Large deviations and the matrix product ansatz

Abstract

We consider probability measures on $A^N$, the set of sequences of symbols on a finite alphabet $A$ of length $N$, that give a weight to each sequence in terms of a collection of matrices with non-negative entries and having rows and columns labeled by a finite or countable set $B$. We prove for such kind of measures large deviations principles for several empirical measures. Our approach is based on a simultaneous combination of an enlargement of the state space to sequences on $A\times B$ and a spectral conjugation that produces a stochastic matrix, as discussed in \cite{GI1}. As a result we describe the measures as hidden Markov measures and can deduce the large deviations results by contraction from the corresponding ones for the enlarged Markov chain. The measure on the enlarged state space is a Markov bridge. The invariant measures of several non equilibrium models of interacting particle systems can be represented by the so called {\it Matrix Product Ansatz} that corresponds to measures of the type that we consider and with matrices labeled by $B$ that is typically countable infinite. The large deviations behavior is different in the cases with $B$ finite or countable. In the finite case we give a variational formula for both the algebraic and the spatial empirical measures, that can be solved in special cases. For the infinite case, we illustrate the method through an example that is the invariant measure of the boundary driven TASEP model in a special regime. We recover in this way the celebrated results in \cite{Der4,Derr7}, and in particular we obtain a variational representation of the rate function similar to that in \cite{Bryc}. Our approach is general and can in principle be applied to any measure represented by the matrix product ansatz with matrices having positive entries.

Figures (3)

• Figure 1: A graphical representation of the transition probabilities of the Markov chain $\mathfrak S$ on $\{0,1\}\times \mathbb N_0$. The coordinates ${\bf c,b}$ on the axis $\mathbb N_0$ are written in boldface to distinguish them from the values of the transition probabilities.
• Figure 2: A graphical representation of the transition probabilities of the effective Markov chain $\mathcal{S}$ on $\{0,1\}\times \mathbb N_0$.
• Figure 3: The dependency structure of the random variables. The arrows exiting from $\zeta_k$ indicate that the variables $\eta_k$ and $\zeta_{k+1}$ can be constructed using the i.i.d variables.

Theorems & Definitions (33)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Theorem 3.4
• proof
• Lemma 3.5
• proof