Coalescing random walks via the coalescence determinant

Abstract

When identical particles on a line collide, they merge and continue as one. Exact determinantal formulas have long been available for particles conditioned never to collide, but collisions change the number of particles, and exact distributions for the survivors have been obtained only in specific settings and by ad hoc methods. Building on the coalescence determinant introduced in a companion paper, we study the wall-particle system: the joint system of survivors and the boundaries between their basins of attraction, starting from every site occupied. Its finite-dimensional distributions are determinants of block matrices built from transition probabilities and their cumulative sums; a finite block matrix suffices even when the initial configuration is infinite. As applications, we recover the Rayleigh spacing density and the joint distribution of consecutive gaps-which are negatively correlated-by new methods, and give a new derivation of the determinantal formula for the joint CDF of finitely many coalescing particles starting from fixed positions. All formulas hold for arbitrary nearest-neighbor random walks and their Brownian scaling limits, with no specific transition kernels required.

Figures (5)

• 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: The $(\mathbf{X}, \mathbf{Y})$ system for coalescing random walks. Paths coalesce on meeting; line weight increases with each merger. Bottom: basin boundaries $\mathbf{X} = (\ldots, \mathbf{x}_{-1/2}, \mathbf{x}_{1/2}, \mathbf{x}_{3/2}, \ldots)$ (triangles) partition the initial line, with $\mathbf{x}_{-1/2} < 0 \leq \mathbf{x}_{1/2}$. Top: survivor positions $\mathbf{Y} = (\ldots, \mathbf{y}_{-1}, \mathbf{y}_0, \mathbf{y}_1, \mathbf{y}_2, \ldots)$ (circles), one per basin.
• Figure 3: Proof of \ref{['thm:xy-correlation']} for $k = 2$. The coalescence determinant applies to the four flanking particles $a_1, b_1, a_2, b_2$ (bold paths: solid for pair $1$, zigzag for pair $2$). Particles $b_1$ and $a_2$ coalesce into survivor $y_1$ (double line); particles $a_1$ and $b_2$ survive as $y_0$ and $y_2$. The intermediate particles (thin gray) cannot cross the flanking paths---the skip-free property traps them in the closing funnel between $b_1$ and $a_2$---so they are absorbed into the same survivors. Adding them does not change the coalescence outcome for the flanking particles.
• Figure 4: Block structure of $\tilde{M}$ for the $1{+}2{+}2{+}1$ pattern ($k = 3$ walls). Rows come in pairs; columns are grouped by survivor: $2 \times 1$ boundary blocks for $y_0$ and $y_3$, and $2 \times 2$ interior blocks for $y_1$ and $y_2$, each containing a $P$ column and an $F$ column. The thick red staircase separates $F{-}1$ blocks (dark blue, solid hatching) from $F$ blocks (orange, dashed hatching); unhatched yellow blocks contain only $P$ entries.
• Figure 5: Joint gap intensity $h(G_1, G_2)$ from \ref{['thm:joint-gap']}, computed by numerical integration. The tilted elliptical contours reflect negative correlation ($\rho \approx -0.163$).

Theorems & Definitions (37)

• Theorem 1: Wall-particle correlation function
• Theorem 2: Discrete gap intensity measure
• Theorem 3: Single gap intensity measure
• Theorem 4: Joint gap intensity
• Definition 2.1: Coalescence matrix
• Theorem 5: Coalescence determinant Sniady2026coalescence
• Example 1: Pattern $2 + 1$
• Remark 1: Labeling convention
• Theorem 6: Wall-particle correlation function
• proof