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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Clusters.ClusterUpdate
import OAI.Probability.CriticalPercolation.Clusters.Earlier
import OAI.Probability.CriticalPercolation.Clusters.HitValue
import OAI.Probability.CriticalPercolation.GraphGeometry.LastInverseDiagonal
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace CriticalPercolation.BondGraph
open CriticalPercolation.FiniteProduct CriticalPercolation.ClusterDeletion
open CriticalPercolation.ClusterColumns
open scoped Classical
variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E)
omit [Fintype V] in
lemma hitting_eq_one_sub_avoidance (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (A : Set V) (s : V) :
expect w (fun ω => hitValue A (component (G.openGraph ω) s)) =
1 - G.avoidedAverage w A s (fun _ => 1) := by
conv_rhs => lhs; rw [← expect_const w hw 1]
unfold avoidedAverage
erw [← expect_sub]
congr 1
funext ω
simp only [hitValue, component_inter_nonempty_iff]
split_ifs <;> norm_num
lemma finite_joint_gluing_list {n : ℕ} (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hpos : ∀ e b, 0 ≤ w e b)
(hclosed : ∀ e, 0 < w e False) (x : Fin (n+1) → V) (hx : Function.Injective x)
(T : Set V) (a : ℝ) (ha : ∀ j : Fin n, a ≤ expect w
(fun ω => hitValue T (component (G.openGraph ω) (x j.castSucc)))) :
expect w (fun ω => hitValue (earlier x (Fin.last n))
(component (G.openGraph ω) (x (Fin.last n)))) * a ≤
expect w (fun ω => jointHitValue (earlier x (Fin.last n)) T
(component (G.openGraph ω) (x (Fin.last n)))) := by
rw [G.hitting_eq_one_sub_avoidance w hw]
apply G.finite_cluster_relay w hw hpos hclosed x hx
(jointHitValue_monotone _ _) (jointHitValue_nonneg _ _)
intro j
have he : (fun ω : E → Prop => jointHitValue (earlier x (Fin.last n)) T
(component (G.openGraph ω) (x j.castSucc))) =
(fun ω => hitValue T (component (G.openGraph ω) (x j.castSucc))) := by
funext ω
apply jointHitValue_eq_hitValue
exact ⟨x j.castSucc, component_self _ _, ⟨j.castSucc, Fin.castSucc_lt_last j, rfl⟩⟩
rw [he]
exact ha j
end CriticalPercolation.BondGraph
open Set MeasureTheory ProbabilityTheory
end
end
end OAI
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