Lean statement

Nonuniqueness phase and critical laws on quasi-transitive graphs

Lean is a language for mathematical statements and proofs that a computer can verify. This page shows one statement file from OpenAI’s release, possible paper matches and independent reports. The code-like names below identify the definitions or theorems selected by the proof-verification settings.

Theorems selected for verification: OAI.Percolation.BenjaminiSchramm.full_main OAI.Percolation.BenjaminiSchramm.cayley_main OAI.Percolation.BenjaminiSchramm.critical_laws

Lean statement file: BenjaminiSchramm.leanShow on the mapOpenAI’s Lean catalogue does not list this as a main result; the release’s Lean notes for family 214 link it.

See the candidate theorem Who ran the proof-verification program Report a comparison with the paper (GitHub issue)

The Lean statement

The lines of the Lean statement that state the theorem, then the definitions in the file that it uses.

theorem full_main {V : Type u} {E : Type v} [Infinite V]    (G : BondGraph V E) (hc : Connected G) (hl : LocallyFinite G)    (hq : QuasiTransitive G) (hh : 0 < hV G) : MainConclusion G := by  sorry⋮theorem critical_laws {V : Type u} {E : Type v} [Infinite V]    (G : BondGraph V E) (hc : Connected G) (hl : LocallyFinite G)    (hq : QuasiTransitive G) (hh : 0 < hV G) : CriticalLawsConclusion G := by  sorry⋮theorem cayley_main {Λ : Type u} [Group Λ]    (S : Finset Λ) (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (_hone : 1 ∉ S)    (hgen : Subgroup.closure (S : Set Λ) = ⊤) (hna : ¬ FolnerAmenable Λ) :    OperatorBounded (cayleyBond S) (pc (cayleyBond S)) ∧      MainConclusion (cayleyBond S) ∧      ∀ p : unitInterval, pc (cayleyBond S) < p → p < pu (cayleyBond S) →        ∀ᵐ ω ∂law p, (infiniteClusters (cayleyBond S) ω).Infinite := by  sorry

A Lean statement states the theorem and leaves its proof as sorry on purpose. OpenAI’s proof is in a separate solution module, and a program called Comparator checks that proof against this statement.

Definitions needed to read it

Local definitions the statement names, followed through the definitions they name in turn. The match is by name in the source text, so a name reached only through notation can be missed; the full file is below.

abbreviation OAI.Percolation.BondGraph line 12

abbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 V

definition OAI.Percolation.Adj line 14

def Adj (G : BondGraph V E) (x y : V) : Prop := ∃ e, G e = s(x, y)

definition OAI.Percolation.LocallyFinite line 16

def LocallyFinite (G : BondGraph V E) : Prop := ∀ x, {e | x ∈ G e}.Finite

definition OAI.Percolation.Connected line 18

def Connected (G : BondGraph V E) : Prop :=  ∀ x y, Relation.ReflTransGen (Adj G) x y

definition OAI.Percolation.IsAut line 21

def IsAut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E) : Prop :=  ∀ e, G (b e) = (G e).map a

definition OAI.Percolation.QuasiTransitive line 24

def QuasiTransitive (G : BondGraph V E) : Prop :=  ∃ representatives : Finset V, ∀ x, ∃ o ∈ representatives,    ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ a o = x

definition OAI.Percolation.vertexBoundary line 28

def vertexBoundary (G : BondGraph V E) (A : Finset V) : Set V :=  {y | y ∉ A ∧ ∃ x ∈ A, Adj G x y}

definition OAI.Percolation.hV line 31

noncomputable def hV (G : BondGraph V E) : ℝ :=  sInf {r | ∃ A : Finset V, A.Nonempty ∧    r = (vertexBoundary G A).ncard / (A.card : ℝ)}

definition OAI.Percolation.Conn line 35

def Conn (G : BondGraph V E) (ω : Set E) (x y : V) : Prop :=  Relation.ReflTransGen (fun a b => ∃ e ∈ ω, G e = s(a, b)) x y

definition OAI.Percolation.cluster line 38

def cluster (G : BondGraph V E) (ω : Set E) (x : V) : Set V :=  {y | Conn G ω x y}

definition OAI.Percolation.infiniteClusters line 41

def infiniteClusters (G : BondGraph V E) (ω : Set E) : Set (Set V) :=  {C | C.Infinite ∧ ∃ x, C = cluster G ω x}

definition OAI.Percolation.law line 44

noncomputable def law (p : unitInterval) : Measure (Set E) :=  setBernoulli Set.univ p

definition OAI.Percolation.pc line 47

noncomputable def pc (G : BondGraph V E) : unitInterval :=  sInf {p | 0 < law p {ω | (infiniteClusters G ω).Nonempty}}

definition OAI.Percolation.pu line 50

noncomputable def pu (G : BondGraph V E) : unitInterval :=  sInf {p | law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1}

definition OAI.Percolation.twoPoint line 53

noncomputable def twoPoint (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=  (law p {ω | Conn G ω x y}).toReal

abbreviation OAI.Percolation.Hilbert line 56

noncomputable abbrev Hilbert (V : Type u) := lp (fun _ : V => ℂ) 2

definition OAI.Percolation.HasKernel line 58

noncomputable def HasKernel [DecidableEq V]    (G : BondGraph V E) (p : unitInterval)    (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=  ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = (twoPoint G p x y : ℂ)

definition OAI.Percolation.OperatorBounded line 63

noncomputable def OperatorBounded (G : BondGraph V E) (p : unitInterval) : Prop := by  classical  exact ∃ A : Hilbert V →L[ℂ] Hilbert V, HasKernel G p A

definition OAI.Percolation.operatorNorm line 67

noncomputable def operatorNorm (G : BondGraph V E) (p : unitInterval) : ℝ≥0∞ := by  classical  exact sInf {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,    HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖}

definition OAI.Percolation.ptwo line 72

noncomputable def ptwo (G : BondGraph V E) : unitInterval :=  sSup {p | OperatorBounded G p}

definition OAI.Percolation.labelLaw line 75

noncomputable def labelLaw : Measure (E → unitInterval) :=  Measure.infinitePi (fun _ => (volume : Measure unitInterval))

definition OAI.Percolation.openEdges line 78

def openEdges (labels : E → unitInterval) (p : unitInterval) : Set E :=  {e | labels e ≤ p}

definition OAI.Percolation.MainConclusion line 81

def MainConclusion (G : BondGraph V E) : Prop :=  (sSup {r | ∃ p : unitInterval, p < pc G ∧ r = operatorNorm G p} =    operatorNorm G (pc G)) ∧  operatorNorm G (pc G) < ⊤ ∧  pc G < ptwo G ∧ ptwo G ≤ pu G ∧  ∃ p₁ p₂ : unitInterval, pc G < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),      ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters G (openEdges labels p)).Infinite

definition OAI.Percolation.openSimpleGraph line 99

def openSimpleGraph (G : BondGraph V E) (ω : Set E) : SimpleGraph V where  Adj x y := x ≠ y ∧ ∃ e ∈ ω, G e = s(x,y)  symm := ⟨by    rintro x y ⟨hxy, e, he, h⟩    exact ⟨hxy.symm, e, he, h.trans (Sym2.eq_swap ..)⟩⟩  loopless := ⟨by simp⟩

18 more definitions are reached the same way; the full file lists them.

Possible paper match, with no recorded review

No paper statement clearly matched the Lean wording. Statement Map, the program that collects Atlas’s source records, selected this possible match using words in OpenAI’s catalogue and Lean note. No review of whether the statements mean the same thing is recorded.

Shared with the catalogue’s wording or OpenAI’s note, none of them matched in the Lean text: almost, bernoulli, bond, cayley, clusters, connection, critical, finite and 17 more.

Corollary 1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs

Corollary (Arbitrary Cayley graphs)

Let \Lambda be a nonamenable finitely generated group, and let S be any finite symmetric generating set with the identity removed. For Bernoulli bond percolation on the simple undirected Cayley graph with edges \{x,xs\}, x\in\Lambda, s\in S, the critical connection operator is bounded on counting-measure \ell^2(\Lambda) and

\pc<\ptwo\le\pu.

The simultaneous nonuniqueness interval in thm:main exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.

Section 1, Introduction Read it in the paper TeX source, line 80

Show the TeX

Let $\Lambda$ be a nonamenable finitely generated group, and let $S$
be any finite symmetric generating set with the identity removed.
For Bernoulli bond percolation on the simple undirected Cayley graph
with edges $\{x,xs\}$, $x\in\Lambda$, $s\in S$, the critical connection
operator is bounded on counting-measure $\ell^2(\Lambda)$ and
\[
 \pc<\ptwo\le\pu.
\]
The simultaneous nonuniqueness interval in \cref{thm:main} exists for
every such fixed generating set. At every fixed $p\in(\pc,\pu)$,
there are almost surely infinitely many infinite open clusters.

Other possible paper matches, closest wording first

Theorem 1.1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: uniform, edge, bernoulli, pu, bond, quasi and eleven more. Found in the paper’s title, which its main theorem states: nonuniqueness. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: coupling, surely, many, infinitely and almost.
Corollary 2 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: pexp, connectivity, integers, dist, laws, exponential and 16 more.
Theorem 2.1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: boundedness, aut, simple, quasi, connected, graph and seven more.
Proposition 9 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: edges, interval, uniform, edge, quasi, connected and seven more. Shared only with the release’s name for the Lean statement or OpenAI’s note on it: coupling, surely, many, infinitely and almost.
Proposition 2 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: radius, event, prob, 1/2, cluster, percolation and three more.
Proposition 1 of Nonuniqueness of percolation on nonamenable quasi-transitive graphs. Words of the Lean statement, its names split into words, that also appear in the theorem: gap, aut, pu, operator, ptwo, bounded and one more.

How Atlas selects possible matches: it compares words in the Lean statement with every statement in the family. It also splits Lean names into words. The release’s label and note receive less weight; a main theorem wins a close match. If the Lean wording yields no clear match, Atlas uses the catalogue description. This selects passages to compare; it does not establish that they mean the same thing.

Reported Comparator runs

two independent groups report that Comparator accepted runs for this Lean statement file: Carlos Toledo’s cert-machine; MathVet. Each group reports its own Comparator run for this statement file. Atlas has not repeated the runs.

For a configured theorem, Comparator tests whether OpenAI’s proof proves the exact Lean statement. Some files instead configure definitions for comparison by name and type. Acceptance does not establish that the statement means the same thing as the paper’s theorem.

Carlos Toledo’s cert-machine

How they ran it, in their words

Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners (.github/workflows/openai-math-kernel.yml), one j…

Accepted Their term: “CERTIFIED”.
Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.
Reported 10 October 2026, at the release version shown here. Original verification report

MathVet

How they ran it, in their words

An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.

Accepted Their term: “comparator-pass”.
Re-run through Comparator: accepted.
Reported 10 October 2026, at the release version shown here. Original verification report

What an accepted Comparator run would establish

If Comparator is run under this configuration and accepts, the solution proves the configured theorem using only the allowed axioms, and the Lean kernel accepts every step.

This configuration lists no definition holes, so every definition the statement uses is compared as it is written in this file.

Allowed axioms: propext, Quot.sound, Classical.choice

Acceptance leaves open whether the Lean statement means the same thing as the paper’s theorem, whether the paper’s other claims are correct, and whether the result is new. Those questions require comparing the definitions and statements, reviewing the paper’s arguments, and reading the earlier literature.

Do family 214’s Lean statements express the advertised result?

These assessments compare all the family’s Lean statements with the whole result OpenAI describes in its catalogue. They do not assess this statement alone. Each group uses its own published rules.

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet
Finds the whole advertised result stated or implied Their term: “full”.
The challenge statements imply the headline claim after unpacking definitions or one routine step. Their record
Their note: full_main (infinite, connected, locally finite, quasi-transitive multigraph with 0 < hV G) concludes MainConclusion G including pc G < ptwo G ∧ ptwo G ≤ pu G and a coupled interval [p₁,p₂] above pc with infinitely many infinite clusters a.s.; critical_laws gives the finite triangle diagram triangleDiagram G (pc G) x ≤ operatorNorm^3 ∧ < ⊤ plus mean-field laws; cayley_main/arbitrary_cayley handle every finite symmetric generating set of a non-Folner-amenable group for every p ∈ (pc, pu). Nonamenability is encoded as positive vertex-isoperimetric constant (graphs) or failure of the Folner condition (groups).
Definitions the group asks readers to compare with the paper: Custom 'amenable': def FolnerAmenable (Λ) : Prop := ∀ K : Finset Λ, ∀ ε > 0, ∃ A : Finset Λ, A.Nonempty ∧ ∀ k ∈ K, ((A.image (· * k)) \ A).card < ε * A.card (and the identical AmenableGroup in CayleyPercolation) is the standard right-Folner criterion, not Mathlib's; for general graphs nonamenability is 0 < hV G with hV G := sInf {|∂A| / |A| : A finite nonempty} (outer vertex boundary). Standard but hand-written.
Carlos Toledo’s cert-machine
Finds the whole advertised result stated or implied Their term: “MATCHES”.
The challenge statements state the headline as OpenAI wrote it. Their record

Read the complete assessments on the family page.

OpenAI’s own note on what this Lean statement covers

“The formalization proves the nonuniqueness phase conjecture for Bernoulli bond percolation on infinite connected locally finite nonamenable quasi-transitive graphs. It establishes $p_c<p_{2\to2}\le p_u$ and a common nonempty coupled interval with infinitely many infinite clusters almost surely.”
Quoted from OpenAI’s Lean note for this family

An AI agent drafted an interpretation of that note, classifying this statement as “states a paper’s theorem as the paper does”. No human review of the draft is recorded.

Supporting material

The complete Lean statement file

240 lines, copied from release version fd4aeeb. A name defined in this file links to its line; pointing at the name shows its definition.

Declared in this fileOpenAI’s library at fd4aeebOther linked names open the current documentation for Mathlib, Lean’s mathematical library.

Links in the file: 150 to its own declarations, 93 to Mathlib’s documentation. 44 declared names also link to the same declaration in OpenAI’s library. OpenAI uses Mathlib version d13f23b. The linked documentation describes the current version, so names that have since changed may be missing there.

import Mathlibnamespace OAIopen MeasureTheory ProbabilityTheoryopen scoped ENNReal NNRealnamespace Percolationuniverse u vvariable {V : Type u} {E : Type v}abbrev BondGraph (V : Type u) (E : Type v) := E → Sym2 Vdef Adj (G : BondGraph V E) (x y : V) : Prop := ∃ e, G e = s(x, y)def LocallyFinite (G : BondGraph V E) : Prop := ∀ x, {e | x ∈ G e}.Finitedef Connected (G : BondGraph V E) : Prop :=  ∀ x y, Relation.ReflTransGen (Adj G) x ydef IsAut (G : BondGraph V E) (a : V ≃ V) (b : E ≃ E) : Prop :=  ∀ e, G (b e) = (G e).map adef QuasiTransitive (G : BondGraph V E) : Prop :=  ∃ representatives : Finset V, ∀ x, ∃ o ∈ representatives,    ∃ a : V ≃ V, ∃ b : E ≃ E, IsAut G a b ∧ a o = xdef vertexBoundary (G : BondGraph V E) (A : Finset V) : Set V :=  {y | y ∉ A ∧ ∃ x ∈ A, Adj G x y}noncomputable def hV (G : BondGraph V E) : ℝ :=  sInf {r | ∃ A : Finset V, A.Nonempty ∧    r = (vertexBoundary G A).ncard / (A.card : ℝ)}def Conn (G : BondGraph V E) (ω : Set E) (x y : V) : Prop :=  Relation.ReflTransGen (fun a b => ∃ e ∈ ω, G e = s(a, b)) x ydef cluster (G : BondGraph V E) (ω : Set E) (x : V) : Set V :=  {y | Conn G ω x y}def infiniteClusters (G : BondGraph V E) (ω : Set E) : Set (Set V) :=  {C | C.Infinite ∧ ∃ x, C = cluster G ω x}noncomputable def law (p : unitInterval) : Measure (Set E) :=  setBernoulli Set.univ pnoncomputable def pc (G : BondGraph V E) : unitInterval :=  sInf {p | 0 < law p {ω | (infiniteClusters G ω).Nonempty}}noncomputable def pu (G : BondGraph V E) : unitInterval :=  sInf {p | law p {ω | ∃! C, C ∈ infiniteClusters G ω} = 1}noncomputable def twoPoint (G : BondGraph V E) (p : unitInterval) (x y : V) : ℝ :=  (law p {ω | Conn G ω x y}).toRealnoncomputable abbrev Hilbert (V : Type u) := lp (fun _ : V => ℂ) 2noncomputable def HasKernel [DecidableEq V]    (G : BondGraph V E) (p : unitInterval)    (A : Hilbert V →L[ℂ] Hilbert V) : Prop :=  ∀ x y, (A (lp.single 2 y (1 : ℂ))) x = (twoPoint G p x y : ℂ)noncomputable def OperatorBounded (G : BondGraph V E) (p : unitInterval) : Prop := by  classical  exact ∃ A : Hilbert V →L[ℂ] Hilbert V, HasKernel G p Anoncomputable def operatorNorm (G : BondGraph V E) (p : unitInterval) : ℝ≥0∞ := by  classical  exact sInf {r | ∃ A : Hilbert V →L[ℂ] Hilbert V,    HasKernel G p A ∧ r = ENNReal.ofReal ‖A‖}noncomputable def ptwo (G : BondGraph V E) : unitInterval :=  sSup {p | OperatorBounded G p}noncomputable def labelLaw : Measure (E → unitInterval) :=  Measure.infinitePi (fun _ => (volume : Measure unitInterval))def openEdges (labels : E → unitInterval) (p : unitInterval) : Set E :=  {e | labels e ≤ p}def MainConclusion (G : BondGraph V E) : Prop :=  (sSup {r | ∃ p : unitInterval, p < pc G ∧ r = operatorNorm G p} =    operatorNorm G (pc G)) ∧  operatorNorm G (pc G) < ⊤ ∧  pc G < ptwo G ∧ ptwo G ≤ pu G ∧  ∃ p₁ p₂ : unitInterval, pc G < p₁ ∧ p₁ < p₂ ∧ p₂ < 1 ∧    ∀ᵐ labels ∂(labelLaw : Measure (E → unitInterval)),      ∀ p ∈ Set.Icc p₁ p₂, (infiniteClusters G (openEdges labels p)).Infiniteend Percolationrun_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}namespace Percolationuniverse u vvariable {V : Type u} {E : Type v}attribute [local instance] Classical.propDecidabledef openSimpleGraph (G : BondGraph V E) (ω : Set E) : SimpleGraph V where  Adj x y := x ≠ y ∧ ∃ e ∈ ω, G e = s(x,y)  symm := ⟨by    rintro x y ⟨hxy, e, he, h⟩    exact ⟨hxy.symm, e, he, h.trans (Sym2.eq_swap ..)⟩⟩  loopless := ⟨by simp⟩end Percolationrun_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}namespace Percolationuniverse u vvariable {V : Type u} {E : Type v}attribute [local instance] Classical.propDecidable Classical.decEqnoncomputable def susceptibility (G : BondGraph V E) (p : unitInterval) (x : V) : ℝ :=  ∑' y, twoPoint G p x yend Percolationrun_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}namespace Percolation.BenjaminiSchrammuniverse u vvariable {V : Type u} {E : Type v}attribute [local instance] Classical.decEqnoncomputable def triangleDiagram (G : BondGraph V E) (p : unitInterval) (x : V) : ℝ≥0∞ :=  ∑' ab : V × V, ENNReal.ofReal    (twoPoint G p x ab.1 * twoPoint G p ab.1 ab.2 * twoPoint G p ab.2 x)noncomputable def graphDistance (G : BondGraph V E) (x y : V) : ℕ :=  (openSimpleGraph G Set.univ).dist x ydef ExponentialDecay (G : BondGraph V E) (p : unitInterval) : Prop :=  ∃ A a : ℝ, 0 < A ∧ 0 < a ∧ ∀ x y,    twoPoint G p x y ≤ A * Real.exp (-a * graphDistance G x y)noncomputable def pexp (G : BondGraph V E) : unitInterval :=  sSup {p | ExponentialDecay G p}def PowerTail (f : ℕ → ℝ) (exponent : ℝ) : Prop :=  ∃ c C : ℝ, 0 < c ∧ 0 < C ∧ ∀ n : ℕ, 0 < n →    c * (n : ℝ) ^ (-exponent) ≤ f n ∧ f n ≤ C * (n : ℝ) ^ (-exponent)def volumeEvent (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=  {ω | (n : ℕ∞) ≤ (cluster G ω x).encard}def intrinsicRadiusEvent (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=  {ω | ∃ y ∈ cluster G ω x, n ≤ (openSimpleGraph G ω).dist x y}def extrinsicRadiusEvent (G : BondGraph V E) (x : V) (n : ℕ) : Set (Set E) :=  {ω | ∃ y ∈ cluster G ω x, n ≤ graphDistance G x y}noncomputable def criticalTail (G : BondGraph V E)    (A : V → ℕ → Set (Set E)) (x : V) (n : ℕ) : ℝ :=  (law (pc G) (A x n)).toRealdef CriticalTailLaws (G : BondGraph V E) : Prop :=  ∀ x, PowerTail (criticalTail G (volumeEvent G) x) (1 / 2) ∧    PowerTail (criticalTail G (intrinsicRadiusEvent G) x) 1 ∧    PowerTail (criticalTail G (extrinsicRadiusEvent G) x) 1def SusceptibilityLaw (G : BondGraph V E) : Prop :=  ∀ x, ∃ c C δ : ℝ, 0 < c ∧ 0 < C ∧ 0 < δ ∧    ∀ p : unitInterval, p < pc G → (pc G : ℝ) - p < δ →      c / ((pc G : ℝ) - p) ≤ susceptibility G p x ∧      susceptibility G p x ≤ C / ((pc G : ℝ) - p)noncomputable def percolationProbability (G : BondGraph V E) (p : unitInterval)    (x : V) : ℝ := (law p {ω | (cluster G ω x).Infinite}).toRealdef PercolationProbabilityLaw (G : BondGraph V E) : Prop :=  ∀ x, ∃ c C δ : ℝ, 0 < c ∧ 0 < C ∧ 0 < δ ∧    ∀ p : unitInterval, pc G < p → (p : ℝ) - pc G < δ →      c * ((p : ℝ) - pc G) ≤ percolationProbability G p x ∧      percolationProbability G p x ≤ C * ((p : ℝ) - pc G)end Percolation.BenjaminiSchrammrun_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}namespace Percolation.BenjaminiSchrammuniverse u vvariable {V : Type u} {E : Type v}def CriticalLawsConclusion (G : BondGraph V E) : Prop :=  (∀ x, triangleDiagram G (pc G) x ≤ operatorNorm G (pc G) ^ 3 ∧    triangleDiagram G (pc G) x < ⊤) ∧  SusceptibilityLaw G ∧ PercolationProbabilityLaw G ∧ CriticalTailLaws G ∧  (pc G < ptwo G ∧ ptwo G ≤ pexp G) ∧  ∀ p : unitInterval, p < ptwo G → ExponentialDecay G pend Percolation.BenjaminiSchrammrun_elab Lean.modifyEnv fun environment => Lean.Meta.auxLemmasExt.setState environment {}namespace Percolation.BenjaminiSchrammuniverse uvariable {Λ : Type u} [Group Λ]attribute [local instance] Classical.decEqabbrev CayleyEdge (S : Finset Λ) := (SimpleGraph.mulCayley (S : Set Λ)).edgeSetdef cayleyBond (S : Finset Λ) : BondGraph Λ (CayleyEdge S) := Subtype.valdef FolnerAmenable (Λ : Type u) [Group Λ] : Prop :=  ∀ K : Finset Λ, ∀ ε : ℝ, 0 < ε → ∃ A : Finset Λ, A.Nonempty ∧    ∀ k ∈ K, (((A.image (fun x => x * k)) \ A).card : ℝ) < ε * A.cardend Percolation.BenjaminiSchrammnamespace Percolation.BenjaminiSchrammuniverse u v/-- Critical operator boundedness, the threshold gap, and simultaneous nonuniqueness. -/theorem full_main {V : Type u} {E : Type v} [Infinite V]    (G : BondGraph V E) (hc : Connected G) (hl : LocallyFinite G)    (hq : QuasiTransitive G) (hh : 0 < hV G) : MainConclusion G := by  sorry/-- Critical mean-field laws and uniform exponential connectivity below the operator threshold. -/theorem critical_laws {V : Type u} {E : Type v} [Infinite V]    (G : BondGraph V E) (hc : Connected G) (hl : LocallyFinite G)    (hq : QuasiTransitive G) (hh : 0 < hV G) : CriticalLawsConclusion G := by  sorry/-- Nonuniqueness for every prescribed finite symmetric Cayley generating set. -/theorem cayley_main {Λ : Type u} [Group Λ]    (S : Finset Λ) (hsym : ∀ s ∈ S, s⁻¹ ∈ S) (_hone : 1 ∉ S)    (hgen : Subgroup.closure (S : Set Λ) = ⊤) (hna : ¬ FolnerAmenable Λ) :    OperatorBounded (cayleyBond S) (pc (cayleyBond S)) ∧      MainConclusion (cayleyBond S) ∧      ∀ p : unitInterval, pc (cayleyBond S) < p → p < pu (cayleyBond S) →        ∀ᵐ ω ∂law p, (infiniteClusters (cayleyBond S) ω).Infinite := by  sorryend Percolation.BenjaminiSchrammend OAI

Every extracted statement of the accompanying papers

The statements the atlas extracted automatically from each accompanying paper’s TeX source, with links to the full sources.

Nonuniqueness of percolation on nonamenable quasi-transitive graphs

Theorem

Let G be an infinite connected locally finite quasi-transitive graph with h_V(G)>0. For Bernoulli bond percolation on G,

\sup_{0\le p<\pc}\norm{T_p}_{2\to2} =\norm{T_{\pc}}_{2\to2}<\infty, \qquad \pc(G)<\ptwo(G)\le\pu(G).

There exist \pc<p_1<p_2<1 such that, in the coupling obtained from independent uniform edge labels, almost surely there are infinitely many infinite open clusters simultaneously for every p\in[p_1,p_2].

Section 1, Introduction Read it in the paper TeX source, line 58

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Let $G$ be an infinite connected locally finite quasi-transitive graph
with $h_V(G)>0$. For Bernoulli bond percolation on $G$,
\[
 \sup_{0\le p<\pc}\norm{T_p}_{2\to2}
       =\norm{T_{\pc}}_{2\to2}<\infty,
 \qquad \pc(G)<\ptwo(G)\le\pu(G).
\]
There exist $\pc<p_1<p_2<1$ such that, in the coupling obtained from
independent uniform edge labels, almost surely there are infinitely
many infinite open clusters simultaneously for every $p\in[p_1,p_2]$.

Corollary (Arbitrary Cayley graphs)

\pc<\ptwo\le\pu.

The simultaneous nonuniqueness interval in thm:main exists for every such fixed generating set. At every fixed p\in(\pc,\pu), there are almost surely infinitely many infinite open clusters.

Section 1, Introduction Read it in the paper TeX source, line 80

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Let $\Lambda$ be a nonamenable finitely generated group, and let $S$
be any finite symmetric generating set with the identity removed.
For Bernoulli bond percolation on the simple undirected Cayley graph
with edges $\{x,xs\}$, $x\in\Lambda$, $s\in S$, the critical connection
operator is bounded on counting-measure $\ell^2(\Lambda)$ and
\[
 \pc<\ptwo\le\pu.
\]
The simultaneous nonuniqueness interval in \cref{thm:main} exists for
every such fixed generating set. At every fixed $p\in(\pc,\pu)$,
there are almost surely infinitely many infinite open clusters.

Corollary (Critical laws and exponential connectivity)

Let G be an infinite connected locally finite quasi-transitive graph with h_V(G)>0. For Bernoulli bond percolation on G,

\nabla_{\pc}(v)\le\norm{T_{\pc}}_{2\to2}^{3}<\infty \qquad\text{for every }v\in V.

For every v\in V,

\begin{aligned}\chi_v(p)&\asymp(\pc-p)^{-1} &&(p\uparrow\pc), \\ \theta_v(p)&\asymp p-\pc &&(p\downarrow\pc), \\ \Prob_{\pc}(|C_v|\ge n)&\asymp n^{-1/2} &&(n\to\infty), \\ \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{int}}(C_v)\ge n\bigr) &\asymp n^{-1} &&(n\to\infty), \\ \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{ext}}(C_v)\ge n\bigr) &\asymp n^{-1} &&(n\to\infty).\end{aligned}

Here n tends through positive integers. The comparison constants and the one-sided neighborhoods of \pc may depend on G and v. Moreover,

\pc(G)<\ptwo(G)\le\pexp(G).

For each fixed p\in[0,\ptwo(G)), there are constants 0<A_{G,p}<\infty and a_{G,p}>0 such that

T_p(x,y)\le A_{G,p}e^{-a_{G,p}\dist_G(x,y)} \qquad\text{for every }x,y\in V.

Section 1, Introduction Read it in the paper TeX source, line 257

Show the TeX

Let $G$ be an infinite connected locally finite quasi-transitive graph
with $h_V(G)>0$. For Bernoulli bond percolation on $G$,
\begin{equation}
 \nabla_{\pc}(v)\le\norm{T_{\pc}}_{2\to2}^{3}<\infty
 \qquad\text{for every }v\in V.
\end{equation}
For every $v\in V$,
\begin{align}
 \chi_v(p)&\asymp(\pc-p)^{-1} &&(p\uparrow\pc), \\
 \theta_v(p)&\asymp p-\pc &&(p\downarrow\pc), \\
 \Prob_{\pc}(|C_v|\ge n)&\asymp n^{-1/2} &&(n\to\infty), \\
 \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{int}}(C_v)\ge n\bigr)
    &\asymp n^{-1} &&(n\to\infty), \\
 \Prob_{\pc}\bigl(\operatorname{rad}_{\mathrm{ext}}(C_v)\ge n\bigr)
    &\asymp n^{-1} &&(n\to\infty). 
\end{align}
Here $n$ tends through positive integers. The comparison constants
and the one-sided neighborhoods of $\pc$ may depend on $G$ and $v$.
Moreover,
\begin{equation}
 \pc(G)<\ptwo(G)\le\pexp(G).
\end{equation}
For each fixed $p\in[0,\ptwo(G))$, there are constants
$0<A_{G,p}<\infty$ and $a_{G,p}>0$ such that
\begin{equation}
 T_p(x,y)\le A_{G,p}e^{-a_{G,p}\dist_G(x,y)}
 \qquad\text{for every }x,y\in V.
\end{equation}

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