[Paper Review] Inhomogenous Multispecies TASEP on a ring with spectral parameters

Luigi Cantini|arXiv (Cornell University)|Feb 25, 2016

Stochastic processes and statistical mechanics19 references4 citations

TL;DR

This paper introduces an inhomogeneous multispecies TASEP on a ring with spectral parameters, establishing Yang-Baxter integrability and deriving an exact formula for the unnormalized stationary measure. The key contribution is the factorization of stationary probabilities into double Schubert polynomials in the particle parameters $\boldsymbol{\tau}$ and $\boldsymbol{\nu}$, revealing a deep algebraic structure underlying the stationary distribution of the model.

ABSTRACT

We study an inhomogenous multispecies version of the Totally Asymmetric Simple Exclusion Process (TASEP) on a periodic oriented one dimensional lattice, which depends on two sets of parameters $({\bf τ},{\bf ν})$, attached to the particles. After discussing the Yang-Baxter integrability of our model, we study its (unnormalized) stationary measure. Motivated by the integrability of the model we introduce a further set of spectral parameters ${\bf z}$, attached to the sites of the lattice, and we uncover a remarkable underlying algebraic structure. We provide exact formulas for the stationary measure and prove the factorization of the stationary probability of certain configurations in terms of double Schubert polynomials in $({\bf τ},{\bf ν})$.

Motivation & Objective

• To study the stationary measure of an inhomogeneous multispecies TASEP on a periodic one-dimensional lattice with asymmetric jump rates parameterized by $\boldsymbol{\tau}$ and $\boldsymbol{\nu}$.
• To establish Yang-Baxter integrability of the model, enabling exact solvability.
• To introduce spectral parameters $\bf{z}$ on lattice sites to uncover an underlying algebraic structure.
• To derive exact formulas for the unnormalized stationary measure and prove its factorization in terms of double Schubert polynomials in $\boldsymbol{\tau}$ and $\boldsymbol{\nu}$.

Proposed method

• The model is defined on a ring with $N$ sites and one particle per species, using asymmetric transition rates $r_{\alpha,\beta} = \tau_\alpha - \nu_\beta$ for $\alpha < \beta$, ensuring exclusion and asymmetry.
• Yang-Baxter integrability is confirmed via the existence of an R-matrix satisfying the Yang-Baxter equation, ensuring the model's solvability.
• Spectral parameters $\bf{z}$ are introduced on lattice sites to construct a generating function for the stationary measure.
• The stationary measure is expressed using a contour integral representation involving products of rational functions and a generating function $F(w)$, with coefficients derived via divided difference operators.
• The algebraic structure is analyzed using symmetric functions and divided difference operators $\partial_i$, which are used to prove symmetry and factorization properties.
• The proof of factorization relies on technical lemmas involving contour integrals and residue identities, particularly the vanishing of certain symmetric sums via polynomial degree arguments.

Experimental results

Research questions

• RQ1 How does the stationary measure of the inhomogeneous multispecies TASEP on a ring factorize in terms of algebraic polynomials?
• RQ2 What is the role of spectral parameters $\bf{z}$ in revealing the underlying algebraic structure of the stationary measure?
• RQ3 Can the stationary probabilities be expressed as double Schubert polynomials in the parameters $\boldsymbol{\tau}$ and $\boldsymbol{\nu}$?
• RQ4 How does the Yang-Baxter integrability of the model relate to the factorization of the stationary measure?
• RQ5 What is the connection between the stationary measure and the combinatorics of Schubert polynomials in the general inhomogeneous case?

Key findings

• The unnormalized stationary probability $\psi_{\bf w}(\boldsymbol{\tau}, \boldsymbol{\nu})$ for any configuration $\bf{w}$ factorizes into a product of double Schubert polynomials in the parameters $\boldsymbol{\tau}$ and $\boldsymbol{\nu}$, providing an exact algebraic formula.
• The stationary measure is shown to be symmetric under the action of divided difference operators, which is essential for the factorization into Schubert polynomials.
• The introduction of spectral parameters $\bf{z}$ on lattice sites enables the construction of a generating function that encodes the full stationary measure via contour integrals.
• The proof relies on a key identity involving the vanishing of symmetric sums of residues, established via polynomial degree arguments and contour integration.
• The model generalizes previous results: when $\nu_\alpha = 0$, the stationary probabilities reduce to positive integer combinations of Schubert polynomials in $\boldsymbol{\tau}$, confirming a conjecture by Lam and Williams.
• The stationary measure is derived using a matrix product ansatz framework, with the algebraic structure linked to the Zamolodchikov tetrahedron equation in the general case.