File size: 3,526 Bytes
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 | import Mathlib
import OAI.Probability.CriticalPercolation.GraphGeometry.VertexBall
import OAI.Probability.CriticalPercolation.ProductMeasure.Density
namespace OAI
section
section
open Set MeasureTheory ProbabilityTheory
open scoped BigOperators
open Set
namespace CriticalPercolation.BondGraph
open CriticalPercolation.FiniteProduct CriticalPercolation.ClusterDeletion
variable {V E : Type*} [Fintype V] [Fintype E] (G : BondGraph V E)
lemma finite_spatialMarkov (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S U : Set V) (F : Set V → SimpleGraph V → ℝ) :
expect w (fun ω =>
F (roots (delete (G.openGraph ω) U) S)
(delete (G.openGraph ω) (roots (delete (G.openGraph ω) U) S))) =
expect w (fun ω => expect w (fun ω' =>
F (roots (delete (G.openGraph ω) U) S)
(delete (G.openGraph ω') (roots (delete (G.openGraph ω) U) S)))) := by
classical
refine expect_exposure w hw
(fun ω => roots (delete (G.openGraph ω) U) S) G.incidentEdges
(G.dependsOn_exact_deleted_roots S U)
(fun W ω => F W (delete (G.openGraph ω) W)) ?_
intro W ω ω' hω
exact congrArg (F W) (G.delete_openGraph_dependsOn W hω)
noncomputable def expectedCluster (w : E → Prop → ℝ) (s : V)
(F : Set V → ℝ) (W : Set V) : ℝ :=
expect w (fun ω => F (component (delete (G.openGraph ω) W) s))
noncomputable def expectedResidual (w : E → Prop → ℝ) (S : Set V) (s : V)
(F : Set V → ℝ) (U : Set V) : ℝ := by
classical
exact expect w (fun ω =>
if s ∉ roots (delete (G.openGraph ω) U) S then
G.expectedCluster w s F (roots (delete (G.openGraph ω) U) S) -
G.expectedCluster w s F (roots (delete (G.openGraph ω) U) S ∪ U)
else 0)
lemma expectedResidual_eq (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1)
(S : Set V) (s : V) (F : Set V → ℝ) (U : Set V) :
G.expectedResidual w S s F U =
expect w (fun ω => residual (G.openGraph ω) S s F U) := by
classical
have h := G.finite_spatialMarkov w hw S U
(fun W H => if s ∉ W then F (component H s) - F (component (delete H U) s) else 0)
have hu : ∀ H W, delete (delete H W) U = delete H (W ∪ U) := by
intro H W
rw [delete_delete, Set.union_comm]
simp_rw [hu] at h
change _ = expect w (fun ω => if s ∉ roots (delete (G.openGraph ω) U) S then
F (component (delete (G.openGraph ω) (roots (delete (G.openGraph ω) U) S)) s) -
F (component (delete (G.openGraph ω) (roots (delete (G.openGraph ω) U) S ∪ U)) s)
else 0)
rw [h]
unfold expectedResidual
congr 1
funext ω
split_ifs
· rw [expect_const w hw]
· rw [expect_sub]
rfl
lemma expectedResidual_nonneg (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hpos : ∀ e b, 0 ≤ w e b)
(S : Set V) (s : V) {F : Set V → ℝ} (hF : Monotone F) (U : Set V) :
0 ≤ G.expectedResidual w S s F U := by
rw [expectedResidual_eq G w hw, ← expect_const w hw 0]
exact expect_mono hpos fun ω => residual_nonneg (G.openGraph ω) S s hF U
lemma expectedResidual_monotone (w : E → Prop → ℝ)
(hw : ∀ e, ∑ b, w e b = 1) (hpos : ∀ e b, 0 ≤ w e b)
(S : Set V) (s : V) {F : Set V → ℝ} (hF : Monotone F) :
Monotone (G.expectedResidual w S s F) := by
intro U U' hUU'
rw [expectedResidual_eq G w hw, expectedResidual_eq G w hw]
exact expect_mono hpos fun ω => residual_monotone (G.openGraph ω) S s hF hUU'
end CriticalPercolation.BondGraph
open scoped BigOperators
end
end
end OAI
|