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Boundary current fluctuations for the half space ASEP and six vertex model

Jimmy He

Abstract

We study fluctuations of the current at the boundary for the half space asymmetric simple exclusion process (ASEP) and the height function of the half space six vertex model at the boundary at large times. We establish a phase transition depending on the asymptotic density of particles at the boundary, with GSE and GOE limits as well as the Baik--Rains crossover distribution near the critical point. This was previously known for half space last passage percolation, and recently established for the half space log-gamma polymer and KPZ equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half space six vertex model and a half space Hall--Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half space ASEP and six vertex model, indicating a hidden free fermionic structure.

Boundary current fluctuations for the half space ASEP and six vertex model

Abstract

We study fluctuations of the current at the boundary for the half space asymmetric simple exclusion process (ASEP) and the height function of the half space six vertex model at the boundary at large times. We establish a phase transition depending on the asymptotic density of particles at the boundary, with GSE and GOE limits as well as the Baik--Rains crossover distribution near the critical point. This was previously known for half space last passage percolation, and recently established for the half space log-gamma polymer and KPZ equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half space six vertex model and a half space Hall--Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half space ASEP and six vertex model, indicating a hidden free fermionic structure.
Paper Structure (31 sections, 49 theorems, 231 equations, 5 figures)

This paper contains 31 sections, 49 theorems, 231 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Overview of results and proof
  3. ASEP
  4. Half space six-vertex model
  5. Algebraic identities
  6. Outline
  7. Notation
  8. The half space six vertex model and half space Hall--Littlewood measure
  9. Half space six vertex model
  10. Symmetric functions
  11. Half space Hall--Littlewood measures
  12. Deformed bosons
  13. Proof of Theorem \ref{['thm: general 6vm HL distr equality']}
  14. An identity of symmetric functions
  15. Sagan--Stanley skew RSK correspondence
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $N(\tau)$ denote the number of particles within the system at time $\tau$ in the half space ASEP started from the empty configuration, and let $\rho$ denote the effective density of particles at $0$. Assume that $t<1$. Then depending on $\rho$, as $\tau\to\infty$, where $F_{GSE}$ and $F_{GOE}$ are the Tracy--Widom GSE and GOE distribution functions, $\Phi$ is the distribution function for a s

Figures (5)

  • Figure 1.1: Configuration of the ASEP along with some possible transitions and their rates. Particles which would jump to an occupied site are blocked. Note that since the site at $0$ is empty, particles enter at rate $\alpha$. If it were occupied, the particle would instead leave at rate $\beta$.
  • Figure 1.2: A configuration of the six vertex model and its height function at $(4,4)$.
  • Figure 1.3: Probabilities for sampling outgoing arrows at a vertex $(i,j)$. The black line represents an arrow and the dashed line represents no arrow.
  • Figure 6.1: The contour $C_{r,R}$.
  • Figure 7.1: The contour for the Gaussian regime.

Theorems & Definitions (103)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 2.1
  • Lemma 2.2
  • ...and 93 more