[Paper Review] A Markov chain on permutations which projects to the PASEP

Sylvie Corteel, Lauren Williams|ArXiv.org|Sep 6, 2006

Stochastic processes and statistical mechanics13 references4 citations

TL;DR

This paper introduces the PT chain, a Markov chain on permutation tableaux that projects to the partially asymmetric exclusion process (PASEP), providing a combinatorial, matrix ansatz-free proof of the PASEP's stationary distribution as a generating function of permutation tableau weights. The PT chain reveals a hidden symmetry extending the PASEP's particle-hole duality and offers a new perspective on the algebraic combinatorics underlying the PASEP via bijections to the symmetric group.

ABSTRACT

The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of N sites. It is partially asymmetric in the sense that the probability of hopping left is q times the probability of hopping right. Additionally, particles may enter from the left with probability alpha and exit from the right with probability beta. It has been observed that the (unique) stationary distribution of the PASEP has remarkable connections to combinatorics -- see for example the papers of Derrida, Duchi and Schaeffer, and Corteel. Most recently we proved that in fact the (normalized) probability of being in a particular state of the PASEP can be viewed as a certain weight generating function for permutation tableaux of a fixed shape. (This result implies the previous combinatorial results.) However, our proof relied on the matrix ansatz of Derrida et al, and hence did not give an intuitive explanation of why one should expect the steady state distribution of the PASEP to involve such nice combinatorics. In this paper we define a Markov chain -- which we call the PT chain -- on the set of permutation tableaux which projects to the PASEP in a very strong sense. This gives a new proof of our previous result which bypasses the matrix ansatz altogether. Furthermore, via the bijection from permutation tableaux to permutations, the PT chain can also be viewed as a Markov chain on the symmetric group. Another nice feature of the PT chain is that it possesses a certain symmetry which extends the "particle-hole symmetry" of the PASEP. More specifically, this is a graph-automorphism on the state diagram of the PT chain which is an involution; this has a simple description in terms of permutations.

Motivation & Objective

• To provide a new, combinatorial proof of the stationary distribution of the PASEP without relying on the matrix ansatz.
• To define a Markov chain on permutation tableaux (the PT chain) that projects to the PASEP, preserving its dynamics.
• To uncover a deeper symmetry in the PASEP's structure by extending the particle-hole duality via an involution on the PT chain.
• To establish a direct connection between the stationary probabilities of the PASEP and the weight-generating function of permutation tableaux of fixed shape.

Proposed method

• Define the PT chain as a Markov process on permutation tableaux, where transitions correspond to particle movements (entry, exit, hops) in the PASEP.
• Construct a surjective projection map from permutation tableaux to PASEP states, ensuring that the PT chain’s dynamics project faithfully to the PASEP’s dynamics.
• Use a bijection between permutation tableaux and permutations (via the map Φ) to reinterpret the PT chain as a Markov chain on the symmetric group S_{N+1}.
• Define an involution on the state diagram of the PT chain that swaps particle and hole roles, generalizing the PASEP’s particle-hole symmetry.
• Assign weights to permutation tableaux as Laurent monomials in α, β, and q, such that the stationary probability of a tableau equals its weight.
• Verify that the transition probabilities of the PT chain are consistent with the projected PASEP dynamics and that the stationary distribution matches the known generating function.

Experimental results

Research questions

• RQ1 Can the stationary distribution of the PASEP be derived combinatorially without the matrix ansatz, via a Markov chain on permutation tableaux?
• RQ2 What is the structural relationship between the PASEP and permutation tableaux, and how can it be formalized as a projection of Markov chains?
• RQ3 Does the PT chain possess a symmetry that generalizes the known particle-hole symmetry of the PASEP?
• RQ4 How does the bijection between permutation tableaux and permutations (Φ) allow the PT chain to be interpreted as a Markov chain on the symmetric group?
• RQ5 What is the precise weight assignment to permutation tableaux that yields the correct stationary probabilities for the PASEP?

Key findings

• The PT chain is a Markov chain on permutation tableaux whose projection to the PASEP state space reproduces the original PASEP dynamics exactly.
• The stationary distribution of the PT chain assigns to each permutation tableau a probability equal to its weight, a Laurent monomial in α, β, and q.
• The PT chain provides a new, combinatorial proof of the main result from Corteel and Williams (2006), which previously relied on the matrix ansatz.
• The PT chain admits an involution that swaps the roles of particles and holes, generalizing the particle-hole symmetry of the PASEP.
• This involution corresponds to a natural transformation on permutation tableaux: conjugation combined with a complementation of entries and shape adjustment.
• Via the bijection Φ, the PT chain is isomorphic to a Markov chain on the symmetric group S_{N+1}, offering a new algebraic interpretation of the PASEP’s stationary distribution.