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Exact determinant formulas for coalescing particle systems

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Pith's one-line read Coalescing particles on a line have exact determinant formulas for every prescribed collision pattern.

desk verdict The ghost method is real and the discrete proof is serious, but Theorem 8.2 as stated is false on reversed-order configurations; that is a fixable support error, not a fatal flaw. read the letter →

arxiv 2602.10782 v3 pith:VS47BDPA submitted 2026-02-11 math.PR math.CO

Piotr Śniady , Ákos Urbán This is my paper · ORCID
classification math.PRmath.CO MSC 05A1505A1915A1560C0560J6582C22
keywords coalescingrandomwalksghostparticlesdeterminantalformulaslatticepathsbirth-deathchainsBrownianmotionvotermodelinteractingparticlesystems
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proves that coalescing particles on a line have exact, closed-form probability formulas for every specified collision pattern—which initial particles merge into which survivors. The trick is to add an invisible 'ghost' particle at each collision, keeping the total count fixed so that a square matrix can be built. The resulting coalescence determinant has transition probabilities in survivor columns and cumulative distributions with a staircase shift in ghost columns. If true, this gives pattern-level exact probabilities wherever the classical non-colliding determinant formula applies: lattice walks, birth-death chains, and continuous diffusions such as Brownian motion.

What carries the argument

The ghost particle method: when two particles collide, one survivor continues as usual and one invisible ghost starts from the collision point as an independent non-interacting walk. This keeps the number of entities at n, so the transition matrix stays square. Ghost columns carry a staircase sign pattern (t+ below the row rank, -t- above), and coefficient extraction selects exactly the terms matching the prescribed collision pattern. Failed path assignments cancel pairwise by segment-swapping at the first spurious crossing, leaving only successful castings; integrating out ghost positions converts transition-probability columns into cumulative-distribution columns.

What would settle it

Enumerate all coalescing path tuples for three simple random walks on Z starting at 0, 1, 2, with the 2+1 coalescence pattern, up to a fixed small time; compute the survivor-position probabilities by exhaustive counting and compare with the determinant formula evaluated at the same transitions. A single mismatch would disprove the discrete claim.

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Extended reading notes

Core claim

The central result is the coalescence determinant: for a composition c1+...+ck=n, the probability that the k survivors land in a measurable set A equals ∫_A det(M̃(y1,...,yk)) dν. The matrix M̃ has transition densities or probabilities in the first column of each block and cumulative distributions F_{x_i}(y_l) - [i<j] in the remaining columns of the block. The same paper proves a finer ghost formula that keeps ghost positions as formal variables; marginalizing over ghost positions converts the transition-probability columns into cumulative columns and yields the coalescence determinant. The proof works by adding one invisible ghost per collision so the total particle count stays n, then canc

Load-bearing premise

The construction assumes particles on a line cannot pass each other without meeting, and that after two particles meet, swapping their future trajectories does not change the probabilities—if either assumption fails, the cancellation at the heart of the determinant identity breaks down.

Editorial extensions

If this is right

  • Any specified coalescence pattern has a closed-form probability that can be evaluated by one determinant, with no simulation or recursion over collision orders.
  • The formula holds for all skip-free Markov processes satisfying planarity and the strong Markov property, so it applies to lattice walks, birth-death chains, and Brownian motion alike.
  • When no collisions occur, the ghost determinant reduces to the classical non-colliding determinant, making the formula a genuine extension of that theory.
  • Summing the pattern-level probabilities over all compositions recovers the previously known determinant for the joint distribution of coalescing Brownian survivors, but with finer information about which particles merged.
  • The measure form of the theorem, valid for both discrete and continuous state spaces, identifies the determinant as a Radon-Nikodym derivative ready for further integration or scaling limits.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Beyond the paper, the same ghost bookkeeping could be applied to absorbing boundaries: replacing the transition kernel by a killed kernel in the staircase columns would give a coalescence determinant for finite intervals, a testable variant.
  • Beyond the paper, the measure identity suggests an exact sampling algorithm: draw survivor positions from the determinant density rather than simulating collisions, which could be considerably faster for large n.
  • Beyond the paper, the density-versus-cumulative column pattern may encode a general operation—'merge rows into survivor columns'—that could extend the formula to higher dimensions or to long-range coalescence wherever a planar cancellation exists.
  • Beyond the paper, the ghost coefficient-extraction formalism may transfer to other reactions, such as annihilation or branching, whenever a sign-reversing involution can be defined on path tuples.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. The paper introduces a 'ghost particle' method for coalescing particle systems on a line. When two particles coalesce, one heir continues and one ghost random walk is created, preserving the total particle count. This restores a square matrix structure and allows the authors to state a determinant formula for the probability of specified coalescence patterns. The main results are Theorem 3.2, a discrete generating-function identity for coalescence performances on planar weighted DAGs; Theorem 7.2, a continuous-time measure identity under Karlin–McGregor assumptions; and Theorem 8.2, the 'coalescence determinant', which marginalizes ghosts and gives a closed-form determinant for survivor positions alone, intended to cover Brownian motion, birth-death chains, and lattice walks.

Significance. The ghost-particle idea is attractive and, if the measure-theoretic statements are corrected, would extend the Karlin–McGregor/LGV determinant formalism to coalescing systems at pattern-level resolution. The discrete proof in Sections 4–6 is self-contained and appears internally consistent: it constructs an explicit sign-reversing involution on castings, has no fitted parameters, and reduces to the classical LGV determinant in the no-coalescence case. The n=2 coalescence example checks out. However, the central continuous and ghost-free statements, as written, fail on non-ordered measurable sets, and the continuous-time proof is only a sketch. These issues are load-bearing for the paper's main claims, though they appear repairable within the manuscript's scope.

major comments (2)
  1. [Theorem 8.2 (§8.3)] Theorem 8.2 asserts P_int(survivor positions ∈ A)=∫_A det(̃M) dν for every measurable A⊆S^k. This is false without an ordering restriction. For n=2, k=2, composition 1+1, ̃M=[p_{x_i}(y_j)]. Take x1<x2 and A={y1>y2}. In the coalescing system two survivors cannot have swapped order, so the left-hand side is 0. The right-hand side is ∫_{y1>y2} det[p_{x_i}(y_j)] dν = P(X1>X2)-P(X1<X2), which is strictly negative for Brownian motion. The marginalization proof in §8 integrates out ghosts but never enforces the chamber y1<...<yk. The theorem must restrict A to that ordered chamber, or include the indicator 1_{y_1<...<y_k} in the integrand. Since the paper explicitly calls det(̃M) a probability density/mass function, this is a statement-level defect, not a cosmetic one.
  2. [Theorems 7.1–7.2 (§7.2, §7.3)] The same support issue appears in the continuous-time ghost formula. Definition 7.1 constrains ghosts relative to their heirs but does not constrain heir positions among themselves; when G=∅ the admissibility condition is vacuous. Taking A={y1>y2} in the two-particle no-coalescence case gives an admissible set for which the RHS of Theorem 7.2 is P(X1>X2)-P(X1<X2)<0 while the LHS is 0. Thus the claimed 'identity between measures' holds only on ordered survivor configurations. In addition, the proof sketch in §7.3 is substantially less complete than the discrete proof: the segment-swap map on the casting space must be shown to be a measurable, measure-preserving, sign-reversing involution, and the claim that each outcome with final positions in A has exactly one successful π needs a continuous analogue of Proposition 5.11 with the same tie-breaking conventions. The paper should either prov
minor comments (5)
  1. [§7.2] The notation 'A⊆R R' in Definition 7.1 appears to be a typo; it should be a subset of the appropriate product state space, e.g. S^R or S^k.
  2. [Abstract and §1.5] The abstract and introduction emphasize 'nearest-neighbor transitions', while Theorem 3.2 is stated for general planar weighted DAGs and §7 covers skip-free birth-death chains. The terminology should be harmonized (e.g., 'skip-free' or 'order-preserving').
  3. [Figure 2 caption] The caption phrase 'heir column below step (i≥rank) above step (i<rank)' is confusing; the figure illustrates ghost columns, not heir columns, and the caption should be reworded to match the matrix display.
  4. [§1.3.2 vs §3.1] The coefficient extraction notation [t^+], [t^-] is used in the introductory example before the formal definition in §3.1. Moving the formal definition earlier, or adding a forward reference, would improve readability.
  5. [§8.1] For discrete state spaces, the matrix entries F_{x_i}(y_l)-[i<j] can be negative, so det(̃M) is a signed mass function on arbitrary sets. Please state explicitly that the formula is a signed measure identity and is a genuine probability mass/density only on the ordered chamber.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the determinant formulas are proved by an independent sign-reversing involution; self-citations are not load-bearing.

full rationale

The central claim (Theorem 3.2, proved in Sections 4–6) is not circular: det(M) is expanded as a signed sum over castings, coefficient extraction defines candidate bijections, and a segment-swap involution cancels non-successful castings. The key lemmas—attribution produces candidates (Prop. 4.7), fixed points are exactly successful castings (Lemma 5.2), the sign identity (Prop. 6.1), and the performance–casting bijection (Prop. 5.11)—do not assume the target probability. No fitted parameter is relabelled as a prediction. Theorem 8.2 is derived from Theorem 3.2 by integrating out ghost positions; the staircase entries F and F−1 are computed explicitly from transition probabilities rather than imposed to match the answer. The cited works by the author and Urbán are used for comparisons, applications, or context, not as premises in the proof; Warren's formula and Karlin–McGregor are external results, and the ghost-free theorem does not reduce to them. The continuous-time proof (Section 7.3) depends on an unproved measure-preservation claim for the segment-swap map, and the statement of Theorem 8.2 for arbitrary measurable A appears to need an order restriction (e.g., the two-particle example where the right-hand side is negative on a reversed-order set). These are correctness concerns, not circularity: the claimed reduction is still from the same determinant/casting expansion, and no equation is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No free parameters appear; the determinants are parameter-free and hold for arbitrary transition probabilities. The axioms are the standard Markov-process and planarity assumptions needed for the Karlin–McGregor framework, plus the new planarity condition (P2). Ghosts are an invented combinatorial entity that integrates out of the final result.

assumptions (4)
  • domain assumption Strong Markov property for the underlying one-particle process
    Used in Sections 5.3 and 7.1 to justify segment-swap measure preservation and the independence of post-meeting trajectories.
  • domain assumption Order preservation / skip-free transitions (nearest-neighbor)
    Ensures the consecutive collision property (P2) and that particles cannot cross without meeting; stated as a key assumption in the abstract and Definition 2.5.
  • domain assumption Planarity properties (P1) crossing property and (P2) consecutive collision property for the spacetime graph
    Definition 2.5; the entire sign-reversing involution relies on these geometric properties.
  • domain assumption Meeting times are stopping times
    Karlin-McGregor assumption (3) in Section 7.1, needed for the strong Markov property at first crossings.
invented entities (1)
  • Ghost particle
    purpose: Preserves the total particle count after a coalescence so the matrix remains square; each collision creates one ghost whose position is later integrated out.
    The ghost is a bookkeeping device internal to the proof; it has no physical meaning and is fully integrated out in the final formula (Theorem 8.2). It is not a new physical entity but a mathematical construction.

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Cite this review

Pith. "Pith review of Exact determinant formulas for coalescing particle systems." pith.science (2026). https://pith.science/paper/VS47BDPA

@misc{pith2026260210782,
  author       = {Pith},
  title        = {Pith review of: Exact determinant formulas for coalescing particle systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VS47BDPA}},
  note         = {Machine review of arXiv:2602.10782}
}
read the original abstract

When particles on a line collide, they may coalesce into one. Such systems arise in the voter model, where boundaries between opinion clusters perform coalescing random walks, and in reaction-diffusion theory, where diffusing particles merge on contact. Computing exact coalescence probabilities has been difficult because collisions reduce the particle count, while classical determinantal methods require a fixed number of particles throughout. We introduce ghost particles: when two particles collide, one survivor continues as usual and one invisible ghost is created alongside it, preserving the total count. This restores the square matrix structure needed for a determinantal formula. We prove that the probability of any specified coalescence pattern - which initial particles merge into which survivors - is given by a determinant whose entries are transition probabilities. Integrating out ghost positions yields a closed-form formula for the surviving particles alone: the coalescence determinant. The only assumptions are the Markov property and nearest-neighbor transitions, so the results apply wherever the classical non-colliding theory does: discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.

Figures

Figures reproduced from arXiv: 2602.10782 by the authors.

Figure 1
Figure 1. Coalescence on the checkerboard lattice (pattern 2+1). Three particles start at x1 < x2 < x3. Particles 1 (solid) and 2 (double) coalesce at c, producing heir 1 (ticked) and a ghost (dotted). Particle 3 (zigzag) does not coalesce; it is heir 2. The ghost shares two edges with heir 2 (shown offset)—ghosts do not interact and may cross any path freely. Final positions: y1 < y2 < z. where p(xi → yj ) is the transition … view at source ↗
Figure 2
Figure 2. The coalescence matrix: staircase structure. Ma￾trix M for the 2+2 coalescence pattern: particles 1 and 2 merge into heir H1; particles 3 and 4 merge into heir H2. Heir columns (yellow) contain transition probabilities p(xi → Hj ). Ghost columns show the staircase pattern: entries with row index i < rank(g) (blue, solid lines) have −p · t −; entries with i ≥ rank(g) (orange, dashed lines) have p · t +. The thick sta… view at source ↗
Figure 3
Figure 3. Structure of the companion papers. · · · · · · · · · · · · time t = 0 t > 0 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: Coalescing random walks starting from every site. Paths [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: Running example: coalescence pattern 3+1. A performance records the collision structure on a spacetime graph— here the lattice Z 2 with North/East steps. Four particles start at xI1 , xI2 , xI3 , xI4 (leaves, colored dots). Particles I2 and I3 meet at z3, merging into …
Figure 6
Figure 6. Figure 6: Interval labeling and the final state. Particles are labeled by unit intervals; final entities by intervals (heirs) or junction points (ghosts). This diagram shows the same example as [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: The consecutive collision property. A forbidden configuration: paths P1 and P3 meet at v and continue as a trunk. Path P2 attempts to “tunnel” around P3 without crossing, then hit the trunk at w. Item (P2) in Definition 2.5 forbids this: P2 must cross P1 or P3 before v…
Figure 8
Figure 8. Figure 8: Successful casting: actor-based view of a perfor￾mance. Same collision as [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: Failed casting: a spurious crossing. Same endpoints as [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: Two-particle coalescence, case ε(g) = +1: ghost left of heir. (a) Schema: four distinct styles indicate entities do not persist past coalescence; the dashed line shows the ghost path. (b) Attribution via path gluing: the swap principle routes the particle from I − to …
Figure 11
Figure 11. Figure 11: Two-particle coalescence, case ε(g) = −1: ghost right of heir. (a) Schema: four distinct styles indicate entities do not persist past coalescence; the dashed line shows the ghost path. (b) Attribution via path gluing: the swap principle routes the particle from I + to…
Figure 12
Figure 12. Figure 12: Full coalescence, bijection π1: junction 3 fires first. Three particles coalesce into one heir H = [1, 4], creating ghosts at junctions 2 and 3. (a) Attribution: following each path through all collisions determines the bijection. Junction 3 fires first (I2 and I3 mee…
Figure 13
Figure 13. Figure 13: Full coalescence, bijection π2: junction 2 fires first. Same final state as [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: Segment swap: the sign-reversing involution. (a) Paths P1 (solid) and P2 (double) cross at vertex c. (b) After the swap, final segments are exchanged: P ′ 1 follows P1 to c, then P2’s tail to y2; P ′ 2 follows P2 to c, then P1’s tail to y1. The bijection updates to π …
Figure 15
Figure 15. Figure 15: An unsuccessful candidate: π3 admits no suc￾cessful casting. The bijection π3: I1 7→ 3, I2 7→ H, I3 7→ 2 satisfies the candidacy condition but admits no successful casting. (a) Any path family has a spurious crossing—the middle particle I2 must collide before reaching…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Determinant and Pfaffian formulas for particle annihilation

    math.PR 2026-02 conditional novelty 7.0 of 10

    Annihilation probabilities for colliding particles on a line are expressed exactly as a determinant with ghost-pair formal variables, collapsing to a Pfaffian for complete annihilation.

Reference graph

Works this paper leans on

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