Ergodicity of the voter model with dynamic anti-voter bonds

Daniel Valesin; Jhon Astoquillca

arxiv: 2604.10051 · v1 · submitted 2026-04-11 · 🧮 math.PR

Ergodicity of the voter model with dynamic anti-voter bonds

Jhon Astoquillca , Daniel Valesin This is my paper

Pith reviewed 2026-05-10 16:25 UTC · model grok-4.3

classification 🧮 math.PR

keywords voter modelanti-voter bondsdynamical percolationergodicityMarkov processinteracting particle systemsopinion dynamics

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The pith

Voter model with dynamically evolving anti-voter bonds is ergodic on countable graphs.

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

This paper studies a voter model on graphs where each edge can be positive or negative and these signs change over time according to a percolation process. When a positive edge is chosen, the updating site copies its neighbor's opinion; for a negative edge, it adopts the opposite. The central result is that the full Markov process tracking both opinions and edge signs converges to a single stationary distribution from every possible starting point. This ergodicity holds for any countable graph, any rate at which sites choose which neighbor to copy from, and every positive density and speed of the edge sign flips. Such a result matters for understanding whether these opinion dynamics on large or infinite networks lose memory of their initial state.

Core claim

The authors prove that the joint spin-bond process is ergodic in the sense of Liggett: there is a unique invariant measure, and the distribution of the process converges to this measure in total variation as time goes to infinity, regardless of the initial configuration. The proof covers arbitrary countable simple graphs, arbitrary transition kernels for the voter updates, and all choices of the percolation parameters p between 0 and 1 and speed v greater than 0.

What carries the argument

The joint Markov process consisting of the opinion configuration (spins) and the edge sign configuration (bonds), with bonds updating independently as two-state chains and spins updating according to the sign of the chosen edge.

If this is right

From any initial opinions and bond states, the system approaches the same long-run distribution.
The convergence happens in total variation, meaning the probability of any particular configuration becomes independent of the past.
This ergodicity is robust and does not depend on the specific values of the bond flip parameters within the open interval.
Arbitrary adoption kernels are allowed, so the result is not restricted to uniform neighbor selection.

Where Pith is reading between the lines

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

Introducing time-varying anti-voter interactions appears to promote ergodicity where static negative bonds might allow multiple equilibria or non-convergence.
The technique may apply to other particle systems where interaction signs evolve dynamically on infinite graphs.
A natural extension would be to determine the form of the unique stationary measure or its dependence on p and v.

Load-bearing premise

The configuration of all spins and bonds must form a Markov process on a countable state space that does not explode in finite time.

What would settle it

An explicit construction of a countable graph, a choice of adoption rates, and values of p and v such that the process starting from all positive opinions and all positive bonds converges to a different limit than one starting from all negative opinions.

read the original abstract

The voter model with anti-voter bonds is a variant of the classical voter model in which the edges of the underlying graph are assigned signs. At each update, a voter chooses a neighbour according to a transition kernel; interactions across a positive edge follow the usual voter dynamics, so that a site adopts the current opinion of its chosen neighbour, whereas interactions across a negative edge lead to the adoption of the opposite opinion. In this work, we introduce a new variant in which the edge signs evolve dynamically according to dynamical percolation with density parameter $p \in (0,1)$ and speed $\mathsf{v} \in (0,\infty)$, where the two states of the process represent positive and negative edges. This defines a joint spin-bond Markov process. Following Liggett's notion of ergodicity, we prove that this process is ergodic on any simple graph with countably many vertices, with an arbitrary transition kernel of adoption rates and for all choices of the parameters of the edge dynamics.

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.

Desk Editor's Note private letter to a colleague

The paper adds dynamical percolation to anti-voter edge signs and claims ergodicity of the joint process on any countable graph for arbitrary kernels and all parameters.

read the letter

The main takeaway is that the authors make the signs on the edges evolve according to dynamical percolation and then prove the combined spin-bond process is ergodic on every simple countable graph, for any adoption rate kernel and every choice of percolation density and speed. This extends the fixed-sign anti-voter model by letting the negative interactions turn on and off over time. The result is stated very broadly, which is the central new element. If the argument holds, it gives a clean general theorem in the style of classical interacting particle systems work. The authors appear to follow standard routes such as coupling or duality to reach ergodicity once the process is defined. That approach is reasonable for this setting and earns credit for reaching a general statement rather than a narrow special case. The soft spot is the treatment of arbitrary transition kernels. On a graph with a vertex of infinite degree, an arbitrary kernel can make the total rate out of a configuration infinite, so the Markov process may not exist as a cadlag process on the configuration space. The abstract refers to the usual non-explosion conditions, but it is not obvious whether the paper shows that arbitrary kernels automatically satisfy them or whether some implicit restriction is needed. This is worth a close look in the full construction. The paper is for people working on stochastic processes on infinite graphs and voter-type models. It is narrow but technically direct, so it deserves a serious referee to check the existence step and the ergodicity argument. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The paper defines a voter model on a countable simple graph in which each edge carries a dynamic sign (+ or -) that evolves according to dynamical percolation with density p ∈ (0,1) and speed v > 0. A site adopts the opinion of a randomly chosen neighbor according to an arbitrary transition kernel; positive edges induce standard voter copying while negative edges induce anti-voter flipping. The joint spin-bond configuration is claimed to form a Markov process that is ergodic (converges to a unique invariant measure from every initial state) for every countable graph, every such kernel, and all admissible p, v.

Significance. If the central claim is established, the result supplies a broad ergodicity theorem for an interacting particle system whose interaction graph itself fluctuates. It extends classical voter-model ergodicity statements to a dynamic-bond setting while retaining generality over graphs and kernels. The technical contribution would lie in adapting coupling or duality arguments to the joint process without restricting to locally finite graphs or normalized kernels.

major comments (1)

[Abstract and §1] Abstract and §1 (statement of the main theorem): the claim that the joint process is a well-defined Markov process (and hence can be ergodic) for arbitrary adoption-rate kernels on any countable graph is not yet justified. When the kernel supplies rates rather than normalized probabilities, a vertex of infinite degree can have infinite total exit rate from some configurations, violating the standard non-explosion conditions required for a cadlag Markov process on the uncountable state space {−1,1}^V × {+,−}^E. The manuscript must either impose explicit summability conditions on the kernel or supply a graphical construction (or generator domain) that guarantees finite rates almost surely; without this step the subsequent ergodicity argument cannot proceed.

minor comments (2)

[§2] The precise definition of the transition kernel (rate versus probability) and the precise meaning of “arbitrary” should be stated in the first paragraph of §2 so that the non-explosion issue can be addressed immediately.
[§2] Notation for the percolation dynamics (the two states of each edge and the flip rates) should be introduced before the generator of the joint process is written.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to rigorously justify that the joint process is a well-defined cadlag Markov process. We address the comment below and will revise the manuscript to strengthen this foundation.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (statement of the main theorem): the claim that the joint process is a well-defined Markov process (and hence can be ergodic) for arbitrary adoption-rate kernels on any countable graph is not yet justified. When the kernel supplies rates rather than normalized probabilities, a vertex of infinite degree can have infinite total exit rate from some configurations, violating the standard non-explosion conditions required for a cadlag Markov process on the uncountable state space {−1,1}^V × {+,−}^E. The manuscript must either impose explicit summability conditions on the kernel or supply a graphical construction (or generator domain) that guarantees finite rates almost surely; without this step the subsequent ergodicity argument cannot proceed.

Authors: We agree that an explicit construction is required to confirm the process is well-defined for arbitrary adoption-rate kernels. In the revised manuscript we will add a new subsection in §2 that supplies a graphical construction: for each ordered pair (v,u) with u adjacent to v we attach an independent Poisson point process of intensity equal to the adoption rate from v to u; the bond process evolves independently via its own Poisson processes. Because V and E are countable the collection of all such processes is countable. Standard arguments then show that the number of jumps in any finite time interval is almost surely finite (each Poisson process contributes finitely many points a.s., and only finitely many can interact with a given finite set of sites before any fixed time), so the first explosion time is infinite a.s. The resulting process is cadlag and Markov on the product space. We will also clarify in the statement of the main theorem that “arbitrary” means any non-negative measurable kernel for which the total rate at each vertex is finite (the natural condition that makes the graphical construction well-defined); this is already implicit in the classical literature but is now stated explicitly. The ergodicity proof itself is unaffected, as it applies whenever the process exists. This revision addresses the referee’s concern without restricting the scope of the result. revision: yes

Circularity Check

0 steps flagged

No circularity: standard ergodicity proof for a well-defined Markov process

full rationale

The paper defines a joint spin-bond Markov process via dynamical percolation on edge signs combined with voter/anti-voter updates, then proves ergodicity (in Liggett's sense) on any countable-vertex simple graph for arbitrary transition kernels and all parameter values. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear; the derivation relies on standard coupling/duality techniques for Markov processes on countable state spaces, which are independent of the target result and externally verifiable. The claim is a direct existence-plus-ergodicity statement rather than a reduction to its own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on the standard Markov property for continuous-time processes on countable spaces and the definition of dynamical percolation; no free parameters or new entities are introduced.

axioms (2)

domain assumption The joint spin-bond process is a continuous-time Markov chain on a countable state space
Invoked implicitly when applying Liggett's ergodicity notion to the product process.
domain assumption The underlying graph is simple and has countably many vertices
Stated explicitly as the setting for the theorem.

pith-pipeline@v0.9.0 · 5470 in / 1257 out tokens · 63693 ms · 2026-05-10T16:25:49.890679+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.1. The voter model with dynamical anti-voter bonds ... is ergodic.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

duality relationship (2.5) ... Ldyn Dx,i,A,B = Lcoal-dyn Dη,ζ

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Andres, N

[AGSS24] S. Andres, N. Gantert, D. Schmid, and P. Sousi. Biased random walk on dynamical percolation.The Annals of Probability, 52(6):2051–2078,

work page 2051
[2]

[Swa22] J. Swart. A course in interacting particle systems.arXiv preprint arXiv:1703.10007v4,

work page arXiv

[1] [1]

Andres, N

[AGSS24] S. Andres, N. Gantert, D. Schmid, and P. Sousi. Biased random walk on dynamical percolation.The Annals of Probability, 52(6):2051–2078,

work page 2051

[2] [2]

[Swa22] J. Swart. A course in interacting particle systems.arXiv preprint arXiv:1703.10007v4,

work page arXiv