File size: 2,852 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Gluing.FreshTrial
import OAI.Probability.CriticalPercolation.Criticality.FiniteEnergy
import OAI.Probability.CriticalPercolation.GraphGeometry.GraphBalls
import OAI.Probability.CriticalPercolation.GraphGeometry.FiniteInduced
namespace OAI
section
section
noncomputable section
open Set Function MeasureTheory ProbabilityTheory
open scoped Classical BigOperators ENNReal
namespace CriticalPercolation.BernoulliProduct
variable {E : Type*}
lemma law_allOpen (q : E → unitInterval) (S : Set E) (hS : S.Finite) :
law q (allOpen S)=∏ e∈hS.toFinset, bitMeasure (q e) {True} := by
have he : allOpen S=⋂ e∈hS.toFinset, {ω : Set E | (e∈ω)=True} := by
ext ω
simp [allOpen,Set.subset_def]
rw [he,(independent_coordinates q).meas_biInter]
· exact Finset.prod_congr rfl (fun e _ => law_coordinate q e True)
· intro e _
exact ⟨{True},MeasurableSet.singleton _,rfl⟩
lemma law_allOpen_lower (q : E → unitInterval) (S : Set E) (hS : S.Finite)
(p : unitInterval) (N : ℕ) (hN : S.ncard ≤ N) (hq : ∀ e∈S, (p : ℝ) ≤ q e) :
(p : ℝ)^N ≤ (law q).real (allOpen S) := by
rw [measureReal_def,law_allOpen q S hS,ENNReal.toReal_prod]
calc
_ ≤ (p : ℝ)^S.ncard := pow_le_pow_of_le_one p.2.1 p.2.2 hN
_ = ∏ e∈hS.toFinset, (p : ℝ) := by simp [Set.ncard_eq_toFinset_card S hS]
_ ≤ _ := by
apply Finset.prod_le_prod₀
· intro edge hedge; exact p.2.1
· intro e he
change (p : ℝ) ≤ (bitMeasure (q e)).real {True}
rw [bitMeasure_real_singleton]
simpa only [bitWeight,ite_true] using hq e (hS.mem_toFinset.mp he)
end BernoulliProduct
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
lemma vertexBalls_disjoint {v w : V} {n : ℕ} (h : w∉G.vertexBall v (n+n)) :
Disjoint (G.vertexBall v n) (G.vertexBall w n) := by
apply Set.disjoint_left.mpr
intro z hz hz'
exact h (G.vertexBall_add hz n (G.vertexBall_comm.mp hz'))
lemma inducedEdges_disjoint {S T : Set V} (h : Disjoint S T) :
Disjoint (G.inducedEdges S) (G.inducedEdges T) := by
apply Set.disjoint_left.mpr
intro e he he'
obtain ⟨⟨x,y⟩,hxy⟩ := Sym2.mk_surjective (G.ends e)
have hx : x∈G.ends e := by rw [← hxy]; simp
exact Set.disjoint_left.mp h (he x hx) (he' x hx)
lemma inward_trials_disjoint (F : V → Set E) (m : ℕ)
(hF : ∀ v, F v⊆G.inducedEdges (G.vertexBall v (2*m+1)))
{v w : V} (h : w∉G.vertexBall v (4*m+2)) : Disjoint (F v) (F w) := by
have h' : w∉G.vertexBall v ((2*m+1)+(2*m+1)) := by
simpa only [show (2*m+1)+(2*m+1)=4*m+2 by omega] using h
exact (G.inducedEdges_disjoint (G.vertexBalls_disjoint h')).mono (hF v) (hF w)
end BondGraph
end CriticalPercolation
end
end
end
end OAI
|