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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 89 90 91 92 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.Leak
import OAI.Probability.CriticalPercolation.Criticality.SharpnessPhi
namespace OAI
section
section CurrentModule7
noncomputable section
open Set MeasureTheory
open scoped Classical BigOperators
namespace CriticalPercolation.Sharpness
open ClusterDeletion
variable {V E : Type*} (G : BondGraph V E)
lemma finite_sum_connection_le [Fintype E] {p : ℝ} (hp : p ∈ Icc 0 1)
(A : Finset V) (S : Set V) (hS : S.Finite) (o : V) (ho : o ∈ S)
(M : ℝ) (hM : ∀ y : V, ∑ v ∈ A,
probability p (fun ω => (G.openGraph ω).Reachable y v) ≤ M) :
∑ v ∈ A, probability p (fun ω => (G.openGraph ω).Reachable o v) ≤
S.ncard + boundaryFlux G p o Sᶜ * M := by
let f := fun x y => probability p (fun ω => (G.openGraph ω).Reachable x y)
let c := fun e => p * probability p (fun ω =>
boundaryReach G Sᶜ (delete (G.openGraph ω) Sᶜ) o e)
have hc : ∀ e, 0 ≤ c e := fun e => mul_nonneg hp.1 (probability_nonneg hp _)
have hout (v) (hv : v ∉ S) : f o v ≤ ∑ e, c e * f (cutOutside G S e) v :=
simon_lieb G hp S o ho v hv
calc
_ ≤ ∑ v ∈ A, ((if v ∈ S then (1:ℝ) else 0) +
∑ e, c e * f (cutOutside G S e) v) := by
apply Finset.sum_le_sum
intro v _
by_cases hv : v ∈ S
· simp only [ite_eq_left hv]
exact (probability_le_one hp _).trans (le_add_of_nonneg_right
(Finset.sum_nonneg fun e _ => mul_nonneg (hc e) (probability_nonneg hp _)))
· simpa only [ite_eq_right hv,zero_add] using hout v hv
_ = (A.filter (· ∈ S)).card + ∑ e, c e * ∑ v ∈ A, f (cutOutside G S e) v := by
rw [Finset.sum_add_distrib,Finset.sum_comm]
simp only [← Finset.mul_sum]
congr 1
simp
_ ≤ S.ncard + (∑ e, c e) * M := by
apply add_le_add
· exact_mod_cast Set.ncard_le_ncard
(show (↑(A.filter (· ∈ S)) : Set V) ⊆ S from fun v hv => (Finset.mem_filter.mp hv).2) hS
· rw [Finset.sum_mul]
exact Finset.sum_le_sum fun e _ => mul_le_mul_of_nonneg_left (hM _) (hc e)
_ = _ := by
rw [boundaryFlux_eq_sum,Finset.mul_sum]
lemma restricted_boundaryFlux_le_phi (hl : G.LocallyFinite) (p : unitInterval)
(S : Set V) (hS : S.Finite) (o : V) (K : Set E) (hK : K.Finite) :
letI := hK.fintype
boundaryFlux (G.restrictEdges K) (p : ℝ) o Sᶜ ≤ phi G p o S := by
let := hK.fintype
let F := fun e => (G.law p).real {ω | boundaryReach G Sᶜ
(delete (G.openGraph ω) Sᶜ) o e}
have hF : Summable F := summable_of_hasFiniteSupport
((finite_incident G hl hS).subset (boundary_measure_support G p o S))
rw [boundaryFlux_eq_sum,phi]
apply mul_le_mul_of_nonneg_left _ p.property.1
calc
_ ≤ ∑ e : K, F e.val := by
apply Finset.sum_le_sum
intro e _
rw [finite_probability_restrict G p K hK]
apply measureReal_mono _ (measure_ne_top _ _)
intro ω hω
change boundaryReach (G.restrictEdges K) Sᶜ
(delete ((G.restrictEdges K).openGraph (Subtype.val ⁻¹' ω)) Sᶜ) o e at hω
rw [G.restrictEdges_openGraph] at hω
have hh : boundaryReach G Sᶜ (delete (G.openGraph (K ∩ ω)) Sᶜ) o e.val := hω
obtain ⟨x,y,he,hx,hy,hr⟩ := hh
have hdel : delete (G.openGraph (K ∩ ω)) Sᶜ ≤ delete (G.openGraph ω) Sᶜ :=
fun _ _ hab => ⟨G.openGraph_mono inter_subset_right hab.1,hab.2⟩
exact ⟨x,y,he,hx,hy,hr.mono hdel⟩
_ ≤ ∑' e, F e := by
simpa only [tsum_fintype,Function.comp_def] using
(tsum_comp_le_tsum_of_inj hF (fun e => measureReal_nonneg)
(Subtype.val_injective : Function.Injective (Subtype.val : K → E)))
end CriticalPercolation.Sharpness
end
end CurrentModule7
end
end OAI
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