Transfer matrices for the totally asymmetric exclusion process

TL;DR

This paper introduces transfer matrices that relate the Markov matrices of TASEP systems of different sizes, providing an algebraic perspective on the matrix-product steady state representation.

Contribution

It demonstrates the existence of transfer matrices that intertwine Markov matrices of consecutive system sizes, revealing a new algebraic structure in TASEP.

Findings

Existence of transfer matrices $T_{L-1,L}$ and $ ilde{T}_{L-1,L}$

Transfer matrices satisfy semi-conjugation property

Provides algebraic insight into matrix-product steady state

Abstract

We consider the totally asymmetric simple exclusion process (TASEP) on a finite lattice with open boundaries. We show, using the recursive structure of the Markov matrix that encodes the dynamics, that there exist two transfer matrices $T_{L-1,L}$ and $\tilde{T}_{L-1,L}$ that intertwine the Markov matrices of consecutive system sizes: $\tilde{T}_{L-1,L}M_{L-1}=M_{L}T_{L-1,L}$. This semi-conjugation property of the dynamics provides an algebraic counterpart for the matrix-product representation of the steady state of the process.