[Paper Review] Inhomogeneous discrete-time exclusion processes

Nicolas Crampé, K. Mallick+2 more|arXiv (Cornell University)|Jun 16, 2015

Stochastic processes and statistical mechanics70 references3 citations

TL;DR

This paper introduces inhomogeneous discrete-time exclusion processes on periodic and open lattices using inhomogeneous transfer matrices derived from integrable quantum spin chains. It establishes a matrix product ansatz for the stationary state, expresses normalization factors as Schur polynomials, and reveals a novel connection between Bethe roots and Lee-Yang zeros, extending integrable stochastic processes beyond homogeneous models.

ABSTRACT

We study discrete time Markov processes with periodic or open boundary conditions and with inhomogeneous rates in the bulk. The Markov matrices are given by the inhomogeneous transfer matrices introduced previously to prove the integrability of quantum spin chains. We show that these processes have a simple graphical interpretation and correspond to a sequential update. We compute their stationary state using a matrix ansatz and express their normalization factors as Schur polynomials. A connection between Bethe roots and Lee-Yang zeros is also pointed out.

Motivation & Objective

• To extend integrable stochastic processes beyond homogeneous exclusion models by introducing inhomogeneous transition rates in the bulk.
• To construct discrete-time Markov processes with periodic and open boundary conditions using inhomogeneous transfer matrices.
• To derive exact expressions for the stationary state using a matrix product ansatz based on Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations.
• To express normalization factors of the stationary measure as Schur polynomials, linking statistical mechanics to symmetric function theory.
• To uncover a surprising connection between the roots of the Bethe equations and the zeros of the normalization function, interpreted as Lee-Yang zeros.

Proposed method

• The dynamics are defined via inhomogeneous transfer matrices, which generate commuting Markov matrices and ensure integrability.
• A sequential updating rule is derived graphically from the transfer matrix structure, enabling explicit construction of stochastic transition rules.
• The matrix product ansatz is applied to the open boundary case using algebraic relations from quantum integrability (Zamolodchikov-Faddeev and Ghoshal-Zamolodchikov relations).
• Normalization factors of the stationary state are computed as determinants involving row vectors derived from $ z_j $-dependent expressions, which are then expressed as Schur polynomials.
• The coefficients in the determinant expansion are shown to satisfy a recurrence $ C_n = (a+b)C_{n-1} - ab C_{n-2} $, with solution $ C_n = \frac{a^n - b^n}{a - b} $, linking to orthogonal polynomial structures.
• The Bethe roots of the integrable model are related to the zeros of the normalization function, interpreted as Lee-Yang zeros in the thermodynamic limit.

Experimental results

Research questions

• RQ1 How can inhomogeneous transition rates be consistently introduced into discrete-time exclusion processes while preserving integrability?
• RQ2 What is the structure of the stationary state for inhomogeneous exclusion processes with open or periodic boundaries?
• RQ3 Can the normalization factor of the stationary measure be expressed in terms of symmetric polynomials, and if so, which ones?
• RQ4 Is there a deeper algebraic or physical connection between the solutions of the Bethe equations and the zeros of the normalization function?
• RQ5 How does the matrix product ansatz generalize to inhomogeneous models, and what algebraic relations underlie its validity?

Key findings

• The stationary state of the inhomogeneous discrete-time exclusion process is constructed exactly using a matrix product ansatz, valid for both open and periodic boundary conditions.
• Normalization factors of the stationary measure are expressed as Schur polynomials in the inhomogeneity parameters $ z_i $, establishing a link to symmetric function theory.
• When all inhomogeneity parameters $ z_i = 1 $, the stationary measure reduces to that of the standard TASEP, confirming consistency with known results.
• A novel connection is found between the roots of the Bethe equations and the zeros of the normalization function, interpreted as Lee-Yang zeros in the context of phase transitions.
• The coefficients in the determinant expansion of the normalization factor satisfy a recurrence relation $ C_n = (a+b)C_{n-1} - ab C_{n-2} $, with closed-form solution $ C_n = \frac{a^n - b^n}{a - b} $, reflecting underlying quantum group symmetry.
• The graphical interpretation of the updating rules via transfer matrix decomposition reveals a sequential update mechanism consistent with stochastic dynamics.