OAI.Percolation.exists_uniform_averaged_decrease: intermediate lemma from OpenAI, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (result family 214)

Proved

OAI.Percolation.exists_uniform_averaged_decrease

by dbenbenn · Oct 27, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Percolation.arbitrary_cayley, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (result family 214), OpenAI Math Release, September 24, 2026, result family 214, github.com/openai/math, Apache License 2.0); in Lean it is the declaration OAI.Percolation.exists_uniform_averaged_decrease of lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Deformation/AveragedDecrease.lean at commit adc7f12, ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Percolation, with open MeasureTheory ProbabilityTheory, open scoped ENNReal NNReal, open MeasureTheory, open scoped ComplexOrder):

theorem exists_uniform_averaged_decrease (G : BondGraph V E) (hG : Function.Injective G)
    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1)
    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
    (hPs : IsSelfAdjoint P) :
    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T → ∀ (k : ℕ), 3 ≤ k →
      ∀ (t : ℝ), 0 ≤ t →
      Real.exp (-t*Real.exp (C*(k:ℝ))) *
        (rawPathAverage G (walkKernel G hloc a) k root weight
          (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
          (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
            (‖P‖^k * (rootTrace root weight (T^3)).re)) ≤
        ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p

Definitions. The statement is written with OAI.Percolation.LocallyFinite, OAI.Percolation.PositiveRealMatrix, OAI.Percolation.QuasiTransitive, OAI.Percolation.RootTransport, OAI.Percolation.matrixCoeff and OAI.Percolation.rootTrace (definitions bundle OAIQ040C000 of this split); OAI.Percolation.deformedDerivativeMass, OAI.Percolation.degree, OAI.Percolation.maxSusceptibility, OAI.Percolation.rawPathAverage, OAI.Percolation.retainedSusceptibilityProduct, OAI.Percolation.vertexAut and OAI.Percolation.walkKernel (definitions bundle OAIQ040C002 of this split); OAI.Percolation.BondGraph, OAI.Percolation.HasKernel, OAI.Percolation.Hilbert, OAI.Percolation.law and OAI.Percolation.pc (the published bundle Def_CayleyPercolation).

Used by. The proof of OAI.Percolation.exists_finite_scale_obstruction of this split imports it.

Preamble

import Mathlib
import Definitions.Def_CayleyPercolation
import Definitions.Def_OAIQ040C002

Formal statement

section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

open MeasureTheory ProbabilityTheory
open scoped ENNReal NNReal

namespace Percolation
universe u v
variable {V : Type u} {E : Type v}
open MeasureTheory
open scoped ComplexOrder
section UniformAveragedDecrease
attribute [local instance] Classical.propDecidable Classical.decEq
variable [Countable V] [Countable E] [Nonempty V]
variable {ι : Type*} [Fintype ι]
attribute [local instance] spectralRealOperatorScalarTower spectralRealOperatorSMulCommClass

theorem exists_uniform_averaged_decrease (G : BondGraph V E) (hG : Function.Injective G)
    (hloc : LocallyFinite G) (hqt : QuasiTransitive G)
    (root : ι → V) (weight : ι → ℝ) (hw : ∀ i, 0 ≤ weight i)
    (Γ : Set (V ≃ V)) (hΓ : Γ ⊆ vertexAut G) (htr : RootTransport root weight Γ)
    (d : ℕ) (hd : ∀ x, degree G x ≤ d) (a : ℝ)
    (hK : ∀ x y, 0 ≤ walkKernel G hloc a x y)
    (hpcrit : 0 < (pc G:ℝ)) (hpcrit₁ : (pc G:ℝ) < 1)
    (P : Hilbert V →L[ℂ] Hilbert V) (hpos : PositiveRealMatrix P)
    (hP : ∀ x y, matrixCoeff P x y = (walkKernel G hloc a x y : ℂ))
    (hPs : IsSelfAdjoint P) :
    ∃ C : ℝ, 0 < C ∧ ∀ (p : unitInterval), (pc G:ℝ)/2 ≤ (p:ℝ) → p < pc G →
      ∀ (T : Hilbert V →L[ℂ] Hilbert V), HasKernel G p T → ∀ (k : ℕ), 3 ≤ k →
      ∀ (t : ℝ), 0 ≤ t →
      Real.exp (-t*Real.exp (C*(k:ℝ))) *
        (rawPathAverage G (walkKernel G hloc a) k root weight
          (retainedSusceptibilityProduct G (walkKernel G hloc a) p t) -
          (1+2*(d:ℝ)/(pc G:ℝ))*(maxSusceptibility G p)^2 *
            (‖P‖^k * (rootTrace root weight (T^3)).re)) ≤
        ∑ i, weight i * ∫ ω, deformedDerivativeMass G (walkKernel G hloc a) p k t Set.univ ω (root i) ∂law p := by
  sorry

end UniformAveragedDecrease
end Percolation

end OAI
end

Source

OpenAI, Nonuniqueness of percolation on nonamenable quasi-transitive graphs (result family 214), OpenAI Math Release, September 24, 2026, https://github.com/openai/math (result family 214; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Nonuniqueness-of-percolation-on-nonamenable-quasi-transitive-graphs-September-24-2026/paper.pdf; Apache License 2.0); Lean: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/lean/OAI/Probability/UnimodularPercolation/UnimodularPercolation/Deformation/AveragedDecrease.lean (declaration OAI.Percolation.exists_uniform_averaged_decrease)

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