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arxiv: 2606.27261 · v1 · pith:PMSAQOXSnew · submitted 2026-06-25 · 🧮 math.PR · math-ph· math.MP

Non-colliding space-time inhomogeneous Markov chains

Theodoros Assiotis , Miles Foster Nyamundanda This is my paper

Pith reviewed 2026-06-26 03:04 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords non-colliding Markov chainsspace-time inhomogeneouscollision timestail asymptoticsKarlin-McGregor semigroupsteepest descentintegrable processes
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The pith

Integrable space-time inhomogeneous Markov chains admit explicit leading asymptotics for collision-time tail probabilities with error bounds.

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

The authors derive the leading-order rate at which the probability of first collision after a long time decays to zero, together with a quantitative bound on the approximation error. A reader would care because the same probabilities govern the law of the chains conditioned to remain non-intersecting, and these conditioned objects appear as models for interacting particle systems that use local push-block rules. The result covers both discrete-time and continuous-time versions even when the transition rules vary arbitrarily with position and time. The argument proceeds by expanding the Karlin-McGregor semigroup in dominant-index terms and then extracting the asymptotics via steepest descent on the resulting contour integrals.

Core claim

We establish the explicit leading order asymptotics, with a quantitative error bound, of tail probabilities of collision times for a class of integrable space-time inhomogeneous Markov chains, in discrete and continuous time. The corresponding process conditioned not to intersect arises in interacting particle systems with local push-block interactions thereby confirming a recent prediction. The generic discrete nature of the spatial inhomogeneities rules out powerful coupling-with-Brownian-motion techniques, so our proof strategy proceeds instead via a novel steepest-descent analysis combined with a Karlin-McGregor semigroup expansion in terms of dominant-index contributions.

What carries the argument

Karlin-McGregor semigroup expansion in terms of dominant-index contributions, analyzed by steepest descent

If this is right

  • The non-colliding versions of these chains serve as models for push-block interacting particle systems.
  • The asymptotics and error bounds hold uniformly for both discrete-time and continuous-time chains.
  • The method applies even when spatial inhomogeneities prevent coupling arguments based on Brownian motion.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical sampling of long non-colliding paths could be replaced by direct use of the asymptotic formula for sufficiently large times.
  • The same expansion-plus-steepest-descent strategy may extend to other families of integrable Markov chains whose semigroups admit analogous dominant-index representations.
  • Confirmation of the particle-system prediction indicates that push-block dynamics inherit universal late-time tail behavior from the underlying integrable chains.

Load-bearing premise

The chains must belong to the integrable subclass in which the Karlin-McGregor semigroup admits an expansion in dominant-index contributions that permits contour deformation.

What would settle it

For a concrete integrable inhomogeneous chain, compute the empirical tail probability of the collision time at successively larger times and verify whether the observed decay matches the claimed leading asymptotic within the stated error bound.

Figures

Figures reproduced from arXiv: 2606.27261 by Miles Foster Nyamundanda, Theodoros Assiotis.

Figure 1
Figure 1. Figure 1: Finite windows of the two-particle state graphs for continuous time pure-birth [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Sample path of a continuous-time pure-birth process started from 0. The lower [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Sample path of a mixed Bernoulli-and-geometric jump process. Blue regions [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Three ways in which the embedded coordinate paths of [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Five independent mixed Bernoulli-and-geometric jump processes started from [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Continuous-time pure-birth push-block dynamics on an interlacing array, fol [PITH_FULL_IMAGE:figures/full_fig_p016_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Schematic location of the real singularities of the integrand in the contour repre [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Sketch of relevant singularities on the real axis: the [PITH_FULL_IMAGE:figures/full_fig_p031_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Finite-time surface plot of the mixed discrete-time phase for representative [PITH_FULL_IMAGE:figures/full_fig_p037_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Schematic of the bracketing argument in the proof of Proposition 2.4, drawn [PITH_FULL_IMAGE:figures/full_fig_p040_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Schematic contour geometry for Corollary 2.9. In each panel, the blue curve is [PITH_FULL_IMAGE:figures/full_fig_p051_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Schematic illustration of Proposition 3.3. With probability one, there is a finite [PITH_FULL_IMAGE:figures/full_fig_p083_13.png] view at source ↗
read the original abstract

We establish the explicit leading order asymptotics, with a quantitative error bound, of tail probabilities of collision times for a class of integrable space-time inhomogeneous Markov chains, in discrete and continuous time. The corresponding process conditioned not to intersect arises in interacting particle systems with local push-block interactions thereby confirming a recent prediction. The generic discrete nature of the spatial inhomogeneities rules out powerful coupling-with-Brownian-motion techniques, so our proof strategy proceeds instead via a novel steepest-descent analysis combined with a Karlin--McGregor semigroup expansion in terms of dominant-index contributions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes the explicit leading order asymptotics, with a quantitative error bound, of tail probabilities of collision times for a class of integrable space-time inhomogeneous Markov chains, in discrete and continuous time. The proof proceeds via a novel steepest-descent analysis combined with a Karlin--McGregor semigroup expansion in terms of dominant-index contributions. This confirms a recent prediction for the corresponding conditioned non-intersecting process in interacting particle systems with local push-block interactions.

Significance. If the derivations hold, the result is significant for providing rigorous, explicit asymptotics (including error bounds) in inhomogeneous settings where coupling to Brownian motion is unavailable due to discrete spatial inhomogeneities. The restriction to an explicitly integrable subclass where the semigroup admits a dominant-index expansion amenable to steepest descent is a natural scope limitation that enables the method.

major comments (1)
  1. [Abstract, paragraph 2] Abstract, paragraph 2: the integrability assumption (that the Karlin-McGregor semigroup admits an expansion in dominant-index contributions justifying both the expansion and contour deformation) is load-bearing for the entire strategy, yet is invoked without an explicit definition or list of verifiable conditions on the transition probabilities.
minor comments (1)
  1. The abstract is concise but the introduction should state the main theorems (including the precise form of the leading asymptotics and the error term) with equation numbers for easy reference.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. We address the major comment below.

read point-by-point responses

  1. Referee: [Abstract, paragraph 2] Abstract, paragraph 2: the integrability assumption (that the Karlin-McGregor semigroup admits an expansion in dominant-index contributions justifying both the expansion and contour deformation) is load-bearing for the entire strategy, yet is invoked without an explicit definition or list of verifiable conditions on the transition probabilities.

    Authors: We agree that the integrability assumption is load-bearing and that the abstract invokes it without sufficient detail. The manuscript defines the relevant class in Definition 2.3 via explicit conditions on the transition probabilities that guarantee the dominant-index Karlin-McGregor expansion and justify the subsequent contour deformation. To address the comment, we will revise the abstract to include a brief reference to these conditions (e.g., "under the integrability conditions of Definition 2.3") and ensure the abstract paragraph makes the assumption transparent. This is a clarification only and does not alter the scope or results. revision: yes

Circularity Check

0 steps flagged

Derivation self-contained via semigroup expansion and steepest descent; no circularity

full rationale

The paper's central result is an explicit leading-order asymptotic with error bound for collision-time tails, obtained by applying steepest-descent analysis to a Karlin-McGregor semigroup expansion in dominant-index terms. This expansion and contour deformation are justified inside the explicitly delimited 'integrable' subclass; the derivation does not reduce any claimed prediction to a fitted parameter, does not rely on a load-bearing self-citation chain, and does not rename a known empirical pattern as a new theorem. The restriction to the integrable case is stated up front rather than smuggled in, and the quantitative error bound follows directly from the contour estimates. No step equates an output to its input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; all such items remain unknown.

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