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The oriented swap process on half line

Yuan Tian

TL;DR

The paper extends the oriented swap process from finite systems to the half-line and analyzes its asymptotic behavior using a synthesis of Harris graphical constructions, Hecke algebra symmetry, and projections to multi-species TASEP. It establishes that $\pi_t^{-1}(1)$ behaves like a Poisson variable with large-time fluctuations that are Gaussian, and derives limiting trajectories for scaled positions with a random initial speed, yielding a unified description of both average and fluctuation scales. Moreover, the work shows that the first $l$ particle fluctuations converge to the GUE corners process, and, at the process level, describes the jump structure of $\pi_t^{-1}$, including the asymptotics of jump counts, jump heights, and the folded-normal scaling of inter-jump times. These results connect infinite random sorting networks to KPZ-type universality via algebraic and probabilistic couplings, broadening the understanding of multi-species interacting particle systems on the half-line.

Abstract

In this paper we study the oriented swap process on the positive integers and its asymptotic properties. Our results extend a theorem by Angel, Holroyd, and Romik regarding the trajectories of particles in the finite oriented swap process. Furthermore, we study the evolution of the type of a particle at the leftmost position over time. Our approach relies on a relationship between multi-species particle systems and Hecke algebras, complemented by a detailed asymptotic analysis.

The oriented swap process on half line

TL;DR

The paper extends the oriented swap process from finite systems to the half-line and analyzes its asymptotic behavior using a synthesis of Harris graphical constructions, Hecke algebra symmetry, and projections to multi-species TASEP. It establishes that behaves like a Poisson variable with large-time fluctuations that are Gaussian, and derives limiting trajectories for scaled positions with a random initial speed, yielding a unified description of both average and fluctuation scales. Moreover, the work shows that the first particle fluctuations converge to the GUE corners process, and, at the process level, describes the jump structure of , including the asymptotics of jump counts, jump heights, and the folded-normal scaling of inter-jump times. These results connect infinite random sorting networks to KPZ-type universality via algebraic and probabilistic couplings, broadening the understanding of multi-species interacting particle systems on the half-line.

Abstract

In this paper we study the oriented swap process on the positive integers and its asymptotic properties. Our results extend a theorem by Angel, Holroyd, and Romik regarding the trajectories of particles in the finite oriented swap process. Furthermore, we study the evolution of the type of a particle at the leftmost position over time. Our approach relies on a relationship between multi-species particle systems and Hecke algebras, complemented by a detailed asymptotic analysis.

Paper Structure

This paper contains 9 sections, 20 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of the process
  4. Hecke algebra and Symmetry
  5. Properties of TASEP and the oriented swap process
  6. GUE corners process
  7. Limiting Trajectories
  8. Fluctuations of $\pi_t$ and $\pi_t^{-1}$
  9. $\pi_t^{-1}$ as a process

Key Result

Theorem 1.3

Let $\alpha ,s \in \mathbb{R}$, one has where $U$ is a uniform random variable on $[-1,1]$, and $(\lambda_1^1, \lambda_2^2, \dots \lambda_l^l)$ is the GUE corners process (see Section sec:Pre).

Figures (2)

  • Figure 1: The oriented swap process at time $0$, $\pi_t(i) = i, \forall i$
  • Figure 2: The oriented swap process after several jumps

Theorems & Definitions (42)

  • Definition 1.1
  • Example 1.2
  • Theorem 1.3: Theorem \ref{['thm3']}, Corollary \ref{['thm4']}
  • Theorem 1.4: Theorem \ref{['thm5.19']}
  • Theorem 1.5: Theorem \ref{['thm5.2']}
  • Definition 2.1: Hecke algebra
  • Proposition 2.2
  • Corollary 2.3
  • Remark 2.4
  • Definition 2.5
  • ...and 32 more