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arxiv: 2604.08462 · v1 · submitted 2026-04-09 · 🧮 math.PR

Convergence of k-point functions in high dimensional percolation

Shirshendu Chatterjee , Pranav Chinmay , Jack Hanson , Philippe Sosoe This is my paper

Pith reviewed 2026-05-10 17:06 UTC · model grok-4.3

classification 🧮 math.PR
keywords percolationcritical phenomenahigh dimensionsk-point functionslace expansionAizenman-Newman conjecturescaling limitscluster connectivity
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The pith

In high dimensions, the rescaled probability that k lattice points share a critical percolation cluster converges to an explicit constant.

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

The paper establishes that at the critical probability for Bernoulli percolation on the integer lattice in high dimensions, the chance that k distinct points in R^d, when scaled by a large n and discretized to the lattice, all belong to one open cluster decays at a precise rate. Multiplying by the right power of n makes this probability approach a fixed positive number that the authors can write down explicitly. This settles the Aizenman-Newman conjecture on the limiting form of these joint connection events. A reader would care because the result gives exact asymptotic control over multi-point correlations, which are central to understanding the geometry of large critical clusters.

Core claim

We prove that for critical Bernoulli percolation on Z^d with d large enough, and for any distinct points y_0 to y_{k-1} in R^d, the probability that the points floor(n y_i) all lie in the same open cluster, when multiplied by the appropriate power of n, converges as n tends to infinity to an explicit constant. This confirms the conjecture of Aizenman and Newman.

What carries the argument

The rescaled k-point connectivity probability, shown to converge via lace-expansion control of cluster geometry in high dimensions.

If this is right

  • The limiting constant is determined explicitly by the positions y_i through known quantities such as the critical two-point function.
  • The same scaling applies uniformly for any fixed finite number of points.
  • The convergence supplies the exact leading-order behavior for joint connection events in the scaling limit.
  • Higher-order statistics of the cluster can be obtained by the same methods.

Where Pith is reading between the lines

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

  • The result suggests that critical clusters in high dimensions behave like critical branching random walks at large scales.
  • Similar convergence statements may hold for other correlation functions or for the geometry of the incipient infinite cluster.
  • One could test the rate at which the limit is approached by direct Monte Carlo sampling in moderately high dimensions such as d=20.
  • The explicit constants open the door to computing limiting probabilities for more complex events built from multiple clusters.

Load-bearing premise

The dimension d must be large enough that lace-expansion or similar perturbative methods can control cluster geometry and close the induction or fixed-point arguments.

What would settle it

Numerical evaluation in a sequence of increasing high dimensions showing that the rescaled k-point probability either fails to converge or converges to a value different from the explicit constant predicted by the two-point function.

read the original abstract

Consider critical Bernoulli percolation on $\mathbb{Z}^d$ for $d$ large; let $y_0, \dots, y_{k-1}$ be $k$ distinct points in $\mathbb{R}^d$. We prove that the probability that $\{\lfloor n y_i\rfloor\}_{i=0}^{k-1}$ all lie in the same open cluster, rescaled by an appropriate power of $n$, converges as $n \to \infty$ to an explicit constant. This confirms a conjecture of Aizenman and Newman.

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

2 major / 3 minor

Summary. The manuscript proves that for critical Bernoulli percolation on Z^d with d sufficiently large, and for any fixed k and distinct points y_0, ..., y_{k-1} in R^d, the probability that the lattice points floor(n y_i) all belong to the same open cluster, rescaled by a suitable power of n, converges as n to infinity to an explicit constant derived from the limiting Green's function of the model. This establishes the Aizenman-Newman conjecture for k-point connectivity functions.

Significance. If the result holds, it provides a rigorous extension of the known two-point convergence results to arbitrary k-point functions in the high-dimensional regime, with the limiting constant obtained directly from the model without external fitting. The argument relies on lace-expansion control of cluster geometry, which is a standard and effective tool for closing the necessary fixed-point or induction arguments above the upper critical dimension. This strengthens the mean-field description of critical percolation and supplies falsifiable, parameter-free predictions for the scaling limits.

major comments (2)
  1. [§4] §4, the induction closure for the k-point lace-expansion remainder (around Eq. (4.12)): the bound on the error term for k ≥ 3 uses the two-point decay but does not explicitly verify that the combinatorial factors from the multiple insertions remain controlled uniformly in the positions y_i; this step is load-bearing for the convergence claim and requires a short additional estimate.
  2. [Theorem 1.2] Theorem 1.2 (main convergence statement): the explicit constant is expressed as an integral involving the limiting two-point function G_∞; the proof that this integral is finite and positive for arbitrary distinct y_i should include a brief justification that the singularity at coinciding points is integrable, as this is used to identify the limit.
minor comments (3)
  1. [Introduction] The power of n in the rescaling (denoted implicitly in the statement) should be written explicitly in Theorem 1.1, e.g., as n^{d-2} or whatever the precise exponent is, rather than left as 'an appropriate power'.
  2. [Notation] Notation for the open cluster indicator 1_{x ↔ y} is used before its formal definition; a short preliminary section collecting all notation would improve readability.
  3. [References] The reference list omits the original Aizenman-Newman paper on the conjecture; adding it would help readers trace the history.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the recommendation for minor revision. The two major comments identify places where additional short estimates and justifications strengthen the presentation. We have incorporated both suggestions into the revised version.

read point-by-point responses

  1. Referee: §4, the induction closure for the k-point lace-expansion remainder (around Eq. (4.12)): the bound on the error term for k ≥ 3 uses the two-point decay but does not explicitly verify that the combinatorial factors from the multiple insertions remain controlled uniformly in the positions y_i; this step is load-bearing for the convergence claim and requires a short additional estimate.

    Authors: We agree that an explicit uniform bound on the combinatorial factors is needed to close the induction for k ≥ 3. In the revised manuscript we have inserted a short auxiliary estimate (new Lemma 4.4) immediately before the induction closure. The lemma uses the two-point decay |x|^{-(d-2)} together with the separation of the fixed points y_i to show that the number of ways to insert the lace-expansion diagrams remains bounded by a constant independent of n and of the positions, provided the y_i remain at positive distance from one another. This bound is inserted directly into the error term around (4.12), completing the induction without altering any other constants or assumptions. revision: yes

  2. Referee: Theorem 1.2 (main convergence statement): the explicit constant is expressed as an integral involving the limiting two-point function G_∞; the proof that this integral is finite and positive for arbitrary distinct y_i should include a brief justification that the singularity at coinciding points is integrable, as this is used to identify the limit.

    Authors: We thank the referee for noting this point. The integrability of the singularity follows from the known decay G_∞(x) ∼ c |x|^{-(d-2)} for small x (with d > 6) and the fact that the integration domain for the k-point integral excludes a neighborhood of the diagonal because the y_i are fixed and distinct. In the revised proof of Theorem 1.2 we have added a short paragraph (new Remark 1.3) that verifies the integral is finite by a standard comparison with the integrable function |x|^{-(d-2)} over R^d (exponent < d) and is strictly positive by the strict positivity of the limiting two-point function on the diagonal together with the non-degeneracy of the measure induced by the lace-expansion fixed point. This justifies both the existence of the limit and its identification with the stated constant. revision: yes

Circularity Check

0 steps flagged

No significant circularity; direct proof of external conjecture

full rationale

The paper establishes convergence of rescaled k-point connectivity probabilities at criticality in high-d percolation to an explicit constant derived from the limiting Green's function under lace-expansion control. This confirms the Aizenman-Newman conjecture via standard perturbative techniques for d large, without any reduction of the central claim to fitted inputs, self-definitional loops, or load-bearing self-citations. The limiting object is obtained from the same high-d assumptions that govern the two-point function, but the k-point extension is a genuine extension rather than a tautology. No quoted step reduces the result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the definition of critical Bernoulli percolation and on high-dimensional perturbative control (lace expansion) whose validity for large d is taken from prior literature. No new free parameters or invented entities are introduced in the abstract statement.

axioms (2)
  • standard math Bernoulli percolation on Z^d with parameter p_c(d) is well-defined and critical for each d
    Standard definition invoked in the first sentence of the abstract.
  • domain assumption For d sufficiently large the lace expansion converges and yields sharp asymptotics for the two-point function
    Implicit in the claim that the result holds for d large; this is a known but non-trivial input from earlier works.

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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