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arxiv: 2605.13371 · v1 · pith:7YN4NAHCnew · submitted 2026-05-13 · 🧮 math.PR

Percolation representations of additive particle systems

Jan M. Swart This is my paper

Pith reviewed 2026-05-14 18:36 UTC · model grok-4.3

classification 🧮 math.PR
keywords percolation representationinteracting particle systemsadditive systemsdistributive latticesgraphical representationcontact processopen paths
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The pith

Additive interacting particle systems with finite distributive lattice state spaces admit percolation representations via open paths.

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

The paper establishes a general construction for percolation representations of additive interacting particle systems when the local state space is any finite distributive lattice. This extends the familiar case of two-state systems, where open paths in a graphical representation capture the dynamics. Such representations matter because they translate the evolution of the particle system into questions about percolation clusters and infinite paths, which often simplify analysis of survival, extinction, or equilibrium behavior. The construction is illustrated explicitly on Krone's two-stage contact process, confirming that the method applies to a nontrivial example beyond the binary case.

Core claim

It is shown how a percolation representation in terms of open paths in a graphical representation can be constructed for additive interacting particle systems when the local state space is a finite distributive lattice, generalizing the well-known two-state case and demonstrated on Krone's two-stage contact process.

What carries the argument

The percolation representation via open paths in the graphical representation of the additive system on the distributive lattice, which encodes the joint evolution through path connectivity.

If this is right

  • Survival or extinction of the process becomes equivalent to the existence of an infinite open percolation cluster starting from the initial configuration.
  • The stationary measures and phase transitions of the particle system can be read off from percolation probabilities on the associated space-time graph.
  • Coupling and monotonicity arguments from percolation theory apply directly to compare different initial conditions or parameters.
  • The construction preserves the additive structure, allowing linear combinations of configurations to correspond to unions of open paths.

Where Pith is reading between the lines

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

  • The approach may extend naturally to infinite distributive lattices or to systems with spatial inhomogeneity if the graphical construction can be localized.
  • Numerical simulation of the particle system could be replaced or accelerated by Monte Carlo sampling of the percolation paths alone.
  • Similar representations might exist for other algebraic structures on the state space, such as semilattices, provided an appropriate notion of additivity is defined.

Load-bearing premise

The interacting particle system must be additive and its local state space must be a finite distributive lattice.

What would settle it

An explicit additive particle system on a finite distributive lattice whose long-term behavior cannot be recovered from the existence or nonexistence of infinite open paths in its standard graphical representation.

Figures

Figures reproduced from arXiv: 2605.13371 by Jan M. Swart.

Figure 1
Figure 1. Figure 1: Percolation representation of the two-stage contact process and its dual, the on-off [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
read the original abstract

It is well-known that additive interacting particle systems with a local state space of cardinality two have a percolation representation in terms of open paths in a graphical representation. In this paper, it is shown how such a percolation representation can be constructed more generally when the local state space is a finite distributive lattice. The theory is demonstrated on Krone's two-stage contact process.

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

0 major / 2 minor

Summary. The paper claims that additive interacting particle systems whose local state space is a finite distributive lattice admit an explicit percolation representation constructed via the lattice join and meet operations, generalizing the classical two-state case. The construction is illustrated by recovering the known percolation behavior of Krone's two-stage contact process.

Significance. If the construction holds, the result is a meaningful extension of graphical methods in interacting particle systems. It supplies a lattice-valued analogue of the standard percolation representation, which may facilitate proofs of survival/extinction criteria and monotonicity properties for models with richer state spaces than the binary case. The explicit recovery of the two-stage contact process provides a useful consistency check.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short paragraph outlining the main steps of the lattice-based construction before the detailed definitions.
  2. [Demonstration] In the demonstration section, explicitly listing the transition rates of Krone's process alongside the recovered percolation events would make the verification easier to follow.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper extends the well-known percolation representation for two-state additive interacting particle systems to the case of finite distributive lattices via an explicit construction using lattice operations. This is demonstrated by recovering the known dynamics of Krone's two-stage contact process. The binary case is treated as an established external fact rather than derived internally, and no load-bearing step reduces by definition, fitted parameter, or self-citation chain to the target result. The derivation is therefore self-contained against the cited independent foundation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result relies on standard definitions from interacting particle systems and lattice theory without introducing new fitted parameters or entities.

axioms (2)
  • domain assumption The interacting particle system is additive
    Additivity is required for the percolation representation to hold via open paths, as referenced from the binary case.
  • domain assumption The local state space is a finite distributive lattice
    This lattice structure is the key condition enabling the generalized construction.

pith-pipeline@v0.9.0 · 5332 in / 1098 out tokens · 126138 ms · 2026-05-14T18:36:30.934832+00:00 · methodology

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Reference graph

Works this paper leans on

6 extracted references · 6 canonical work pages

  1. [1]

    E. Foxall. Duality and complete convergence for multi-type additive growth models. Adv.\ Appl.\ Probab. 48(1) (2016), 32--51

    work page 2016

  2. [2]

    Griffeath

    D. Griffeath. Additive and Cancellative Interacting Particle Systems. Lecture Notes in Math. 724, Springer, Berlin, 1979

    work page 1979

  3. [3]

    T.E. Harris. Additive set-valued Markov processes and graphical methods. Ann.\ Probab. 6 (1978), 355--378

    work page 1978

  4. [4]

    S. Krone. The two-stage contact process. Ann.\ Appl.\ Probab. 9(2) (1999), 331--351

    work page 1999

  5. [5]

    Sturm and J.M

    A. Sturm and J.M. Swart. Pathwise duals of monotone and additive Markov processes. J.\ Theor.\ Probab. 31(2) (2018), 932--983

    work page 2018

  6. [6]

    J.M. Swart. A Course in Interacting Particle Systems. To be published by Cambridge University Press. Preprint (2025) arXiv:1703.10007v5

    work page arXiv 2025