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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.FiniteInterpolationUniform
import OAI.Probability.CriticalPercolation.ProductMeasure.MonotoneIndicatorUpper
import OAI.Probability.CriticalPercolation.Clusters.ClusterTail
namespace OAI
section
section
noncomputable section
open Set MeasureTheory ProbabilityTheory
open scoped Classical
namespace CriticalPercolation
namespace BernoulliProduct
variable {E : Type*} [Fintype E]
lemma prob_eq_real (q : E → unitInterval) (A : Set (Set E)) :
FiniteProduct.prob (fun e => bitWeight (q e)) (fun ω => ω ∈ A) = (law q).real A := by
rw [real_eq_expect_indicator]
unfold FiniteProduct.prob
rfl
end BernoulliProduct
namespace BondGraph
variable {V E : Type*} (G : BondGraph V E)
def twoClusterEvent (u v : V) (s : ℕ) : Set (Set E) :=
{ω | ¬ (G.openGraph ω).Reachable u v} ∩ G.clusterTail u s ∩ G.clusterTail v s
lemma measurableSet_twoClusterEvent [Countable V] [Countable E] (u v : V) (s : ℕ) :
MeasurableSet (G.twoClusterEvent u v s) := by
exact ((G.measurable_reachable u v).setOf.compl.inter
(G.measurableSet_clusterTail u s)).inter (G.measurableSet_clusterTail v s)
lemma clusterTail_iff_card [Fintype V] (ω : Set E) (u : V) (s : ℕ) :
ω ∈ G.clusterTail u s ↔ s ≤ (G.cluster ω u).toFinset.card := by
constructor
· rintro ⟨K,hK,hsub,hs⟩
exact hs.trans (by simpa only [Set.ncard_eq_toFinset_card'] using Set.ncard_le_ncard hsub)
· intro hs
exact ⟨G.cluster ω u, Set.toFinite _, Set.Subset.rfl, by simpa only [Set.ncard_eq_toFinset_card'] using hs⟩
lemma twoClusterEvent_iff_endpoint [Fintype V] (ω : Set E) (u v : V) (s : ℕ) :
ω ∈ G.twoClusterEvent u v s ↔ BinaryLaw.endpointEvent G u v s ω := by
simp only [twoClusterEvent, Set.mem_inter_iff, Set.mem_ofPred_eq,
clusterTail_iff_card, BinaryLaw.endpointEvent, and_assoc]
lemma finite_endpoint_prob [Fintype V] [Fintype E] (p : unitInterval) (u v : V) (s : ℕ) :
FiniteProduct.prob (fun _ : E => BernoulliProduct.bitWeight p)
(BinaryLaw.endpointEvent G u v s) = (G.law p).real (G.twoClusterEvent u v s) := by
rw [law, ← BernoulliProduct.law_const, ← BernoulliProduct.prob_eq_real]
congr 1
funext ω
exact propext (G.twoClusterEvent_iff_endpoint ω u v s).symm
theorem finite_interpolation_measure [Fintype V] [Fintype E]
(d : ℕ) (x : ℕ → V) (hd : 2≤d) (p : unitInterval)
(hp : (p : ℝ) ∈ Set.Ioo 0 1) {c ε : ℝ} (hc : 0<c) (hε : 0<ε)
(hεp : ε≤p) (hεq : ε≤1-(p:ℝ))
(hconn : ∀ j k, j<k → k≤d → k-j<d →
c ≤ (G.law p).real {ω | (G.openGraph ω).Reachable (x j) (x k)})
(s : ℕ) (hδ : 0 < (G.law p).real (G.twoClusterEvent (x 0) (x d) s)) :
∃ j<d, 0 < (G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s) ∧
-Real.log ((G.law p).real (G.twoClusterEvent (x j) (x (j+1)) s)) ≤
interpolationConstant c ε *
(Real.log (1/(G.law p).real (G.twoClusterEvent (x 0) (x d) s))+Nat.clog 2 d+1) := by
have h := BinaryLaw.finite_interpolation_uniform G d x hd hp
(fun _ : E => BernoulliProduct.bitWeight p)
(fun _ => by simp [BernoulliProduct.bitWeight])
(fun _ => by simp [BernoulliProduct.bitWeight]) hc hε hεp hεq
simp_rw [G.finite_endpoint_prob p] at h
apply h
intro j k hj hk hjk
change c ≤ FiniteProduct.prob (fun _ : E => BernoulliProduct.bitWeight p)
(fun ω => ω ∈ {η | (G.openGraph η).Reachable (x j) (x k)})
rw [BernoulliProduct.prob_eq_real, BernoulliProduct.law_const]
exact hconn j k hj hk hjk
exact hδ
end BondGraph
end CriticalPercolation
end
end
end
end OAI
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