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f205175 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | import OAI.Probability.CriticalZ3.Subunit
namespace OAI
open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical Filter Topology Function
open scoped ENNReal BigOperators Matrix
namespace CriticalZ3
open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory Classical
/-- Vanishing origin order parameter at the exact bond critical infimum. -/
theorem bond_critical_origin : bondLaw bondCritical (bondInfiniteAt 0) = 0 := by
by_cases hc : bondCritical = 0
· simp only [hc,bondLaw_zero_infinite]
· have hp : 0 < bondCritical := lt_of_le_of_ne bondCritical_mem.1 (Ne.symm hc)
by_contra hn
have hpos : 0 < bondLaw bondCritical (bondInfiniteAt 0) := bot_lt_iff_ne_bot.mpr hn
obtain ⟨q,hq,hqc,hqpos⟩ := Coarse.lower_percolating_parameter .bond hp bondCritical_lt_one hpos
exact (ne_of_gt hqpos) (bond_subcritical_origin hq hqc)
/-- Vanishing origin order parameter at the separate site critical infimum. -/
theorem site_critical_origin : siteLaw siteCritical (siteInfiniteAt 0) = 0 := by
by_cases hc : siteCritical = 0
· simp only [hc,siteLaw_zero_infinite]
· have hp : 0 < siteCritical := lt_of_le_of_ne siteCritical_mem.1 (Ne.symm hc)
by_contra hn
have hpos : 0 < siteLaw siteCritical (siteInfiniteAt 0) := bot_lt_iff_ne_bot.mpr hn
obtain ⟨q,hq,hqc,hqpos⟩ := Coarse.lower_percolating_parameter .site hp siteCritical_lt_one hpos
exact (ne_of_gt hqpos) (site_subcritical_origin hq hqc)
namespace SignedSymmetry
theorem bondConnected_map (S : SignedSymmetry) (ω : BondConfig) (x y : Vertex) :
BondConnected (fun e => ω (S.bond e)) x y ↔ BondConnected ω (S.vertex x) (S.vertex y) :=
rtg_equiv_iff S.vertex (S.bondOpen_map ω) x y
theorem siteConnected_map (S : SignedSymmetry) (ω : SiteConfig) (x y : Vertex) :
SiteConnected (fun e => ω (S.vertex e)) x y ↔ SiteConnected ω (S.vertex x) (S.vertex y) :=
and_congr_right (fun _ => rtg_equiv_iff S.vertex (S.siteOpen_map ω) x y)
theorem bondInfiniteAt_map (S : SignedSymmetry) (ω : BondConfig) (x : Vertex) :
(fun e => ω (S.bond e)) ∈ bondInfiniteAt x ↔ ω ∈ bondInfiniteAt (S.vertex x) := by
have he : bondCluster (fun e => ω (S.bond e)) x = S.vertex ⁻¹' bondCluster ω (S.vertex x) := by
ext y
exact S.bondConnected_map ω x y
change (bondCluster (fun e => ω (S.bond e)) x).Infinite ↔ _
rw [he]
constructor
· intro hi hf
exact hi (Set.Finite.preimage S.vertex.injective.injOn hf)
· intro hi
exact hi.preimage (fun y _ => S.vertex.surjective y)
theorem siteInfiniteAt_map (S : SignedSymmetry) (ω : SiteConfig) (x : Vertex) :
(fun e => ω (S.vertex e)) ∈ siteInfiniteAt x ↔ ω ∈ siteInfiniteAt (S.vertex x) := by
have he : siteCluster (fun e => ω (S.vertex e)) x = S.vertex ⁻¹' siteCluster ω (S.vertex x) := by
ext y
exact S.siteConnected_map ω x y
change (siteCluster (fun e => ω (S.vertex e)) x).Infinite ↔ _
rw [he]
constructor
· intro hi hf
exact hi (Set.Finite.preimage S.vertex.injective.injOn hf)
· intro hi
exact hi.preimage (fun y _ => S.vertex.surjective y)
end SignedSymmetry
theorem bondInfiniteAt_probability (p : ℝ) (x : Vertex) :
bondLaw p (bondInfiniteAt x) = bondLaw p (bondInfiniteAt 0) := by
let S := faceSymmetry x canonicalFace
have he : (fun ω e => ω (S.bond e)) ⁻¹' bondInfiniteAt 0 = bondInfiniteAt x := by
ext ω
simp only [Set.mem_preimage,S.bondInfiniteAt_map,faceSymmetry_zero,S]
have h := ProductCylinder.reindex_probability S.bond (parameter p) (bondInfiniteAt 0)
(measurableSet_bondInfiniteAt 0)
rw [he] at h
exact h
theorem siteInfiniteAt_probability (p : ℝ) (x : Vertex) :
siteLaw p (siteInfiniteAt x) = siteLaw p (siteInfiniteAt 0) := by
let S := faceSymmetry x canonicalFace
have he : (fun ω e => ω (S.vertex e)) ⁻¹' siteInfiniteAt 0 = siteInfiniteAt x := by
ext ω
simp only [Set.mem_preimage,S.siteInfiniteAt_map,faceSymmetry_zero,S]
have h := ProductCylinder.reindex_probability S.vertex (parameter p) (siteInfiniteAt 0)
(measurableSet_siteInfiniteAt 0)
rw [he] at h
exact h
/-- Nearest-neighbor bond percolation on Z³ has almost surely no infinite cluster at criticality. -/
theorem bond_critical_no_infinite :
∀ᵐ ω ∂bondLaw bondCritical, ∀ x : Vertex, (bondCluster ω x).Finite := by
rw [ae_all_iff]
intro x
rw [ae_iff]
exact (bondInfiniteAt_probability bondCritical x).trans bond_critical_origin
/-- Nearest-neighbor site percolation on Z³ has almost surely no infinite cluster at its own criticality. -/
theorem site_critical_no_infinite :
∀ᵐ ω ∂siteLaw siteCritical, ∀ x : Vertex, (siteCluster ω x).Finite := by
rw [ae_all_iff]
intro x
rw [ae_iff]
exact (siteInfiniteAt_probability siteCritical x).trans site_critical_origin
theorem critical_no_infinite :
(∀ᵐ ω ∂bondLaw bondCritical, ∀ x : Vertex, (bondCluster ω x).Finite) ∧
(∀ᵐ ω ∂siteLaw siteCritical, ∀ x : Vertex, (siteCluster ω x).Finite) :=
⟨bond_critical_no_infinite, site_critical_no_infinite⟩
end CriticalZ3
end OAI
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