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1 About this webpage

This webpage was created with Ximera, an interactive textbook platform hosted by Ohio State University. The Ximera Project is funded 2024-2026 (with no other external funding) by a $2,125,000 Open Textbooks Pilot Program grant from the federal Department of Education.

2 Introduction

The Matrix Product Ansatz is a method for finding stationary measures in an interacting particle system with open boundary conditions. Let the generator be written in the form \[ B_1 + \sum _{l=1}^l w_{l,l+1} + \bar {B}_L \] where we use the subscript notation. In other words, \(w\) is a local two–site generator, while \(B\) acts on the left–most lattice site, and \(\bar {B}\) acts on the right–most lattice site.

Denote a particle configuration on \(L\) sites by \[ \mathcal {C} = (\tau _1,\ldots ,\tau _L), \] where each \(tau_i\) are particle variables equally either 0 or 1, indicating the number of particles. The Ansatz takes the form \[ \mathcal {S}(\mathcal {C}) = \frac {1}{Z_L} \langle W | \prod _{i=1}^L ((1-\tau _i)E + \tau _i D) | V \rangle \] where \(E\) and \(D\) are elements of a non–commutative algebra, and the product is ordered from the index \(i=1\) on the left to \(i=L\) on the right.

In [KS97], it was proven that there is a solution of the form \[ w\left [ \binom {E}{D} \otimes \binom {E}{D} \right ] \]

References