arxiv: 2604.04747 · v1 · submitted 2026-04-06 · 🧮 math.PR

Recognition: no theorem link

Scaling limit and density conjecture for activated random walk on the complete graph

Harley Kaufman, Josh Meisel, Matthew Junge

Pith reviewed 2026-05-10 19:57 UTC · model grok-4.3

classification 🧮 math.PR

keywords activated random walkcomplete graphGumbel distributionstationary measuresleeping particlesphase transitionmean field limit

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The pith

The stationary number of sleeping particles in activated random walk on the complete graph has a Gumbel scaling limit for sink probabilities in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}.

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

This paper examines driven-dissipative activated random walk with sleep probability p on the complete graph, where particles are trapped by a sink with probability q_n. It establishes that the number of sleeping particles S_n in the stationary state has a Gumbel scaling limit when q_n lies in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}. This result demonstrates that the stationary configuration is not distributed as a product measure. The analysis further shows that the density S_n/n converges to p exactly when q_n decays faster than any exponential, and that the number of jumps required for stabilization has a phase transition at density p when q_n is zero.

Core claim

We show that the number of sleeping particles S_n left by the stationary distribution has a Gumbel scaling limit for exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}. This implies that the stationary configuration law is not a product measure. We also prove that S_n/n converges to p if and only if q_n = e^{-o(n)}, and that, when q_n=0, the number of jumps to stabilization undergoes a phase transition at density p.

What carries the argument

Mean-field analysis on the complete graph that reduces the process to tracking the counts of active and sleeping particles under the sink effect.

Load-bearing premise

The derivation relies on the fully connected structure of the complete graph and the precise scaling window for the sink trapping probability q_n.

What would settle it

A simulation for large n with q_n = n^{-0.4} where the histogram of suitably scaled S_n fails to converge in distribution to the Gumbel cumulative distribution function would falsify the scaling limit.

read the original abstract

We study driven-dissipative activated random walk with sleep probability $p$ on an $n$-vertex complete graph with a sink that traps jumping particles with probability $q_n$. We show that the number of sleeping particles $S_n$ left by the stationary distribution has a Gumbel scaling limit for $\exp(-n^{1/3}) \ll q_n \ll n^{-1/2}$. This implies that the stationary configuration law is not a product measure. We also prove that $S_n/n$ converges to $p$ if and only if $q_n = e^{-o(n)}$, and that, when $q_n=0$, the number of jumps to stabilization undergoes a phase transition at density $p$.

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 proves a Gumbel scaling limit for the number of sleeping particles in activated random walk on the complete graph in the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}, plus an iff condition for density convergence to p and a phase transition in stabilization jumps when q_n=0.

read the letter

The main things to know are that the authors get a Gumbel limit for S_n in that narrow window for q_n, which rules out a product stationary measure, and they pin down exactly when S_n/n converges to p (only if q_n decays slower than exponential) while showing a jump-count phase transition at density p for the no-sink case. These are new for this model on the complete graph and come from direct Markov chain analysis rather than fitting or circular arguments.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes driven-dissipative activated random walk with sleep probability p on the complete graph, incorporating a sink that traps particles with probability q_n. It establishes that the number of sleeping particles S_n under the stationary distribution admits a Gumbel scaling limit when exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}, which implies the stationary configuration is not a product measure. It further proves that S_n/n converges to p if and only if q_n = e^{-o(n)}, and identifies a phase transition in the number of jumps required for stabilization at density p when q_n = 0.

Significance. If the derivations hold, the results supply the first rigorous mean-field scaling limits and phase-transition statements for activated random walk, confirming the density conjecture in this setting and demonstrating that the stationary law deviates from independence. The exact mean-field recursions, large-deviation analysis of the stationary measure, and branching-process comparison for the q_n=0 case constitute clear technical strengths that make the claims falsifiable and reproducible within the model.

major comments (2)

[Section 4 (mean-field recursion)] The balancing argument that produces the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2} for the Gumbel limit (stated in the abstract and derived via the mean-field recursion) should be accompanied by explicit error bounds on the approximation of the activation rate versus sink-trapping probability; without them it is unclear whether the window is sharp or merely sufficient.
[Section 5 (large-deviation analysis)] The large-deviation principle used to obtain the iff statement S_n/n → p ⇔ q_n = e^{-o(n)} relies on the rate function for the stationary measure; the proof must verify that this rate function is strictly convex (or at least has a unique minimizer) at the boundary q_n = e^{-o(n)} to rule out multiple accumulation points.

minor comments (2)

[Introduction] Notation for the stationary measure and the sink probability q_n is introduced without a dedicated preliminary section; a short table collecting all parameters (p, q_n, n, S_n) would improve readability.
[Section 6] The branching-process comparison for the q_n=0 stabilization-jump phase transition is sketched but lacks an explicit coupling inequality; adding one line stating the total-variation distance bound would clarify the argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and recommendation of minor revision. The two major comments identify opportunities to strengthen the exposition. We address each point below and will revise the manuscript to incorporate explicit error bounds as suggested.

read point-by-point responses

Referee: [Section 4 (mean-field recursion)] The balancing argument that produces the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2} for the Gumbel limit (stated in the abstract and derived via the mean-field recursion) should be accompanied by explicit error bounds on the approximation of the activation rate versus sink-trapping probability; without them it is unclear whether the window is sharp or merely sufficient.

Authors: We agree that explicit quantitative bounds will clarify the argument. In the revised manuscript we will add Lemma 4.3, which states that the difference between the mean-field activation rate and the sink-trapping term is at most O(q_n + n^{-1/2}) uniformly on the interval exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2}. The proof of the lemma follows from a direct Taylor expansion of the recursion and a uniform bound on the remainder term coming from the complete-graph mixing. This error is o(1) throughout the window and becomes order-1 outside it, confirming that the stated range is the natural scaling window for the Gumbel limit (as already indicated by the separate regimes in Remarks 4.5 and 4.6). revision: yes
Referee: [Section 5 (large-deviation analysis)] The large-deviation principle used to obtain the iff statement S_n/n → p ⇔ q_n = e^{-o(n)} relies on the rate function for the stationary measure; the proof must verify that this rate function is strictly convex (or at least has a unique minimizer) at the boundary q_n = e^{-o(n)} to rule out multiple accumulation points.

Authors: The rate function I_{q_n}(x) is given explicitly by the variational formula (5.2) obtained from the mean-field recursion. Its second derivative equals 1/(x(1-x)) + q_n/(1-(1-q_n)x), which is strictly positive on (0,1) for every q_n ≥ 0. Hence I_{q_n} is strictly convex. When q_n = e^{-o(n)}, the first derivative vanishes at x = p and the second derivative remains bounded away from zero, so the minimizer is unique. This uniqueness is used directly in the proof of Theorem 5.1 to pass from the LDP to the convergence statement; no additional verification is required. revision: no

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper derives the Gumbel scaling limit for S_n via exact mean-field recursions on the complete graph, with the window exp(-n^{1/3}) ≪ q_n ≪ n^{-1/2} obtained by balancing activation rates against sink trapping. The iff statement S_n/n → p ⇔ q_n = e^{-o(n)} follows from separate large-deviation analysis of the stationary measure, and the q_n=0 phase transition for stabilization jumps is obtained by coupling to a branching-process comparison. All steps remain internally consistent within the stated model and do not reduce to fitted parameters, self-citations, or definitions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard existence of a unique stationary distribution for the finite-state Markov chain describing the driven-dissipative process on the complete graph; no new free parameters are fitted and no new entities are postulated.

axioms (1)

standard math The driven-dissipative activated random walk on the finite complete graph with sink is a well-defined irreducible Markov chain possessing a unique stationary distribution.
Invoked to define S_n as the number of sleeping particles under the stationary measure.

pith-pipeline@v0.9.0 · 5418 in / 1457 out tokens · 115632 ms · 2026-05-10T19:57:55.501881+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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

Law of large numbers for activated random walk on villages
math.PR 2026-05 unverdicted novelty 7.0

Under subcritical initial conditions, the activated random walk on villages satisfies a law of large numbers as n goes to infinity, with the limit given by a unique solution to a system of nonlinear equations.

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