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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 | import Mathlib
import OAI.Probability.CriticalPercolation.Clusters.ComponentSelf
import OAI.Probability.CriticalPercolation.Clusters.ExpectedCluster
import OAI.Probability.CriticalPercolation.ProductMeasure.Distribution
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped BigOperators
namespace CriticalPercolation.BondGraph
open CriticalPercolation.FiniteProduct CriticalPercolation.ClusterDeletion
variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E)
open scoped Classical
noncomputable def clusterUpdate (w : E → Prop → ℝ) (S : Set V) (s : V)
(F : Set V → ℝ) (U : Set V) : ℝ :=
expect w (fun ω => G.expectedCluster w s F (roots (delete (G.openGraph ω) U) S))
omit [Fintype V] in
lemma expectedCluster_antitone (w : E → Prop → ℝ) (hpos : ∀ e b, 0 ≤ w e b)
(s : V) {F : Set V → ℝ} (hF : Monotone F) :
Antitone (G.expectedCluster w s F) := by
intro U U' hU
exact expect_mono hpos fun ω => hF (component_mono (delete_antitone _ hU) s)
omit [Fintype V] in
lemma clusterUpdate_monotone (w : E → Prop → ℝ) (hpos : ∀ e b, 0 ≤ w e b)
(S : Set V) (s : V) {F : Set V → ℝ} (hF : Monotone F) :
Monotone (G.clusterUpdate w S s F) := by
intro U U' hU
exact expect_mono hpos fun ω => G.expectedCluster_antitone w hpos s hF
(roots_mono (delete_antitone _ hU) S)
omit [Fintype V] in
lemma clusterUpdate_nonneg (w : E → Prop → ℝ) (hpos : ∀ e b, 0 ≤ w e b)
(S : Set V) (s : V) {F : Set V → ℝ} (hF : ∀ U, 0 ≤ F U) (U : Set V) :
0 ≤ G.clusterUpdate w S s F U :=
expect_nonneg hpos fun _ => expect_nonneg hpos fun _ => hF _
noncomputable def avoidedAverage (w : E → Prop → ℝ) (S : Set V) (s : V)
(F : Set V → ℝ) : ℝ := by
classical
exact expect w (fun ω => if s ∉ roots (G.openGraph ω) S then
F (component (G.openGraph ω) s) else 0)
lemma clusterUpdate_exposure (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S : Set V) (s : V) (F g : Set V → ℝ) :
(expect w (fun ω => if s ∉ roots (G.openGraph ω) S then
g (component (G.openGraph ω) s) * G.expectedCluster w s F (roots (G.openGraph ω) S)
else 0)) =
G.avoidedAverage w S s (fun U => g U * G.clusterUpdate w S s F U) := by
classical
have h := G.finite_spatialMarkov w hw {s} ∅
(fun U H => if Disjoint U S then g U * G.expectedCluster w s F (roots H S) else 0)
simp only [delete_empty, roots_singleton] at h
have hleft : (fun ω : E → Prop =>
if Disjoint (component (G.openGraph ω) s) S then
g (component (G.openGraph ω) s) * G.expectedCluster w s F
(roots (delete (G.openGraph ω) (component (G.openGraph ω) s)) S) else 0) =
(fun ω => if s ∉ roots (G.openGraph ω) S then
g (component (G.openGraph ω) s) * G.expectedCluster w s F
(roots (G.openGraph ω) S) else 0) := by
funext ω
rw [← not_mem_roots_iff_disjoint]
split_ifs with hs
· rfl
· rw [roots_delete_component _ S s hs]
rw [hleft] at h
erw [h]
unfold avoidedAverage
congr 1
funext ω
rw [← not_mem_roots_iff_disjoint]
split_ifs
· rw [expect_zero]
· rw [expect_const_mul]
rfl
lemma avoidedAverage_exposure (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S : Set V) (s : V) (F : Set V → ℝ) :
G.avoidedAverage w S s F =
expect w (fun ω => if s ∉ roots (G.openGraph ω) S then
G.expectedCluster w s F (roots (G.openGraph ω) S) else 0) := by
classical
have h := G.finite_spatialMarkov w hw S ∅
(fun W H => if s ∉ W then F (component H s) else 0)
simp only [delete_empty] at h
have hleft : (fun ω : E → Prop => if s ∉ roots (G.openGraph ω) S then
F (component (delete (G.openGraph ω) (roots (G.openGraph ω) S)) s) else 0) =
(fun ω => if s ∉ roots (G.openGraph ω) S then F (component (G.openGraph ω) s) else 0) := by
funext ω
split_ifs with hs
· rfl
· rw [delete_roots_component _ S s hs]
rw [hleft] at h
change expect w _ = _
erw [h]
congr 1
funext ω
split_ifs
· rw [expect_zero]
· rfl
lemma clusterUpdate_stationary (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S : Set V) (s : V) (F : Set V → ℝ) :
G.avoidedAverage w S s (G.clusterUpdate w S s F) = G.avoidedAverage w S s F := by
have h := G.clusterUpdate_exposure w hw S s F (fun _ => 1)
simp only [one_mul] at h
exact h.symm.trans (G.avoidedAverage_exposure w hw S s F).symm
end CriticalPercolation.BondGraph
end
end
end OAI
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