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arxiv: 2604.18558 · v1 · submitted 2026-04-20 · 🧮 math.PR · math-ph· math.MP

Uniform analyticity of local observables in FK-percolation and analyticity of the Ising spontaneous magnetisation

Lucas D'Alimonte , Lo\"ic Gassmann This is my paper

Pith reviewed 2026-05-10 03:40 UTC · model grok-4.3

classification 🧮 math.PR math-phmath.MP
keywords FK-percolationanalyticityIsing modelPotts modelspontaneous magnetizationmixing conditionspercolation parameterconnectivity probabilities
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The pith

Local event probabilities in FK-percolation are uniformly analytic in the bond probability p under mixing assumptions.

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

The paper shows that probabilities of local events in the FK-percolation model depend analytically on the percolation parameter p whenever the measure satisfies suitable mixing conditions, and that these probabilities obey a uniform exponential growth bound in the size of the event. This uniform analyticity transfers directly to the associated Potts model, establishing that the spontaneous magnetization remains analytic in a range of parameters that includes the full supercritical regime of the Ising model in every dimension d at least 3. The same technique also yields analyticity of the susceptibility throughout the subcritical interval for any Potts model with two or more colors, as well as analyticity of multi-point and truncated connectivity functions in the FK measure itself.

Core claim

The probabilities of local events in the FK-percolation model are uniformly analytic in the percolation parameter p and satisfy a uniform exponential growth bound under suitable mixing assumptions on the measure. This result implies that the magnetisation of the Potts model is analytic in a suitable range of parameters, including the Ising case in all dimensions d ≥ 3 throughout the supercritical regime. Analyticity likewise holds for the susceptibility of the Potts model with q colours for any q ≥ 2 in the whole subcritical interval, and for various multi-point and truncated connectivity probabilities in the FK-percolation measure.

What carries the argument

Uniform analyticity of local observables with respect to the percolation parameter p, obtained from mixing assumptions and transferred to Potts and Ising quantities.

If this is right

  • The spontaneous magnetization of the Ising model is analytic throughout the supercritical regime in every dimension d ≥ 3.
  • The susceptibility of the Potts model is analytic across the entire subcritical interval for every number of colors q ≥ 2.
  • Multi-point and truncated multi-point connectivity probabilities in FK-percolation are analytic in the percolation parameter p.

Where Pith is reading between the lines

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

  • The same mixing-based argument could be tested on other dependent percolation models that obey comparable spatial mixing to obtain parallel analyticity statements.
  • Analyticity of magnetization and susceptibility opens the possibility of convergent power-series representations that could be used to compute these quantities away from criticality.
  • The absence of singularities in these local and global observables suggests that the phase transition in the Ising and Potts models manifests only through changes in decay rates rather than through non-analyticities in the functions themselves.

Load-bearing premise

The FK-percolation measure satisfies suitable mixing assumptions that support the uniform analyticity and exponential growth bounds.

What would settle it

An explicit example or numerical computation exhibiting a non-analytic local event probability as a function of p inside a mixing FK-percolation regime, or a singularity in the Ising spontaneous magnetization within the supercritical phase for some d ≥ 3.

read the original abstract

We prove that, in the FK-percolation model, the probabilities of local events are uniformly analytic in the percolation parameter $p$ under suitable mixing assumptions on the measure, and satisfy a uniform exponential growth bound. This result allows us to prove that the magnetisation of the Potts model is analytic in a suitable range of parameters, including the Ising case in all dimensions $d \geq 3$ in the whole supercritical regime. We also provide a proof of the analyticity of the susceptibility of the Potts model with $q$ colours, for any $q \geq 2$ in the whole subcritical interval. Finally, we prove the analyticity of various quantities in the FK-percolation measure, including the multi-point and truncated multi-point connectivity probabilities.

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 / 2 minor

Summary. The manuscript proves that, in the FK-percolation model, the probabilities of local events are uniformly analytic in the percolation parameter p under suitable mixing assumptions on the measure and satisfy a uniform exponential growth bound. This is used to establish analyticity of the spontaneous magnetization for the Potts model (including the Ising case q=2) in d ≥ 3 throughout the supercritical regime, analyticity of the susceptibility for q ≥ 2 in the whole subcritical interval, and analyticity of multi-point and truncated multi-point connectivity probabilities in FK-percolation.

Significance. If the mixing assumptions hold with p-uniform constants, the results would provide a rigorous route to analyticity of key observables across phase transitions in percolation and Potts models, extending known subcritical analyticity into the supercritical regime for magnetization. The technique of deriving uniform bounds on local events to control global quantities is potentially useful for other lattice models with exponential mixing.

major comments (2)
  1. [Abstract and §1] Abstract and §1 (statement of main results): the claim that the Ising spontaneous magnetization is analytic in the whole supercritical regime for d ≥ 3 rests on the mixing assumptions holding with constants independent of p down to p_c. No verification or citation is given that exponential mixing (or the required decay of correlations) remains uniform for the FK-Ising measure in d=3 as p ↓ p_c; without this, the analyticity radius may shrink to zero and the application fails.
  2. [Application to magnetization (§4–5)] Theorem on magnetization (application section, e.g. §4–5): the passage from uniform analyticity of local events plus the exponential growth bound to analyticity of the magnetization uses a series expansion whose radius is controlled by the mixing rate. If the mixing constant deteriorates near p_c (as is possible in d=3), the bound does not remain uniform and the claimed analyticity in the entire supercritical interval is not justified.
minor comments (2)
  1. [Preliminaries] The precise statement of the 'suitable mixing assumptions' (e.g., the form of the exponential decay and the constants) should be isolated in a dedicated preliminary subsection with explicit dependence on p and d.
  2. [Notation and definitions] Notation for local events and the percolation parameter p could be standardized earlier; occasional shifts between wired and free boundary conditions are not always flagged when the mixing assumption is invoked.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of the mixing assumptions in the applications. We address the two major comments below and will make revisions to clarify the conditional nature of the results.

read point-by-point responses

  1. Referee: [Abstract and §1] Abstract and §1 (statement of main results): the claim that the Ising spontaneous magnetization is analytic in the whole supercritical regime for d ≥ 3 rests on the mixing assumptions holding with constants independent of p down to p_c. No verification or citation is given that exponential mixing (or the required decay of correlations) remains uniform for the FK-Ising measure in d=3 as p ↓ p_c; without this, the analyticity radius may shrink to zero and the application fails.

    Authors: The main theorem on uniform analyticity of local observables (and the associated exponential growth bound) is explicitly conditional on the mixing assumptions holding with constants independent of p. The application to the spontaneous magnetization of the Potts model, including the Ising case q=2 in d ≥ 3, is presented under this hypothesis. We agree that the manuscript does not provide a dedicated citation or verification that the mixing rate for the FK-Ising measure remains uniform down to p_c in d=3. In the revised version we will add a clarifying remark in the introduction and in the application section stating that the analyticity of the magnetization holds throughout the supercritical regime provided the p-uniform mixing assumption is satisfied, and we will note that while such uniformity is established in high dimensions, its verification for d=3 as p ↓ p_c is a separate question. revision: partial

  2. Referee: [Application to magnetization (§4–5)] Theorem on magnetization (application section, e.g. §4–5): the passage from uniform analyticity of local events plus the exponential growth bound to analyticity of the magnetization uses a series expansion whose radius is controlled by the mixing rate. If the mixing constant deteriorates near p_c (as is possible in d=3), the bound does not remain uniform and the claimed analyticity in the entire supercritical interval is not justified.

    Authors: The series expansion argument indeed derives its radius from the mixing rate. When the mixing constants are p-uniform, the argument yields analyticity on the entire open interval (p_c,1]. If the mixing rate deteriorates as p ↓ p_c, the radius at each fixed p > p_c remains positive (so local analyticity holds pointwise), but the proof as written relies on uniformity to control the expansion simultaneously across the interval. We will revise §4–5 to distinguish the uniform case from the p-dependent case and to state explicitly that the claimed analyticity throughout the supercritical regime is conditional on p-uniform mixing. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external mixing assumptions

full rationale

The paper proves uniform analyticity of local event probabilities in FK-percolation under suitable mixing assumptions on the measure, then applies this to establish analyticity of Potts/Ising magnetization in the supercritical regime. No equations or sections reduce the central claims by construction to fitted parameters, self-definitions, or load-bearing self-citations; the mixing conditions are external inputs, and the proofs use standard percolation techniques without tautological reduction. The result is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of probability measures on lattices together with the key domain assumption of mixing; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • domain assumption Suitable mixing assumptions on the FK-percolation measure
    Invoked to guarantee uniform analyticity and exponential growth bounds for local event probabilities.

pith-pipeline@v0.9.0 · 5436 in / 1364 out tokens · 43579 ms · 2026-05-10T03:40:59.569995+00:00 · methodology

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