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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Gluing.MoveExit
import OAI.Probability.CriticalPercolation.Gluing.RelayMoves
import OAI.Probability.CriticalPercolation.ProductMeasure.FiniteOpenWitness
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical ENNReal
namespace CriticalPercolation.EuclideanEllipse
lemma finite_circle_cover (zeta : ℝ) (hzeta : 0<zeta) :
∃ S : Finset Circle, ∀ c : Circle, ∃ d∈S, ‖(c : ℂ)-(d : ℂ)‖<zeta := by
obtain ⟨S,_hS,hSf,hcover⟩ := (isCompact_univ : IsCompact (Set.univ : Set Circle)).finite_cover_balls hzeta
refine ⟨hSf.toFinset,?_⟩
intro c
obtain ⟨d,hd,hcd⟩ := Set.mem_iUnion₂.mp (hcover (Set.mem_univ c))
change dist (c : ℂ) (d : ℂ)<zeta at hcd
exact ⟨d,hSf.mem_toFinset.mpr hd,by simpa only [dist_eq_norm] using hcd⟩
lemma exists_radial_circle (z : ℂ) (hz : z≠0) :
∃ c : Circle, z=‖z‖•(c : ℂ) := by
let c : Circle := ⟨‖z‖⁻¹•z,by
apply mem_sphere_zero_iff_norm.mpr
exact norm_smul_inv_norm (𝕜 := ℝ) hz⟩
refine ⟨c,?_⟩
change z=‖z‖•(‖z‖⁻¹•z)
rw [smul_smul,mul_inv_cancel₀ (norm_ne_zero_iff.mpr hz),one_smul]
lemma annulus_covered (S : Finset Circle) (zeta : ℝ)
(hcover : ∀ c : Circle, ∃ d∈S, ‖(c : ℂ)-(d : ℂ)‖<zeta)
(z : ℂ) (hlo : 1≤‖z‖) (hhi : ‖z‖≤1+zeta) :
∃ d∈S, ‖z-(d : ℂ)‖≤2*zeta := by
have hz : z≠0 := by intro h; subst z; norm_num at hlo
obtain ⟨c,hc⟩ := exists_radial_circle z hz
obtain ⟨d,hd,hcd⟩ := hcover c
have he : z-(c : ℂ)=(‖z‖-1)•(c : ℂ) := by rw [sub_smul,one_smul,←hc]
have hdiff : ‖z-(c : ℂ)‖=‖z‖-1 := by
rw [he,norm_smul,Real.norm_eq_abs,Circle.norm_coe,mul_one,abs_of_nonneg (by linarith)]
refine ⟨d,hd,?_⟩
have htri : ‖z-(d : ℂ)‖≤‖z-(c : ℂ)‖+‖(c : ℂ)-(d : ℂ)‖ := by
simpa only [sub_add_sub_cancel] using norm_add_le (z-(c : ℂ)) ((c : ℂ)-(d : ℂ))
linarith
end CriticalPercolation.EuclideanEllipse
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
def ellipseExit (Psi : V → ℂ) (A : Set V) (zeta : ℝ) (c : Circle) : Set (Set E) :=
G.restrictedSeed A {x | ‖Psi x-(c : ℂ)‖≤2*zeta ∧ ‖Psi x‖≤1+zeta}
(G.inducedEdges {x | ‖Psi x‖≤1+zeta})
lemma ellipseExit_iff (Psi : V → ℂ) (A : Set V) (zeta : ℝ) (c : Circle) (ω : Set E) :
ω∈G.ellipseExit Psi A zeta c ↔ ∃ a∈A, ∃ x, ∃ w : (G.openGraph ω).Walk a x,
(∀ v∈w.support, ‖Psi v‖≤1+zeta) ∧ ‖Psi x-(c : ℂ)‖≤2*zeta := by
constructor
· intro h
obtain ⟨a,ha,x,⟨w⟩,hx⟩ := Set.mem_iUnion₂.mp h
refine ⟨a,ha,x,w.mapLe (G.openGraph_mono Set.inter_subset_left),?_,hx.1⟩
intro v hv
rw [SimpleGraph.Walk.support_mapLe_eq_support] at hv
exact G.walk_induced_support w hx.2 v hv
· rintro ⟨a,ha,x,w,hw,hx⟩
exact Set.mem_iUnion₂.mpr ⟨a,ha,x,G.walk_reachable_induced _ ω w hw,hx,hw x w.end_mem_support⟩
lemma ellipseExit_upper (Psi : V → ℂ) (A : Set V) (zeta : ℝ) (c : Circle) :
IsUpperSet (G.ellipseExit Psi A zeta c) := G.restrictedSeed_upper _ _ _
lemma ellipseExit_finiteWitness (Psi : V → ℂ) (A : Set V) (zeta : ℝ) (c : Circle) :
BernoulliProduct.HasFiniteOpenWitness (G.ellipseExit Psi A zeta c) := G.restrictedSeed_finiteWitness _ _ _
lemma full_infinite_cluster_exit (Psi : V → ℂ) (ω : Set E) (a : V)
(hi : (G.cluster ω a).Infinite)
(hbound : (G.cluster (ω∩G.inducedEdges {x | ‖Psi x‖≤1}) a).Finite) :
∃ b∈G.cluster ω a, 1<‖Psi b‖ := by
by_contra hn
push Not at hn
apply hi
apply hbound.subset
rintro b ⟨w⟩
apply G.walk_reachable_induced _ ω w
intro x hx
exact hn x (w.takeUntil x hx).reachable
theorem infinite_cluster_ellipseExit (Psi : V → ℂ) (A : Set V) (zeta D : ℝ)
(hD : 0≤D) (hDz : D≤zeta)
(hPsi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖Psi x-Psi y‖≤D)
(hseed : ∀ a∈A, ‖Psi a‖<1) (S : Finset Circle)
(hcover : ∀ c : Circle, ∃ d∈S, ‖(c : ℂ)-(d : ℂ)‖<zeta)
{ω : Set E} (hi : ∃ a∈A, (G.cluster ω a).Infinite)
(hbound : ∀ a∈A, (G.cluster (ω∩G.inducedEdges {x | ‖Psi x‖≤1}) a).Finite) :
ω∈⋃ c∈S, G.ellipseExit Psi A zeta c := by
obtain ⟨a,ha,hi⟩ := hi
obtain ⟨b,⟨w⟩,hb⟩ := G.full_infinite_cluster_exit Psi ω a hi (hbound a ha)
have hstep : ∀ ⦃x y⦄, (G.openGraph ω).Adj x y → |‖Psi x‖-‖Psi y‖|≤D := by
intro x y hxy
exact (abs_norm_sub_norm_le _ _).trans (hPsi (G.openGraph_mono (Set.subset_univ _) hxy))
obtain ⟨x,p,_hp,hpb,hpx⟩ := _root_.OAI.SimpleGraph.Walk.bounded_first_crossing (fun x => ‖Psi x‖) D 1 hD hstep w (hseed a ha) hb.le
obtain ⟨c,hc,hcx⟩ := EuclideanEllipse.annulus_covered S zeta hcover (Psi x) hpx (by linarith [hpb x p.end_mem_support])
apply Set.mem_iUnion₂.mpr
refine ⟨c,hc,(G.ellipseExit_iff Psi A zeta c ω).mpr ⟨a,ha,x,p,?_,hcx⟩⟩
intro v hv
linarith [hpb v hv]
end CriticalPercolation.BondGraph
end
end
end
end OAI
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