle for an integer partition, $\lambda$, of $30$ when $n=2$ and $n=-3$.

Theorems & Definitions (97)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 1.2
  • Lemma 1.2
  • Theorem 1.3
  • Proposition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.6
  • ...and 87 more