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Pfaffian point processes for coalescing particles via checkerboard duality

Piotr Śniady

Abstract

Coalescing particles on a line merge when they meet. When every site is initially occupied, only finitely many particles survive at any positive time, and their positions form a Pfaffian point process: all correlation functions are determined by pairwise quantities arranged in antisymmetric matrices. Previous proofs of this structure relied on analytic methods specific to time-homogeneous dynamics. We identify the checkerboard duality as the structural reason for the Pfaffian: on a discrete planar graph, binary random choices create two complementary non-crossing forests, one tracing ancestral lineages backward and the other carrying the coalescing particles forward as domain boundaries. This duality converts the absence of particles in an interval into coalescence of ancestral lineages at its endpoints. A cancellative labeling then converts coalescence into annihilation, for which a companion paper provides a Pfaffian formula. The resulting empty-interval formula holds for any discrete graph with the checkerboard structure and arbitrary inhomogeneous edge probabilities, requiring no symmetry or specific distributions. This covers settings beyond existing methods, including totally asymmetric dynamics and position-dependent transition rules, and yields an explicit Pfaffian point process in each setting. For Brownian motion, the formula recovers the known Pfaffian point process and the empty-interval probabilities previously derived by PDE methods.

Pfaffian point processes for coalescing particles via checkerboard duality

Abstract

Coalescing particles on a line merge when they meet. When every site is initially occupied, only finitely many particles survive at any positive time, and their positions form a Pfaffian point process: all correlation functions are determined by pairwise quantities arranged in antisymmetric matrices. Previous proofs of this structure relied on analytic methods specific to time-homogeneous dynamics. We identify the checkerboard duality as the structural reason for the Pfaffian: on a discrete planar graph, binary random choices create two complementary non-crossing forests, one tracing ancestral lineages backward and the other carrying the coalescing particles forward as domain boundaries. This duality converts the absence of particles in an interval into coalescence of ancestral lineages at its endpoints. A cancellative labeling then converts coalescence into annihilation, for which a companion paper provides a Pfaffian formula. The resulting empty-interval formula holds for any discrete graph with the checkerboard structure and arbitrary inhomogeneous edge probabilities, requiring no symmetry or specific distributions. This covers settings beyond existing methods, including totally asymmetric dynamics and position-dependent transition rules, and yields an explicit Pfaffian point process in each setting. For Brownian motion, the formula recovers the known Pfaffian point process and the empty-interval probabilities previously derived by PDE methods.
Paper Structure (31 sections, 5 theorems, 23 equations, 2 figures)

This paper contains 31 sections, 5 theorems, 23 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Why Pfaffians?
  3. Checkerboard duality
  4. Main results
  5. The empty-interval formula
  6. Brownian motion
  7. Comparison with analytic approaches
  8. Related work and organization
  9. The checkerboard lattice and duality
  10. The dual forests
  11. The lattice and the voter model
  12. The backward opinion forest
  13. The forward boundary forest
  14. Checkerboard duality
  15. Cancellative labeling (coalescence to annihilation)
  16. ...and 16 more sections

Key Result

Theorem 1

On diagonal $u + v = T$, let $a_1, b_1, \ldots, a_n, b_n$ be $\mathbb{Z}'^2$-vertices with $a_1 < b_1 < a_2 < \cdots < a_n < b_n$. Then where $A$ is the $2n \times 2n$ antisymmetric matrix indexed by the $2n$ endpoints $a_1, b_1, \ldots, a_n, b_n$. For $k < l$, the entry $A_{kl}$ is the probability that independent backward particles from endpoints $k$ and $l$ cross or meet Sniady2026annihilation

Figures (2)

  • Figure 1: Coalescing random walks starting from every site of a lattice segment. Paths merge on contact; the surviving population thins over time.
  • Figure 2: Checkerboard duality on the $(u,v)$ lattice. At each $\mathbb{Z}'^2$-vertex, a random binary choice (copy from West or South) determines two complementary forests. The backward opinion forest (double blue arrows, on $\mathbb{Z}'^2$) traces ancestral lineages in the direction of decreasing $u+v$. The forward boundary forest (on $\mathbb{Z}^2$) has thick red arrows at boundary vertices (where neighboring opinions differ) and thin red arrows at non-boundary vertices. The thick diagonal line marks $u + v = 0$, where all opinions are initially distinct and every $\mathbb{Z}^2$-vertex carries a boundary. An interval on a later diagonal is free of boundaries if and only if the backward lineages from its endpoints share a common ancestor (\ref{['prop:checkerboard-duality']}).

Theorems & Definitions (12)

  • Theorem 1: Pfaffian empty-interval formula
  • Proposition 1: Checkerboard duality
  • proof
  • Proposition 2: Pairwise coalescence equals total annihilation
  • proof
  • Theorem 2: Pfaffian empty-interval formula
  • proof
  • Remark 1: Touching intervals
  • Proposition 3: Discrete Pfaffian point process
  • proof
  • ...and 2 more