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arxiv: 1906.09949 · v1 · pith:IISGTTBOnew · submitted 2019-06-24 · 🧮 math.PR

Time Scales of the Fredrickson-Andersen Model on Polluted mathbb{Z}² and mathbb{Z}³

Assaf Shapira , Erik Slivken This is my paper

Pith reviewed 2026-05-25 17:03 UTC · model grok-4.3

classification 🧮 math.PR
keywords Fredrickson-Andersen modelkinetically constrained modelspolluted latticesinfection timequenched pollutiontime scalesZ^2Z^3
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The pith

Bounds on the infection time of the origin are given for the Fredrickson-Andersen model with low-density quenched pollution on Z^2 and Z^3.

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

The paper investigates the Fredrickson-Andersen model, a type of kinetically constrained model, on polluted versions of the two- and three-dimensional integer lattices. It considers a fixed set of blocked sites with low density and provides bounds on how long it takes for the origin to become infected under the two-neighbor constraint. A reader would care because this helps understand how defects influence the time scales of relaxation in models of glassy dynamics.

Core claim

For a quenched polluted environment with low pollution density we give bounds on the infection time of the origin in the two-neighbor constrained Fredrickson-Andersen model on the polluted square and cubic lattices.

What carries the argument

Quenched low-density pollution consisting of fixed blocked sites that interact with the two-neighbor facilitation rule to determine infection propagation.

If this is right

  • The infection can propagate to the origin from distant locations despite the blocked sites.
  • The time scales for the model are characterized in terms of the pollution density.
  • The results apply to both two and three dimensions.

Where Pith is reading between the lines

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

  • These bounds may help in designing numerical experiments to study glassy dynamics with defects.
  • The approach could extend to other kinetically constrained models with different facilitation rules.
  • It implies that low levels of fixed disorder do not prevent the system from reaching equilibrium states in reasonable times.

Load-bearing premise

The density of the fixed blocked sites is low enough to allow the two-neighbor facilitation to enable infection propagation across the lattice to the origin.

What would settle it

A specific low-density pollution configuration where the measured infection time at the origin lies outside the provided bounds would falsify the result.

Figures

Figures reproduced from arXiv: 1906.09949 by Assaf Shapira, Erik Slivken.

Figure 4.1
Figure 4.1. Figure 4.1: Illustration of the proof of Proposition 21. We see here how an infected column could propagate in a good box. i stands for infected sites. Other sites could be either infected or healthy, according to their initial state. Proof. By Corollary 14 the origin belongs to an infinite cluster of good boxes with probability greater than 1 − 16q ε/3 . In particular, it is contained in a self-avoiding path of len… view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Illustration of the proof of Proposition 21. We see here how to rotate an empty column in a good box. i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i [PITH_FULL_IMAGE:figures/full_fig_p009_4_2.png] view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Illustration of the proof of Proposition 21. We see here how to propagate infection through a good path. For the lower bound, we will use the ideas of [11] in order to construct a path that empties the origin and satisfies the hypotheses of Lemma 8. In contrast to the infection mechanism in the bootstrap percolation ([11]), we will need a finer path, that not only allows us to empty the origin, but also … view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: B0 = A0 is the left turquoise brick, B1 and A1 are represented by the pink brick, B2 and A2 are both represented by the orange brick (with different sail orientations), and B3 and A3 are the turquoise translations to the right of B0 . The red outlines are where the tip of a brick coincides with the base othe another brick. Finally, the sequence rev(Γ) is a legal sequence of moves starting from η X1∪X2∪X0… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: A sequence of eight bricks {C i}, that facilitate cleaning outside A3 or B3 . The tips of B3 and A3 point to the bottom section and 3rd section, respectively, of C 0 . Starting from B3 this example satisfies, B3 .0 C 0 .0 C 1 .0 C 2 .3 C 3 .3 C 4 .2 C 5 .3 C 6 .0 C 7 .0 B0 . The colors correspond to orientation. Some unused sections of bricks were removed or made transparent to show hidden bricks. Inters… view at source ↗
read the original abstract

We study the Kinetically Constrained Model on the polluted square lattice, with two-neighbor constraints. For a quenched polluted environment with low pollution density we give bounds on the infection time of the origin.

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 / 0 minor

Summary. The manuscript studies the Fredrickson-Andersen kinetically constrained model with two-neighbor facilitation rules on polluted versions of the square and cubic lattices. For a quenched (fixed) polluted environment whose density is low enough that facilitation still permits propagation, it claims to establish bounds on the time until the origin becomes infected.

Significance. If the claimed bounds are rigorous, explicit, and derived from standard KCM techniques such as bootstrap percolation or renormalization, the result would extend existing time-scale analyses from clean lattices to disordered settings and could inform the effect of fixed obstacles on relaxation. The quenched low-density regime is a natural and technically nontrivial extension.

major comments (1)
  1. [Abstract] Abstract: the claim that bounds exist is stated without indicating their form (e.g., polynomial, exponential, or stretched-exponential in the density), the proof strategy, or any supporting calculation or reference to a theorem number. This prevents evaluation of whether the central assertion is substantiated.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report. We address the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim that bounds exist is stated without indicating their form (e.g., polynomial, exponential, or stretched-exponential in the density), the proof strategy, or any supporting calculation or reference to a theorem number. This prevents evaluation of whether the central assertion is substantiated.

    Authors: We agree that the abstract is too concise and does not indicate the form of the bounds, the proof strategy, or a theorem reference. This limits the ability to assess the claim from the abstract alone. In the revised manuscript we will expand the abstract to state the form of the bounds obtained on the infection time, briefly note the use of bootstrap percolation and renormalization, and include a reference to the main theorem. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives rigorous bounds on infection times for the Fredrickson-Andersen KCM under quenched low-density pollution on Z^2 and Z^3. The abstract and structure indicate standard probabilistic/combinatorial arguments for propagation under facilitation rules when density is below threshold, with no equations, self-definitions, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the central claim to its inputs by construction. The derivation chain is self-contained against external benchmarks in KCM literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract; the claim rests on standard probabilistic techniques for kinetically constrained models whose details are not supplied.

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Reference graph

Works this paper leans on

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