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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Model
import OAI.Probability.CriticalPercolation.Clusters.ThreeClusterAvoidanceSecond
import OAI.Probability.CriticalPercolation.Exploration.PivotalComparison
import OAI.Probability.CriticalPercolation.GraphGeometry.IsolateThree
import OAI.Probability.CriticalPercolation.ProductMeasure.ClosedThin
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_general (w : E → Prop → ℝ)
(hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z)
(a s t b : V) (has : a ≠ s) (hat : a ≠ t) :
prob w (fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a 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_mul_le_of_strict w hw hp _ _ _ _ (fun w' hw' hp' hc' =>
G.threeCluster_avoidance w' hw' hp' hc' a s t b has hat)
lemma threeCluster_avoidance_second_general (w : E → Prop → ℝ)
(hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z)
(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_general w hw hp 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
theorem residual_converted_variance_general (w : E → Prop → ℝ)
(hw : ∀ e, ∑ z, w e z = 1) (hp : ∀ e z, 0 ≤ w e z)
(a s t b : V) (has : a ≠ s) (hat : a ≠ t) (hst : s ≠ t)
{p : ℝ} (hparam : p ∈ Set.Ioo 0 1) :
let u := prob w (G.separateThree a s t)
let v := prob w (G.joinedPair a s t)
let α := prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable t b)
let β := prob w (fun ω => G.separateThree a s t ω ∧ (G.openGraph ω).Reachable a b)
let γ := prob w (fun ω => G.joinedPair a s t ω ∧ (G.openGraph ω).Reachable a b)
let σ := prob w (fun ω => G.joinedPair t a s ω ∧ (G.openGraph ω).Reachable t b)
p*(1-p)*α*(α+σ) ≤
p*(1-p)*(u+v)*(u+v-β-γ)/((1-p)*(u+v)+p*(u+v-β-γ)) *
((v+α-γ)/(u+v-β-γ)-v/(u+v))^2 := by
dsimp only
have h1 := G.threeCluster_avoidance_general w hw hp a s t b has hat
have h2 := G.threeCluster_avoidance_second_general w hw hp a s t b hat.symm hst.symm
have h3 := G.threeCluster_harris w hw hp a s t
have h4 := G.threeCluster_event_bounds w hp a s t b
apply DFS.converted_variance hparam (prob_nonneg w hp _) (prob_nonneg w hp _)
(prob_nonneg w hp _) (prob_nonneg w hp _) (prob_nonneg w hp _) (prob_nonneg w hp _)
h4.1 h4.2 _ h2 h3
simpa only [mul_comm] using h1
end CriticalPercolation.BondGraph
end
end
end OAI
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