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Determinant and Pfaffian formulas for particle annihilation

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read Ghost pairs make annihilating particles exactly solvable on a line: every outcome probability is a single determinant, and complete extinction is a Pfaffian.

desk verdict New ghost-pair determinant and Pfaffian for annihilation; the central proof does not cover its advertised random-walk case because the consecutive-collision axiom P2 is false for that graph. read the letter →

arxiv 2602.13183 v3 pith:UANSBTFD submitted 2026-02-13 math.PR math.CO

Piotr Śniady This is my paper · ORCID
classification math.PRmath.CO MSC 05A1505A1915A1560C0560J6582C22
keywords annihilatingrandomwalksghostparticlesdeterminantalformulaPfaffianpairwisecoalescenceplanarspacetimegraphexactfinite-timeprobabilities
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

Annihilating particles on a line have resisted exact probability calculations because every collision removes two walkers, leaving more starting points than final positions. This paper establishes that if destroyed particles keep walking as invisible ghost pairs, the number of trajectories stays fixed and every prescribed outcome—how many collisions, where survivors end, where ghosts end—has probability given by one determinant. For complete extinction, the determinant collapses to a Pfaffian built solely from pairwise annihilation probabilities, explaining the Pfaffian structure earlier work found through differential equations. The same formula yields an exact Pfaffian for the event that prescribed consecutive pairs coalesce. If correct, it gives finite-time, configuration-level probabilities uniformly for lattice walks, birth-death chains, and diffusions including Brownian motion.

What carries the argument

The central object is the ghost pair: when two particles annihilate, both trajectories continue as anonymous invisible walkers joined as a numbered pair, so the entity count never drops below n. This restores the square-matrix structure needed for determinant methods. The proof operates on a planar directed acyclic graph with a crossing property and a consecutive-collision property, using a sign-reversing involution that swaps path segments at the first spurious crossing; the consecutive-collision property guarantees the crossing particles are adjacent, so the swap preserves the coefficient-extraction constraints.

What would settle it

On a small planar lattice, e.g., four walkers on Z with T = 3 steps, enumerate all annihilation evolutions directly and compare the right-hand side of Theorem 3.1 for every final state; a single mismatch refutes the formula. A sharper test targets the hidden axiom: construct a planar DAG satisfying the crossing property but violating the consecutive-collision property and check whether the first-crossing involution still cancels failed castings; if a pairing of non-adjacent particles survives, the determinant includes spurious terms.

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Extended reading notes

Core claim

On a planar spacetime graph with n walkers, fix k annihilations, s = n−2k survivors at prescribed positions, and k ghost-pair endpoints. The paper proves that the probability of this exact outcome equals 1/k! times the coefficient of formal variables t_j^{±1} in det(M), where M's survivor columns are transition weights and its ghost columns carry variables whose product rule records which initial particle is higher-indexed. The proof's sign-reversing involution cancels all nonphysical assignments pairwise, leaving exactly the physical performances. When s = 0, marginalizing over ghost positions collapses the determinant into Pf(A), the Pfaffian of pairwise annihilation weights. The same mech

Load-bearing premise

The consecutive-collision property—whenever paths from two non-adjacent starting positions meet, every intermediate path must cross one of them before the meeting—is load-bearing for the sign-reversing involution; the paper cites its verification for lattice walks, birth-death chains, and Brownian motion to a companion paper rather than proving it here.

Editorial extensions

If this is right

  • Exact finite-time probabilities for any prescribed annihilation outcome on any process satisfying the two planarity axioms: lattice walks, birth-death chains, and Brownian motion.
  • Complete extinction is determined by pairwise annihilation probabilities alone, so many-particle extinction becomes computable from two-particle data.
  • The Pfaffian coalescence formula follows from an annihilation theorem, giving exact probabilities for prescribed pairwise mergers and connecting coalescence and annihilation at the configuration level.
  • The determinant-to-Pfaffian collapse explains Pfaffian point-process structure for annihilating systems by a single combinatorial argument, without differential equations.
  • For Ising–Glauber domain walls and A + A → ∅ reactions, outcome-level probabilities—not just densities or correlations—are now available.

Reading between the lines

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

  • If the consecutive-collision property fails but the crossing property holds, the involution may still pair first crossings; testing the determinant formula on non-nearest-neighbor walks would show whether the adjacency hypothesis is truly necessary or just a proof convenience.
  • The appendix's n = 3 inconsistency suggests a structural barrier: no Karlin–McGregor-style expression can specify which particles annihilated, so exact identity of annihilating pairs is not a determinantal observable in this framework—a warning for any attempt to refine ghost labels.
  • Because marginalizing ghost positions turns the determinant into a Pfaffian only when survivors are absent, partial annihilation formulas have mixed determinant/Pfaffian structure; similar partial-collapse identities may hold in Pfaffian point process theory.
  • The method's applicability to arbitrary space-time-varying transition weights suggests exact formulas for disordered or inhomogeneous systems where no generator-based approach is available.
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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 / 4 minor

Summary. The paper develops a combinatorial 'ghost pair' method for annihilating particles on a line. When two particles annihilate, both trajectories continue as invisible ghosts, so the number of trajectories remains n and a square determinant can be written. Theorem 3.1 expresses the weight of a fixed final state—survivor positions and ghost-pair positions—as a coefficient of a determinant whose ghost columns carry formal variables. The proof is organized through castings, attribution, rehearsal, and a sign-reversing involution. Section 5 uses a cancellative labeling to convert pairwise coalescence into complete annihilation and derives a Pfaffian formula, with applications to biased random walks on Z. The paper is self-consciously combinatorial and positions itself as complementary to the analytic Pfaffian results of Tribe–Zaboronski and Garrod et al.

Significance. If the central theorem is correct, it gives exact finite-time probabilities for prescribed annihilation outcomes, generalizes the Karlin–McGregor/LGV determinant to a setting with a varying number of visible particles, and provides a purely combinatorial explanation of the Pfaffian structure of annihilating systems. The manuscript has real strengths: the casting/rehearsal/involution architecture is spelled out in detail, the two- and four-particle examples are helpful, there are no fitted parameters, and the paper is honest about the role of ghost anonymity and about the computational nature of Appendix A. However, the consecutive-collision axiom P2 is both essential to the proof and, as stated, false for a graph used in the paper's own applications. Until that axiom is corrected and verified, the advertised scope is not covered by the proof.

major comments (2)
  1. [Definition 2.4; Lemma 4.15; Proposition 4.18; Corollary 5.5] The consecutive collision property (P2) is false as stated for the spacetime graph used in Corollary 5.5. Let vertices be (t,z) with edges (t,z) -> (t+1,z±1). Take sources x=-1, x'=0, x''=1. The paths P1: (0,-1)->(1,0) and P3: (0,1)->(1,0) meet at v=(1,0). The path P2: (0,0)->(1,1)->(2,2) neither passes through v nor intersects P1 or P3 before time 1. Thus P2 fails. The proof uses P2 precisely where the involution needs the first crossing to be between adjacent active particles: Lemma 4.15 assumes 'I<J adjacent in the active set', and Proposition 4.18 says 'By the consecutive collision property (P2), they are adjacent in the active set'. Without a correct P2, the sign-reversing involution on a non-adjacent crossing can break candidacy for a ghost pair involving an intervening particle, so the cancellation argument in Sections 4.6-4.8 is not valid for that graph. Since Theorem 5.3 and Cor
  2. [Section 5.4, proof of Theorem 5.3] The derivation of the Pfaffian sign is not given. After marginalizing over ghost positions and ghost sign patterns, each perfect matching receives contributions from k! ghost-pair numberings and 2^k sign patterns. The text says that the 1/k! factor absorbs the numberings and that summing over matchings gives Pf(A), but it never shows the sign of an individual matching after the determinant sign and the formal-variable signs are combined. Since the entire combinatorial content of the Pfaffian is the alternating sign (see Example 5.4's counterterm A13A24), this step needs an explicit lemma or calculation. This is likely fixable, but as written Theorem 5.3 is not fully proved.
minor comments (4)
  1. [Section 3.1, Eq. (3.1)] The formal-variable algebra is under-specified. The paper should state explicitly that the variables are commutative (or define the noncommutative ordering rules), define t_j^+ and t_j^- as independent formal monomials, and give the relation for products such as τ^-_J τ^+_I that occur in the Leibniz expansion when row order is not the index order.
  2. [Example 5.4] The sentence 'The matching {(1,3),(2,4)} is not physically realizable' is too strong. Under the discrete-time random walk model of Corollary 5.5, particles 1 and 3 can meet at a vertex while particle 2 is elsewhere; the physical realizability claim fails unless an additional convention is imposed. Please rephrase this as a statement about cancellation in the Pfaffian expansion rather than an a priori physical impossibility.
  3. [Appendix A] The appendix reports an exact-arithmetic inconsistency but does not include the matrix or a reproducibility statement. Since the claim is used to support the statement that ghost anonymity is essential, please provide the system, the code, or at least the full set of equations checked.
  4. [Section 1.3.3 vs Section 5.4] The introduction announces a Pfaffian formula for complete annihilation, while Section 5 states and proves a Pfaffian formula for pairwise coalescence. The connection via the cancellative labeling should be stated more explicitly at the point where the complete-annihilation version is first introduced, so the reader knows the two formulations are the same identity.

Circularity Check

1 steps flagged · score 4.0 of 10

Discrete core is self-contained, but the advertised random-walk/Brownian scope rests on the same-author companion paper for the P2 axiom that powers the sign-reversing involution.

  1. self citation load bearing [Definition 2.4 and Section 2.4; used in Lemma 4.15 and Proposition 4.18; relevant to Corollary 5.5]
    "The consecutive collision property (P2), introduced in the companion paper [Śni26a], ensures that non-adjacent particles cannot collide without involving intermediate ones. Both properties hold for lattice paths, random walks on Z, birth-death chains, and Brownian motion; see [Śni26a] for verification."

    The proof of the annihilation formula needs P2 to run the sign-reversing involution: Lemma 4.15 assumes 'I < J adjacent in the active set,' and Section 4.6 adds 'The consecutive collision property (P2) guarantees that the first crossing among active paths always involves adjacent particles'; Proposition 4.18 again derives the essential sign identity from P2. The present paper supplies no verification of P2 for the advertised models, only a citation to the same author's companion paper. Thus the claimed applications to lattice paths, random walks, birth-death chains, and Brownian motion are not derived here; they are inherited from a self-citation, and the cancellation argument collapses if P2 is not granted. The determinant-to-Pfaffian algebra itself is not circular.

full rationale

The combinatorial core is a genuine bijection between performances and successful castings, followed by a weight-preserving, sign-reversing involution. No parameter is fitted, and no probability is defined as the determinant coefficient; instead, the coefficient of the determinant is proved to equal the performance generating function. Given planarity axioms P1 and P2, Theorem 3.1 follows from Lemmas 4.14-4.18 and Proposition 4.9 without reference to the companion paper. Theorem 5.3 is likewise derived from Theorem 3.1 plus the cancellative labeling of Griffeath and ben-Avraham-Brunet, not from the target Pfaffian, and it is benchmarked against known external results [TZ11; GPTZ18]. So there is no equation-level circularity or fitted-input-called-prediction. The main circularity-adjacent issue is the load-bearing self-citation for P2 and for the continuous-time extension: Section 1.6 and Section 2.4 defer the verification that makes the abstract's broad claims true to [Śni26a]. This is a real dependence on the author's own companion work. There is also an independent correctness risk: as literally stated, P2 appears false on the spacetime graph used in Corollary 5.5 (take x=-1, x'=0, x''=1; the paths -1->0 and 1->0 meet at v, while the path 0->1->2 avoids v and both earlier paths), so the advertised biased-random-walk application is not secured by this paper's proof. That is a correctness gap rather than a circular reduction, and it is why the score is 4 rather than higher.

Assumptions & free parameters 0 free parameters · 5 assumptions · 1 invented entities

The central derivation relies on the combinatorial spacetime-graph model and standard LGV theory; it introduces no fitted parameters. The notable load-bearing external inputs are: the companion paper's verification of planarity for concrete processes, and the cancellative-duality literature in Section 5. Ghosts are an invented auxiliary entity lacking independent falsifiable consequences.

assumptions (5)
  • domain assumption Planarity axioms (P1) crossing property and (P2) consecutive collision property (Definition 2.4).
    Used throughout Section 4; (P2) is essential for Lemma 4.15 and Proposition 4.18. Verification for lattice paths, random walks, birth-death chains, and Brownian motion is deferred to [Śni26a].
  • standard math Weight-preserving segment swap (Lemma 4.14(ii)).
    Follows from path weights being products of edge weights in a commutative ring; no new physical assumption.
  • domain assumption Cancellative duality / parity labeling (Griffeath, ben-Avraham–Brunet, Athreya–Swart).
    Used in Section 5.3 to convert pairwise coalescence into complete annihilation; cited to [Gri79], [AB05], [AS12] and not proved in the paper.
  • standard math Karlin–McGregor / Lindström–Gessel–Viennot determinant theorem.
    Used as the k=0 reduction of the annihilation formula (Section 3.3) and as the survival determinant in the Laplace expansion.
  • domain assumption Strong Markov property and meeting times being stopping times for continuous processes.
    Invoked in Section 1.6 for Brownian motion and birth-death chains; the continuous-time treatment is deferred to [Śni26a] and [KM59].
invented entities (1)
  • Ghost pairs (ordered pairs of invisible walkers emerging from an annihilation event)
    purpose: Restore the constant entity count n after annihilations remove particles, enabling an n×n determinantal formula.
    A mathematical device with no direct observable signature; its validity is judged through the derived formulas, not through an independent prediction.

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Cite this review

Pith. "Pith review of Determinant and Pfaffian formulas for particle annihilation." pith.science (2026). https://pith.science/paper/UANSBTFD

@misc{pith2026260213183,
  author       = {Pith},
  title        = {Pith review of: Determinant and Pfaffian formulas for particle annihilation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UANSBTFD}},
  note         = {Machine review of arXiv:2602.13183}
}
read the original abstract

We consider systems of particles on a line in which colliding particles annihilate each other and vanish. Computing exact annihilation probabilities is difficult because every collision reduces the particle count, while determinantal methods require a fixed count throughout. The ghost particle method, introduced in a companion paper for coalescence, removes the obstacle: destroyed particles continue walking as invisible ghosts, so the number of trajectories never changes. Applied to annihilation, the method yields an exact determinantal formula for the probability of any prescribed outcome - the number of annihilations, the survivor positions, and the positions of the ghosts. For complete annihilation, where no particle survives, the determinant collapses to a Pfaffian, an algebraic relative of the determinant built from pairwise quantities: although the particles interact, the extinction probability is determined by pairwise annihilation probabilities alone. This gives a combinatorial explanation of the Pfaffian structure of annihilating systems, previously derived through differential equations for specific dynamics. The annihilation formula also yields results about coalescence: the event that prescribed pairs of particles have merged can be reinterpreted as complete annihilation, producing a Pfaffian coalescence formula. All formulas are exact for any finite initial configuration and apply to discrete lattice paths, birth-death chains, and continuous diffusions including Brownian motion.

Figures

Figures reproduced from arXiv: 2602.13183 by the authors.

Figure 1
Figure 1. Annihilation on the checkerboard lattice. Four particles start at x1 < x2 < x3 < x4. Particles 2 (double) and 3 (zigzag) annihilate at c; both are destroyed and an ordered pair of ghosts emerges (dashed paths). Particles 1 (solid) and 4 (tick marks) survive. Ghost paths freely cross survivor paths (shown offset)—ghosts do not interact. Final positions: a < y1 < b < y2. When particles annihilate, the count decreases.… view at source ↗
Figure 2
Figure 2. An annihilation performance on the lattice [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Two-particle annihilation, case εj = +1: aj ⪯ bj . (a) Schema: particles I (thick) and J (wavy) collide; both are destroyed and two ghost paths emerge (dashed). Four distinct styles emphasize that no identity persists through the collision. (b) Attribution via the swap principle: particle I (left, thick) is glued to the rightward ghost at bj (thick dashed); particle J (right, wavy) is glued to the leftward ghost at … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Successful and failed castings (annihilation, [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]
Figure 5
Figure 5. Figure 5: The segment swap operation. (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 paths still cross at c, but now go to swa…
Figure 6
Figure 6. Figure 6: The sign-reversing involution in action: a matched pair [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

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Works this paper leans on

2 extracted references · 1 linked inside Pith

  1. [2]

    Nonintersecting paths, Pfaffians, and plane par- titions

    arXiv:2602.22885 [math.PR]. [Ste90] John R. Stembridge. “Nonintersecting paths, Pfaffians, and plane par- titions”. In:Adv. Math.83 (1990), pp. 96–131.doi: 10.1016/0001- 8708(90)90070-4. [TZ11] RogerTribeandOlegZaboronski.“Pfaffianformulaeforone-dimensional coalescing and annihilating systems”. In:Electron. J. Probab.16 (2011), pp. 2080–2103.doi:10.1214/E...

    arXiv 1990

  2. [2026]

    [Śni26b] Piotr Śniady.Pfaffian structure of basin walls for coalescing particles

    arXiv:2602.10782 [math.PR]. [Śni26b] Piotr Śniady.Pfaffian structure of basin walls for coalescing particles

    arXiv

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Reviewed August 2, 2026 · model on record in the stance chip above.