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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 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 | import Mathlib
import OAI.Probability.CriticalPercolation.Criticality.Weight
import OAI.Probability.CriticalPercolation.GraphGeometry.Link2
namespace OAI
section
section CurrentModule2
noncomputable section
open Set
open scoped Classical BigOperators
namespace CriticalPercolation.Sharpness
open FiniteProduct ClusterDeletion
lemma expect_exposure_finiteRange {I A B : Type*} [Fintype I] [Fintype A]
(w : I → A → ℝ) (hw : ∀ e, ∑ a, w e a = 1)
(r : (I → A) → B) (D : B → Set I)
(hr : ∀ b, DependsOn (fun ω => r ω = b) (D b))
(f : B → (I → A) → ℝ) (hf : ∀ b, DependsOn (f b) (D b)ᶜ) :
expect w (fun ω => f (r ω) ω) =
expect w (fun ω => expect w (f (r ω))) := by
let : Fintype (Set.range r) := (Set.finite_range r).fintype
let r' : (I → A) → Set.range r := fun ω => ⟨r ω,ω,rfl⟩
exact expect_exposure w hw r' (fun b => D b.val)
(fun b ω ω' hω => by
apply propext
simpa only [r',Subtype.ext_iff] using (hr b.val hω).to_iff)
(fun b => f b.val) (fun b => hf b.val)
variable {V E : Type*} [Fintype E] (G : BondGraph V E)
lemma spatialMarkov (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(T : Set V) (F : Set V → SimpleGraph V → ℝ) :
expect w (fun ω => F (roots (G.openGraph ω) T)
(delete (G.openGraph ω) (roots (G.openGraph ω) T))) =
expect w (fun ω => expect w (fun ω' => F (roots (G.openGraph ω) T)
(delete (G.openGraph ω') (roots (G.openGraph ω) T)))) := by
refine expect_exposure_finiteRange w hw
(fun ω => roots (G.openGraph ω) T) G.incidentEdges
(G.dependsOn_exact_roots T)
(fun R ω => F R (delete (G.openGraph ω) R)) ?_
intro R ω ω' hω
exact congrArg (F R) (G.delete_openGraph_dependsOn R hω)
def boundaryCount (o : V) (R : Set V) (H : SimpleGraph V) : ℝ :=
∑ e, if boundaryReach G R H o e then 1 else 0
def boundaryFlux (p : ℝ) (o : V) (R : Set V) : ℝ :=
p * expect (fun _ : E => weight p)
(fun ω => boundaryCount G o R (delete (G.openGraph ω) R))
omit [Fintype E] in
lemma closedPivot_reachesTarget (o : V) (T : Set V) (ω : Set E) (e : E) :
closedPivot (reachesTarget G o T) e ω ↔
o ∉ roots (G.openGraph ω) T ∧
boundaryReach G (roots (G.openGraph ω) T)
(delete (G.openGraph ω) (roots (G.openGraph ω) T)) o e := by
constructor
· rintro ⟨_,ho,he⟩
exact ⟨ho,(target_creating_iff G o T ω e ho).mp he⟩
· rintro ⟨ho,he⟩
exact ⟨boundaryReach_closed G o T ω e he,ho,
(target_creating_iff G o T ω e ho).mpr he⟩
lemma sum_closedPivot_eq (p : ℝ) (o : V) (T : Set V) :
∑ e, probability p (closedPivot (reachesTarget G o T) e) =
expect (fun _ : E => weight p) (fun ω =>
if o ∉ roots (G.openGraph ω) T then
expect (fun _ : E => weight p) (fun ω' => boundaryCount G o
(roots (G.openGraph ω) T)
(delete (G.openGraph ω') (roots (G.openGraph ω) T))) else 0) := by
have hs := spatialMarkov G (fun _ : E => weight p) (fun _ => weight_sum p)
T (fun R H => if o ∉ R then boundaryCount G o R H else 0)
unfold probability
rw [← expect_sum]
calc
_ = expect (fun _ : E => weight p) (fun ω =>
if o ∉ roots (G.openGraph ω) T then
boundaryCount G o (roots (G.openGraph ω) T)
(delete (G.openGraph ω) (roots (G.openGraph ω) T)) else 0) := by
congr 1
funext ω
have he (e : E) : closedPivot (reachesTarget G o T) e ω ↔
o ∉ roots (G.openGraph ω) T ∧ boundaryReach G (roots (G.openGraph ω) T)
(delete (G.openGraph ω) (roots (G.openGraph ω) T)) o e :=
closedPivot_reachesTarget G o T ω e
simp only [he]
by_cases ho : o ∈ roots (G.openGraph ω) T
· simp [ho]
· simp [ho,boundaryCount]
_ = _ := by
rw [hs]
congr 1
funext ω
split_ifs <;> simp only [expect_zero]
lemma exposedFlux_identity (p : ℝ) (o : V) (T : Set V) :
p * (∑ e, probability p (closedPivot (reachesTarget G o T) e)) =
expect (fun _ : E => weight p) (fun ω =>
if o ∉ roots (G.openGraph ω) T then
boundaryFlux G p o (roots (G.openGraph ω) T) else 0) := by
rw [sum_closedPivot_eq, ← expect_const_mul]
congr 1
funext ω
split_ifs <;> simp only [boundaryFlux,mul_zero]
theorem derivative_identity {p : ℝ} (hp : p ∈ Ioo 0 1) (o : V) (T : Set V) :
p * (1-p) * deriv (fun q => probability q (reachesTarget G o T)) p =
expect (fun _ : E => weight p) (fun ω =>
if o ∉ roots (G.openGraph ω) T then
boundaryFlux G p o (roots (G.openGraph ω) T) else 0) := by
have hd := hasDerivAt_probability_closedPivot hp (reachesTarget G o T)
(fun _ _ h => reachesTarget_mono G o T h)
rw [hd.deriv, ← exposedFlux_identity]
field_simp [(sub_pos.mpr hp.2).ne']
theorem differential_inequality {p a : ℝ} (hp : p ∈ Ioo 0 1) (o : V) (T : Set V)
(hflux : ∀ R : Set V, T ⊆ R → o ∉ R → a ≤ boundaryFlux G p o R) :
a * (1-probability p (reachesTarget G o T)) ≤
p * (1-p) * deriv (fun q => probability q (reachesTarget G o T)) p := by
rw [derivative_identity G hp o T]
have hcomp : expect (fun _ : E => weight p)
(fun ω => if o ∉ roots (G.openGraph ω) T then (1:ℝ) else 0) =
1-probability p (reachesTarget G o T) := by
calc
_ = expect (fun _ : E => weight p) (fun ω =>
1-(if reachesTarget G o T ω then (1:ℝ) else 0)) := by
congr 1
funext ω
unfold reachesTarget
by_cases ho : o ∈ roots (G.openGraph ω) T <;> simp [ho]
_ = _ := by
rw [expect_sub,expect_const _ (fun _ => weight_sum p)]
rfl
rw [← hcomp, ← expect_const_mul]
apply expect_mono (fun _ b => weight_nonneg ⟨hp.1.le,hp.2.le⟩ b)
intro ω
by_cases ho : o ∈ roots (G.openGraph ω) T
· simp [ho]
· simp only [ho,not_false_eq_true,ite_true,mul_one]
exact hflux (roots (G.openGraph ω) T) (roots_self _ _) ho
end CriticalPercolation.Sharpness
end
end CurrentModule2
end
end OAI
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