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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.ProductMeasure.BitMeasure
import OAI.Probability.CriticalPercolation.GraphGeometry.EuclideanEllipse
import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseSeed
import OAI.Probability.CriticalPercolation.GraphGeometry.EllipseExit
import OAI.Probability.CriticalPercolation.Gluing.CorridorGraph
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory Filter
open scoped Classical Topology ENNReal
namespace CriticalPercolation.BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma normalized_edge_bound (pi : V → ℂ) (R W D : ℝ) (hW : 0<W) (hWR : W≤R)
(c : Circle) (hpi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖pi x-pi y‖≤D) :
∀ ⦃x y⦄, G.fullGraph.Adj x y →
‖EuclideanEllipse.normalize R W c (pi x)-EuclideanEllipse.normalize R W c (pi y)‖≤D/W := by
intro x y hxy
rw [←EuclideanEllipse.normalize_sub]
exact (EuclideanEllipse.normalize_norm_le R W hW hWR c _).trans
(div_le_div_of_nonneg_right (hpi hxy) hW.le)
lemma normalized_clusters_finite (pi : V → ℂ) (R W : ℝ) (hW : 0<W) (hWR : W≤R)
(c : Circle) (mu : Measure (Set E))
(hbox : ∀ U W : ℝ, ∀ᵐ ω ∂mu, ∀ a,
(G.cluster (ω∩G.inducedEdges {x | |(pi x).re|≤U ∧ |(pi x).im|≤W}) a).Finite) :
∀ᵐ ω ∂mu, ∀ a,
(G.cluster (ω∩G.inducedEdges {x | ‖EuclideanEllipse.normalize R W c (pi x)‖≤1}) a).Finite := by
have hR : 0<R := hW.trans_le hWR
have hsub : {x | ‖EuclideanEllipse.normalize R W c (pi x)‖≤1}⊆
{x | |(pi x).re|≤R ∧ |(pi x).im|≤R} := by
intro x hx
change ‖EuclideanEllipse.normalize R W c (pi x)‖≤1 at hx
have hn : ‖pi x‖≤R := by
calc
_ = ‖EuclideanEllipse.expand R W c (EuclideanEllipse.normalize R W c (pi x))‖ := by
rw [EuclideanEllipse.expand_normalize R W hR.ne' hW.ne']
_ ≤ R*‖EuclideanEllipse.normalize R W c (pi x)‖ := EuclideanEllipse.expand_norm_le R W hW.le hWR c _
_ ≤ R := by nlinarith
exact ⟨(Complex.abs_re_le_norm _).trans hn,(Complex.abs_im_le_norm _).trans hn⟩
filter_upwards [hbox R R] with ω hω a
exact (hω a).subset (G.cluster_mono (Set.inter_subset_inter_right _ (G.inducedEdges_mono hsub)) a)
theorem all_ellipses_favored [Countable E]
(pi : V → ℂ) (A : Set V) (W B D zeta : ℝ) (hD : 0≤D) (hW : 0<W)
(hBW : B<W) (hDz : D/W≤zeta)
(hpi : ∀ ⦃x y⦄, G.fullGraph.Adj x y → ‖pi x-pi y‖≤D)
(hA : ∀ a∈A, ‖pi a‖≤B) (S : Finset Circle)
(hcover : ∀ c : Circle, ∃ d∈S, ‖(c : ℂ)-(d : ℂ)‖<zeta)
(q : E → unitInterval) (a : ℝ) (ha : 0≤a)
(hi : 1-a^S.card<(BernoulliProduct.law q).real (G.seedInfiniteEvent A))
(hbox : ∀ U W : ℝ, ∀ᵐ ω ∂BernoulliProduct.law q, ∀ x,
(G.cluster (ω∩G.inducedEdges {x | |(pi x).re|≤U ∧ |(pi x).im|≤W}) x).Finite) :
∀ R : ℝ, W≤R → ∀ c : Circle, ∃ d∈S,
1-a<(BernoulliProduct.law q).real (G.ellipseExit (EuclideanEllipse.normalize R W c ∘ pi) A zeta d) := by
intro R hWR c
have hseed : ∀ x∈A, ‖EuclideanEllipse.normalize R W c (pi x)‖<1 := by
intro x hx
exact (EuclideanEllipse.normalize_norm_le R W hW hWR c (pi x)).trans_lt
((div_lt_one hW).mpr ((hA x hx).trans_lt hBW))
have hbound : ∀ᵐ ω ∂BernoulliProduct.law q, ∀ x∈A,
(G.cluster (ω∩G.inducedEdges {x | ‖EuclideanEllipse.normalize R W c (pi x)‖≤1}) x).Finite := by
filter_upwards [G.normalized_clusters_finite pi R W hW hWR c (BernoulliProduct.law q) hbox] with ω hω x _hx
exact hω x
exact G.exists_favored_ellipse _ A zeta (D/W) (by positivity) hDz
(G.normalized_edge_bound pi R W D hW hWR c hpi) hseed S hcover q a ha hi hbound
end CriticalPercolation.BondGraph
end
end
end
end OAI
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