[Paper Review] Askey-Wilson signed measures and open ASEP in the shock region

Yizao Wang, Jacek Wesołowski+1 more|arXiv (Cornell University)|Jul 13, 2023

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TL;DR

This paper introduces a family of multi-dimensional Askey–Wilson signed measures to provide a unified, rigorous description of the stationary measure of open asymmetric simple exclusion process (ASEP) across the entire phase diagram, including the shock region. By expressing the joint generating function via integrations over these signed measures, the authors derive the macroscopic density profile and limit fluctuations in the shock region—confirming physics postulations and establishing Brownian motion limits in high/low density phases and random shock profiles on the coexistence line.

ABSTRACT

We introduce a family of multi-dimensional Askey-Wilson signed measures. We offer an explicit description of the stationary measure of the open asymmetric simple exclusion process (ASEP) in the full phase diagram, in terms of integrations with respect to these Askey-Wilson signed measures. Using our description, we provide a rigorous derivation of the density profile and limit fluctuations of open ASEP in the entire shock region, including the high and low density phases as well as the coexistence line. This in particular confirms the existing physics postulations of the density profile.

Motivation & Objective

• To provide a unified, rigorous description of the stationary measure of open ASEP across the full phase diagram, including the shock region where previous methods failed due to sign issues in Askey–Wilson measures.
• To extend the matrix product ansatz and Askey–Wilson process framework—previously limited to the fan region—to the shock region by introducing signed measures instead of positive measures.
• To confirm long-standing physics postulations regarding the macroscopic density profile in the high and low density phases and on the coexistence line.
• To derive the limit fluctuations of the stationary measure in the shock region, showing they are given by Brownian motion, consistent with the fan region.
• To lay a foundation for future studies of currents, correlation functions, large deviations, and scaling limits of open KPZ equation and fixed point measures.

Proposed method

• Introduce a new family of multi-dimensional Askey–Wilson signed measures to replace the positive measures used in prior work, enabling treatment of the shock region.
• Express the joint generating function of the open ASEP stationary measure as integrals with respect to these signed measures, generalizing the approach in [BW17] beyond the fan region.
• Use the parameterization $ A = \kappa_+(\beta,\delta), B = \kappa_-(\beta,\delta), C = \kappa_+(\alpha,\gamma), D = \kappa_-(\alpha,\gamma) $ to map boundary and bulk parameters to the measure parameters.
• Establish uniform boundedness of atom masses in the signed measures by proving that denominators in the mass formulas remain bounded away from zero under appropriate parameter constraints.
• Apply asymptotic analysis under weakly asymmetric scaling $ t_k = e^{-s_k/n^\theta} $ to control the growth of total variation norms of the measures, showing they grow at most as $ n^{2\theta d} $.
• Leverage the convergence of the finite-time transition kernels to the stationary measure via the signed measure representation, ensuring the validity of macroscopic limits.

Experimental results

Research questions

• RQ1 Can the stationary measure of open ASEP in the shock region be rigorously described using a generalization of the Askey–Wilson process framework?
• RQ2 What is the macroscopic density profile in the high and low density phases and on the coexistence line of open ASEP, and does it match physics predictions?
• RQ3 What are the limit fluctuations of the stationary measure in the shock region, and do they coincide with those in the fan region?
• RQ4 Can the signed measure framework be extended to compute multi-point Laplace transforms of the open KPZ equation stationary measure beyond the $ u+v \geq 0 $ regime?
• RQ5 What are the conditions under which the total variation of the signed measures remains uniformly bounded in the scaling limit?

Key findings

• The stationary measure of open ASEP is rigorously described across the entire phase diagram using integrations with respect to multi-dimensional Askey–Wilson signed measures.
• The macroscopic density profile in the high and low density phases is constant, confirming physics postulations based on the matrix product ansatz.
• On the coexistence line, the density profile is a random process depending on a uniformly distributed shock position, consistent with physics predictions.
• The limit fluctuations of the stationary measure in the high and low density phases are given by Brownian motion, matching results from the fan region.
• The total variation of the signed measures grows at most as $ n^{2\theta d} $ under weakly asymmetric scaling, ensuring convergence in the macroscopic limit.
• The method provides a unified framework that extends previous results in the fan region to the shock region, enabling future studies of currents, correlation functions, and KPZ fixed point measures.