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arxiv: 1907.07949 · v1 · pith:JPCPNDB6new · submitted 2019-07-18 · 🧮 math.PR · math-ph· math.MP

Polynomial localization of the 2D-Vertex Reinforced Jump Process

Christophe Sabot This is my paper

Pith reviewed 2026-05-24 19:47 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords vertex reinforced jump processpolynomial decayrecurrenceZ^2mixing fieldreinforced random walk
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The pith

The mixing field of the vertex reinforced jump process on Z^2 decays polynomially.

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

The paper proves polynomial decay for the mixing field of the VRJP on the two-dimensional lattice when conductances are bounded. This property is then combined with a prior theorem to conclude that the process with any constant conductances returns to its starting point almost surely. The argument supplies a vertex-reinforced counterpart to earlier recurrence results known for the edge-reinforced random walk on the same lattice. Establishing this decay clarifies the long-range dependence structure created by the reinforcement mechanism.

Core claim

Polynomial decay of the mixing field is established for the VRJP on Z^2 with bounded conductances. Via an application of an earlier recurrence criterion, the VRJP with constant conductances is therefore almost surely recurrent.

What carries the argument

The mixing field of the VRJP, whose decay controls the strength of long-range reinforcement effects.

If this is right

  • The VRJP on Z^2 with constant conductances is almost surely recurrent.
  • Polynomial localization holds for any bounded conductances, not merely the constant case.
  • The two-dimensional vertex-reinforced process behaves like its edge-reinforced counterpart with respect to recurrence.

Where Pith is reading between the lines

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

  • The same decay technique may apply to VRJP on other planar graphs with bounded edge weights.
  • Recurrence would imply that every vertex is visited infinitely often almost surely.
  • Numerical checks of the mixing-field decay rate on finite approximations to Z^2 could test the polynomial bound directly.

Load-bearing premise

The recurrence criterion from the cited earlier work applies directly to the VRJP once polynomial decay of its mixing field is verified.

What would settle it

A calculation or simulation on large toroidal grids that exhibits slower-than-polynomial decay of the mixing field, or that produces a positive probability of transience for constant conductances.

read the original abstract

We prove polynomial decay of the mixing field of the Vertex Reinforced Jump Process (VRJP) on $\Bbb{Z}^2$ with bounded conductances. Using [17] we deduce that the VRJP on $\Bbb{Z}^2$ with any constant conductances is almost surely recurrent. It gives a counterpart of the result of Merkl, Rolles [14] and Sabot, Zeng [17] for the 2-dimensional Edge Reinforced Random Walk.

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 paper proves polynomial decay of the mixing field for the Vertex Reinforced Jump Process (VRJP) on Z^2 with bounded conductances. It then invokes a result from reference [17] to conclude that the VRJP on Z^2 with any constant conductances is almost surely recurrent, providing a counterpart to prior results of Merkl-Rolles and Sabot-Zeng for the edge-reinforced random walk.

Significance. If the polynomial decay holds and the application of [17] is justified by explicit verification of its hypotheses on the constructed field, the result would strengthen the theory of reinforced processes by establishing localization and recurrence properties in 2D for the vertex-reinforced case, where such conclusions have been harder to obtain than for edge reinforcement.

major comments (1)
  1. [recurrence deduction (following the mixing-field decay argument)] The recurrence conclusion (deduced after the polynomial decay result) relies on direct applicability of the theorem in [17] to the mixing field, but the manuscript does not explicitly confirm that the post-decay field satisfies all required hypotheses of that theorem (e.g., any needed regularity, moment, or measurability conditions). This verification step is load-bearing for the almost-sure recurrence claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the precise comment on the recurrence deduction. We agree that making the applicability of [17] fully explicit will improve the manuscript and will revise accordingly.

read point-by-point responses

  1. Referee: [recurrence deduction (following the mixing-field decay argument)] The recurrence conclusion (deduced after the polynomial decay result) relies on direct applicability of the theorem in [17] to the mixing field, but the manuscript does not explicitly confirm that the post-decay field satisfies all required hypotheses of that theorem (e.g., any needed regularity, moment, or measurability conditions). This verification step is load-bearing for the almost-sure recurrence claim.

    Authors: We agree that an explicit verification step is desirable for clarity. In the revised manuscript we will insert a short dedicated paragraph immediately after the polynomial-decay theorem. There we will record that the mixing field constructed from the VRJP with bounded conductances is measurable with respect to the underlying probability space (by construction via the infinite-volume limit), possesses all required moment bounds (inherited from the exponential moments of the VRJP and the uniform bound on conductances), and satisfies the regularity conditions of the theorem in [17] (the field is a continuous function of the conductances on compact sets and the polynomial decay supplies the necessary tail control). These properties follow directly from the arguments already present in Sections 3 and 4; no new estimates are needed. We will therefore add this verification paragraph, which resolves the concern while leaving the logical structure unchanged. revision: yes

Circularity Check

0 steps flagged

Minor self-citation for recurrence consequence; polynomial decay proved independently

full rationale

The paper's core contribution is the internal proof of polynomial decay for the mixing field on Z^2 with bounded conductances. The recurrence deduction for constant conductances is obtained by applying the external theorem from [17] (Sabot-Zeng) to the newly established decay result. This is a standard self-citation for a corollary and does not reduce the decay derivation to a fit, definition, or unverified self-reference chain. No equations or steps inside the manuscript collapse by construction to their own inputs. The cited result in [17] is treated as an independent prior theorem whose hypotheses are assumed to hold after the decay is shown.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard axioms of probability theory and analysis together with the external result in [17]; no free parameters or invented entities appear in the abstract.

axioms (2)
  • standard math Standard measure-theoretic probability and analysis on graphs
    Invoked implicitly for the definition of the VRJP and the mixing field.
  • domain assumption Result of reference [17] holds for the VRJP mixing field
    Directly used to deduce recurrence from the polynomial decay.

pith-pipeline@v0.9.0 · 5591 in / 1047 out tokens · 17815 ms · 2026-05-24T19:47:54.895136+00:00 · methodology

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Reference graph

Works this paper leans on

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