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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.GraphGeometry.IsolateThree
import OAI.Probability.CriticalPercolation.ProductMeasure.ProbDisjoint
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open Set
open scoped BigOperators
namespace CriticalPercolation.BondGraph
open FiniteProduct
open scoped Classical
variable {V E : Type*} (G : BondGraph V E) [Fintype V] [Fintype E]
lemma threeCluster_avoidance_second (w : E → Prop → ℝ)
(hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z) (hc : ∀ e, 0 < w e False)
(a s t b : V) (hat : t ≠ a) (hst : t ≠ s) :
prob w (fun ω => G.joinedPair t a s ω ∧ (G.openGraph ω).Reachable t b) *
prob w (G.separateThree a s t) ≤
prob w (G.joinedPair t a s) *
prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) := by
have h := G.threeCluster_avoidance w hw hp hc t a s b hat hst
rw [prob_congr w (G.separateThree_rotate a s t),
prob_congr w (fun ω => and_congr_left (fun _ => G.separateThree_rotate a s t ω))] at h
exact h
omit [Fintype V] in
lemma threeCluster_event_bounds (w : E → Prop → ℝ) (hp : ∀ e z, 0 ≤ w e z)
(a s t b : V) :
prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) ≤
prob w (G.separateThree a s t) ∧
prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b) ≤
prob w (G.separateThree a s t) + prob w (G.joinedPair a s t) -
prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable a b) -
prob w (fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a b) := by
constructor
· exact prob_mono w hp (fun _ h => h.1)
· have h1 := prob_disjoint_le w hp
(fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b)
(fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable a b)
(G.separateThree a s t) (fun _ h => h.1) (fun _ h => h.1)
(fun _ h => h.1.1.1.2 (h.2.2.trans h.1.2.symm))
have h2 := prob_mono w hp (P := fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a b)
(Q := G.joinedPair a s t) (fun _ h => h.1)
linarith
end CriticalPercolation.BondGraph
end
end
end OAI
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