| 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 |
|
|
