| 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 |
|
|
