[Paper Review] The combinatorics of hopping particles and positivity in Markov chains

Lauren Williams|arXiv (Cornell University)|Feb 1, 2022

Advanced Combinatorial Mathematics4 citations

TL;DR

This paper explores the stationary distribution of the asymmetric simple exclusion process (ASEP) with open boundaries, showing it can be expressed via combinatorial objects like staircase tableaux and multiline queues, and connects it to special functions such as Askey-Wilson and Macdonald polynomials. A key contribution is the demonstration that while the Markov Chain Tree Theorem yields a manifestly positive formula for the stationary distribution, it is often not compact, whereas combinatorial formulas (e.g., via tableaux) are compact but may involve large ratios of terms.

ABSTRACT

The asymmetric simple exclusion process (ASEP) is a model for translation in protein synthesis and traffic flow; it can be defined as a Markov chain describing particles hopping on a one-dimensional lattice. In this article I give an overview of some of the connections of the stationary distribution of the ASEP to combinatorics (tableaux and multiline queues) and special functions (Askey-Wilson polynomials, Macdonald polynomials, and Schubert polynomials). I also make some general observations about positivity in Markov chains.

Motivation & Objective

• To investigate the stationary distribution of the open-boundary ASEP using combinatorial and algebraic methods.
• To establish connections between the ASEP's stationary distribution and special functions, including Askey-Wilson, Macdonald, and Schubert polynomials.
• To analyze the positivity and compactness of stationary distribution formulas derived from the Markov Chain Tree Theorem and combinatorial objects.
• To compare the complexity of different representations of the stationary distribution, particularly in terms of the number of terms and common factors.
• To clarify the conditions under which stationary distribution formulas are manifestly positive or compact, and to explore the implications for Markov chain analysis.

Proposed method

• The stationary distribution of the ASEP is derived using the global balance equations and expressed as unnormalized probabilities Ψ(τ) proportional to the stationary measure.
• Combinatorial formulas for Ψ(τ) are constructed via staircase tableaux and multiline queues, which yield compact, manifestly positive expressions.
• The Markov Chain Tree Theorem is applied to express the stationary distribution as a sum over rooted spanning trees, yielding a manifestly positive but often non-compact formula.
• The ratio between the tree-based and tableaux-based formulas is analyzed, revealing that the latter is more compact and has significantly fewer terms.
• Polynomial factorization is used to extract common factors from tree-based formulas, leading to a compact representation that may no longer be manifestly positive.
• Theoretical analysis compares the structure of stationary distribution formulas across different combinatorial models, emphasizing positivity and minimality.

Experimental results

Research questions

• RQ1 Can the stationary distribution of the open-boundary ASEP be expressed in a manifestly positive way using combinatorial objects like tableaux or multiline queues?
• RQ2 How do the number of terms in the stationary distribution formula derived from the Markov Chain Tree Theorem compare to those from combinatorial tableaux constructions?
• RQ3 Is there a polynomial factor common to all unnormalized stationary probabilities in the tree-based formula, and what does its removal reveal about compactness?
• RQ4 Why does the Markov Chain Tree Theorem produce a formula that is manifestly positive but not compact, while combinatorial constructions yield compact but non-manifestly positive forms?
• RQ5 What is the nature of the ratio between the tree-based and tableaux-based formulas, and how does it grow with system size n?

Key findings

• The stationary distribution of the open-boundary ASEP is manifestly positive when expressed via staircase tableaux, with all coefficients in the unnormalized probabilities Ψ(τ) being positive.
• The number of terms in the tree-based formula (from the Markov Chain Tree Theorem) grows rapidly with n, and for n=6, the ratio of terms between the tree and tableaux formulas exceeds 10^25.
• The unnormalized probabilities derived from the Markov Chain Tree Theorem are not compact, as they share a common factor (e.g., (q+1) in Example 4.2), which can be factored out to yield a more compact form.
• After removing the common factor, the compact formula may no longer be manifestly positive, as seen in the reduced unnormalized probabilities in Table 5, which contain negative coefficients when expanded.
• For n=3, the partition function Z₃(α,β,q) is a polynomial with 24 terms (counted with multiplicity), and it is manifestly positive, supporting the conjecture that Zₙ is a positive polynomial with (n-1)! terms.
• The ratio Qₙ(α,β,q) = Ψ_tree(τ)/Ψ_tab(τ) is a polynomial in α,β,q, and for n=6, Q₆(1,1,1) is approximately 1.13 × 10^25, indicating a massive increase in complexity when using the tree-based formula.