pith. sign in

arxiv: 2508.07601 · v2 · submitted 2025-08-11 · 🧮 math-ph · math.MP· math.NT· math.PR

Adelic Models of Percolation

Matilde Marcolli This is my paper

Pith reviewed 2026-05-19 00:28 UTC · model grok-4.3

classification 🧮 math-ph math.MPmath.NTmath.PR
keywords percolationadelic geometryhierarchical latticespower meantoric volume formfunction fieldsnumber fields
0
0 comments X

The pith

Adelic product formulas and power mean deformations relate percolation models on lattices to those on hierarchical lattices.

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

The paper establishes relations between long-range percolation on ordinary lattices and on hierarchical lattices by inserting three intermediate geometric constructions. A one-parameter deformation using the power mean function converts the lattice model into one governed by a toric volume form. The adelic product formula for function fields then maps the hierarchical lattice model onto an adelic percolation model, while the corresponding formula for number fields maps the toric model on the ring of integers (in its Minkowski embedding) onto a second adelic model. These bridges are constructed so that connectivity and scaling properties carry across the different geometries.

Core claim

Models of long range percolations on lattices and on hierarchical lattices are related through the use of three intermediate geometries: a 1-parameter deformation based on the power mean function, relating lattice percolation to a percolation model governed by the toric volume form; the adelic product formula for a function field, relating the hierarchical lattice model to an adelic percolation model; and the adelic product formula for number fields that relates the toric percolation model on the lattice given by the ring of integers in the Minkowski embedding to another adelic percolation model.

What carries the argument

The adelic product formulas for function fields and number fields, combined with the one-parameter power-mean deformation that produces the toric volume form model.

If this is right

  • Percolation measures on adelic spaces inherit the connectivity and scaling behavior of the lattice models they are built from.
  • The toric volume form supplies a continuous parameter that interpolates between discrete lattice percolation and adelic models.
  • Arithmetic tools such as product formulas become available for studying critical phenomena in these unified percolation settings.
  • The same chain of deformations and product formulas can be applied to other statistical-mechanics models defined on lattices.

Where Pith is reading between the lines

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

  • Results about percolation thresholds known for hierarchical lattices could be transferred to adelic models and then read back in arithmetic terms.
  • Varying the power-mean parameter continuously might locate transition points where long-range connectivity changes character.
  • Adelic percolation models might provide a setting in which global information from number fields influences local connectivity statistics.

Load-bearing premise

The adelic product formulas can be used to define percolation measures on the corresponding adelic spaces that preserve connectivity and scaling properties of the original lattice and hierarchical models.

What would settle it

An explicit computation for a concrete function field or number field in which the percolation threshold or the existence of an infinite cluster in the adelic model differs from the threshold or cluster property in the corresponding hierarchical or toric lattice model.

Figures

Figures reproduced from arXiv: 2508.07601 by Matilde Marcolli.

Figure 1
Figure 1. Figure 1: The Bruhat–Tits tree of Q2, from [5] [PITH_FULL_IMAGE:figures/full_fig_p020_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The p-adic norm and the Bruhat–Tits tree, from [5]. The p-adic valuation νK(x − y) for x, y ∈ K is computed by the number d⊥(x, y) of steps on the tree that connect the root to the vertex where the paths from the root to x and to y bifurcate, and p-adic distance is therefore (5.13) |x − y|K = p −d⊥(x,y) , see [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
read the original abstract

Models of long range percolations on lattices and on hierarchical lattices are related through the use of three intermediate geometries: a 1-parameter deformation based on the power mean function, relating lattice percolation to a percolation model governed by the toric volume form; the adelic product formula for a function field, relating the hierarchical lattice model to an adelic percolation model; and the adelic product formula for number fields that relates the toric percolation model on the lattice given by the ring of integers in the Minkowski embedding to another adelic percolation model.

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 claims to relate models of long-range percolation on lattices and hierarchical lattices via three intermediate geometries: a one-parameter power-mean deformation connecting lattice percolation to a toric-volume-form percolation model; the adelic product formula for a function field, which maps the hierarchical lattice model to an adelic percolation model; and the adelic product formula for number fields, which maps the toric percolation model on the ring of integers (in the Minkowski embedding) to another adelic percolation model.

Significance. If the constructions are made rigorous and the induced measures on adelic spaces are shown to preserve connectivity, long-range decay, and scaling properties of the source models, the work would supply a novel geometric bridge between discrete percolation and adelic/number-theoretic settings. The explicit use of product formulas and the toric deformation is a distinctive feature that could, in principle, allow transfer of results across these geometries.

major comments (2)
  1. [Section introducing the adelic percolation model via the function-field product formula] The central construction applies the adelic product formula (for function fields) to obtain an adelic percolation model from the hierarchical lattice. However, the product formula supplies a multiplicative measure on the adele ring; the manuscript does not specify the rule that selects which pairs of adelic points are joined by a bond or the probability law on those bonds. Without this step, it is impossible to verify that the resulting random graph inherits the long-range decay or one-arm probabilities of the original hierarchical model. This definition is load-bearing for the claimed equivalence.
  2. [Section on the number-field adelic product formula and the toric-to-adelic map] The number-field construction likewise invokes the adelic product formula to relate the toric percolation model on the Minkowski-embedded ring of integers to an adelic model. No explicit check or calculation is supplied showing that critical exponents, almost-sure connectivity, or scaling limits survive the passage through the product formula. Because the equivalence of the lattice and hierarchical models rests on both intermediate adelic models being faithful, this omission directly affects the main claim.
minor comments (2)
  1. [Section on the power-mean deformation] The power-mean deformation parameter is introduced but its range and limiting cases (recovering the original lattice metric) should be stated explicitly, together with a brief verification that the toric volume form reduces to the standard lattice measure at the appropriate value.
  2. Notation for the adelic measures and the induced bond probabilities should be introduced with a short table or diagram contrasting the original lattice/hierarchical measures with the adelic versions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments correctly identify places where the manuscript would benefit from greater explicitness in defining the adelic bond rules and in verifying the transfer of percolation properties. We address each point below and will revise the manuscript to incorporate the requested clarifications.

read point-by-point responses

  1. Referee: [Section introducing the adelic percolation model via the function-field product formula] The central construction applies the adelic product formula (for function fields) to obtain an adelic percolation model from the hierarchical lattice. However, the product formula supplies a multiplicative measure on the adele ring; the manuscript does not specify the rule that selects which pairs of adelic points are joined by a bond or the probability law on those bonds. Without this step, it is impossible to verify that the resulting random graph inherits the long-range decay or one-arm probabilities of the original hierarchical model. This definition is load-bearing for the claimed equivalence.

    Authors: We agree that an explicit rule for bond formation and the associated probability law must be stated clearly if the inheritance of long-range decay and one-arm probabilities is to be verified. The construction in the manuscript defines the adelic model by transporting the hierarchical connection probabilities via the product formula: two adelic points are joined by a bond with probability equal to the product, over all places, of the local connection probabilities determined by the valuations and the hierarchical distance at the corresponding place. This multiplicative structure is intended to preserve the decay. We nevertheless acknowledge that the manuscript presents this rule only implicitly. In the revision we will add a precise definition of the bond-selection map together with a short argument showing that the one-arm event probabilities are bounded above and below by the corresponding quantities on the hierarchical lattice. revision: yes

  2. Referee: [Section on the number-field adelic product formula and the toric-to-adelic map] The number-field construction likewise invokes the adelic product formula to relate the toric percolation model on the Minkowski-embedded ring of integers to an adelic model. No explicit check or calculation is supplied showing that critical exponents, almost-sure connectivity, or scaling limits survive the passage through the product formula. Because the equivalence of the lattice and hierarchical models rests on both intermediate adelic models being faithful, this omission directly affects the main claim.

    Authors: The referee is right that the manuscript does not contain explicit calculations confirming the survival of critical exponents or scaling limits under the number-field product formula. Our reasoning rests on the fact that the toric volume form is continuous with respect to the Minkowski embedding and that the product formula yields a measure whose finite-dimensional marginals are comparable to those of the original toric model, thereby preserving almost-sure connectivity at the critical parameter. We concede, however, that these comparisons are only sketched. The revised version will include a lemma establishing that the probability of the one-arm event in the adelic model is sandwiched between the corresponding probabilities in the toric model, together with a brief discussion of how this implies equality of the critical exponents. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper relates lattice and hierarchical long-range percolation models via three intermediate constructions: a power-mean deformation to a toric-volume model, then adelic product formulas (for function fields and number fields) to produce percolation measures on adele rings. These product formulas are standard, externally established results in algebraic geometry and number theory; the paper invokes them to transport connectivity and scaling properties rather than deriving the formulas from the percolation data itself. No self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain appears in the abstract or described structure. The central claim therefore rests on independent mathematical input and does not reduce to its own outputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The construction rests on the standard adelic product formula from algebraic number theory and on the existence of a toric volume form; the 1-parameter power-mean deformation is introduced without independent justification in the abstract.

free parameters (1)
  • power-mean deformation parameter
    A single real parameter that interpolates between the lattice and toric-volume percolation models.
axioms (1)
  • standard math Adelic product formula holds for the function field and number field cases used to define the intermediate percolation models.
    Invoked to relate hierarchical and toric models to their adelic counterparts.
invented entities (1)
  • adelic percolation model no independent evidence
    purpose: Intermediate object that equates the hierarchical lattice percolation with an adelic construction via the product formula.
    New model introduced by the paper; no independent evidence or falsifiable prediction supplied in the abstract.

pith-pipeline@v0.9.0 · 5606 in / 1484 out tokens · 44018 ms · 2026-05-19T00:28:10.230234+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

  1. [1]

    Aizenman, C.M

    M. Aizenman, C.M. Newman, Tree graph inequalities and critical behavior in percolation models , Journal of Statistical Physics, Vol. 36 (1984) N.1-2, 107–143

    work page 1984

  2. [2]

    Beilinson, Yu.I

    A.A. Beilinson, Yu.I. Manin, The Mumford form and the Polyakov measure in string theory . Comm. Math. Phys. 107 (1986), no. 3, 359–376

    work page 1986

  3. [3]

    Chen, M.D

    H.H. Chen, M.D. Huang, On p-adic expansions of algebraic integers . ISSAC’15—Proceedings of the 2015 ACM International Symposium on Symbolic and Algebraic Computation, 109–116, ACM, 2015

    work page 2015

  4. [4]

    D. A. Dawson and L. G. Gorostiza, Percolation in an ultrametric space, Electron. J. Probab., 18:no. 12, 26, 2013

    work page 2013

  5. [5]

    Heydeman, Matthew; Marcolli, Matilde; Saberi, Ingmar A.; Stoica, Bogdan, Tensor networks, p-adic fields, and algebraic curves: arithmetic and the AdS3/CFT2 correspondence , Adv. Theor. Math. Phys. 22 (2018), no. 1, 93–176

    work page 2018

  6. [6]

    Heydeman, M

    M. Heydeman, M. Marcolli, S. Parikh, I. Saberi, Nonarchimedean holographic entropy from networks of perfect tensors. Adv. Theor. Math. Phys. 25 (2021), no. 3, 591–721

    work page 2021

  7. [7]

    Heydenreich, R

    M. Heydenreich, R. van der Hofstad, Progress in high-dimensional percolation and random graphs . CRM Short Courses. Centre de Recherches Math´ ematiques, Montreal, QC, Springer, 2017

    work page 2017

  8. [8]

    Tom Hutchcroft, The critical two-point function for long-range percolation on the hierarchical lattice, Ann. Appl. Probab. 34 (2024), no. 1B, 986–1002

    work page 2024

  9. [9]

    Tom Hutchcroft, Critical cluster volumes in hierarchical percolation, Proc. Lond. Math. Soc. (3) 130 (2025), no. 1, Paper No. e70023, 88 pp

    work page 2025

  10. [10]

    Tom Hutchcroft, Sharp hierarchical upper bounds on the critical two-point function for long-range percolation on Zd. J. Math. Phys. 63 (2022), no. 11, Paper No. 113301, 18 pp

    work page 2022

  11. [11]

    Kadiri, Explicit zero-free regions for Dedekind zeta functions, Int

    H. Kadiri, Explicit zero-free regions for Dedekind zeta functions, Int. J. Number Theory 8 (2012), no. 1, 125–147

    work page 2012

  12. [12]

    Koval, R

    V. Koval, R. Meester, P. Trapman, Long-range percolation on the hierarchical lattice, Electron. J. Probab., 17:no. 57, 21, 2012

    work page 2012

  13. [13]

    Leinster, Entropy and Diversity , Cambridge University Press, 2021

    T. Leinster, Entropy and Diversity , Cambridge University Press, 2021

    work page 2021

  14. [14]

    Lyons, Random walks and percolation on trees , Ann

    R. Lyons, Random walks and percolation on trees , Ann. Prob. Vol.18 (1990) N.3, 931–958

    work page 1990

  15. [15]

    Manin, p-adic automorphic functions , Current problems in mathematics, Vol

    Yu.I. Manin, p-adic automorphic functions , Current problems in mathematics, Vol. 3, pp. 5–92, 259, Itogi Nauki i Tekhniki, Akad. Nauk SSSR Vsesojuz. Inst. Nauchn. i Tehn. Informacii, Moscow, 1974

    work page 1974

  16. [16]

    Manin, The partition function of the Polyakov string can be expressed in terms of theta-functions , Phys

    Yu.I. Manin, The partition function of the Polyakov string can be expressed in terms of theta-functions , Phys. Lett. B 172 (1986), no. 2, 184–185

    work page 1986

  17. [17]

    Conformal invariance and string theory (Poiana Bra¸ sov, 1987)

    Yu.I. Manin, Reflections on arithmetical physics , in “Conformal invariance and string theory (Poiana Bra¸ sov, 1987)”, 293–303, Perspect. Phys., Academic Press, 1989. ADELIC MODELS OF PERCOLATION 29

    work page 1987

  18. [18]

    Manin, Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry , Invent

    Yu.I. Manin, Three-dimensional hyperbolic geometry as ∞-adic Arakelov geometry , Invent. Math. 104 (1991), no. 2, 223–243

    work page 1991

  19. [19]

    Manin, M

    Yu.I. Manin, M. Marcolli, Holography principle and arithmetic of algebraic curves . Adv. Theor. Math. Phys. 5 (2001), no. 3, 617–650

    work page 2001

  20. [20]

    M. Mustata, Zeta functions in algebraic geometry , https://public.websites.umich.edu/~mmustata/zeta_book.pdf Department of Mathematics and Department of Computing and Mathematical Sciences, California Institute of Technology, Pasadena, CA 91125, USA Email address: matilde@caltech.edu

    work page