| import Mathlib |
| import OAI.Probability.CriticalPercolation.GraphGeometry.EraseBond |
| import OAI.Probability.CriticalPercolation.GraphGeometry.WiredEvent |
|
|
| namespace OAI |
|
|
| section |
| section |
| open Set MeasureTheory ProbabilityTheory |
| open Set |
| open scoped BigOperators |
| open scoped Classical |
|
|
| namespace CriticalPercolation.BondGraph |
| open scoped Classical |
| open FiniteProduct |
|
|
| variable {V E : Type*} (G : BondGraph V E) |
|
|
| def connects (a t : V) (ω : Set E) : Prop := (G.openGraph ω).Reachable a t |
|
|
| lemma connects_mono (a t : V) : ∀ (ω ω' : Set E), ω ⊆ ω' → G.connects a t ω → G.connects a t ω' := |
| fun _ _ h => fun hr => hr.mono (G.openGraph_mono h) |
|
|
| lemma eraseBond_force (f : E) (a : V) (ω : Set E) (b : Prop) : |
| (G.eraseBond f a).openGraph (Function.update ω f b) = (G.eraseBond f a).openGraph ω := by |
| erw [G.eraseBond_openGraph,G.eraseBond_openGraph] |
| congr 1 |
| ext e |
| change (e ≠ f ∧ Function.update ω f b e) ↔ (e ≠ f ∧ e ∈ ω) |
| by_cases h : e = f |
| · subst e; simp |
| · erw [Function.update_of_ne h] |
| rfl |
| |
| lemma connects_eq_wired_deleted (f : E) (a v z t : V) (ω : Set E) |
| (hf : f ∈ ω) (he : G.ends f = s(v,z)) |
| (hv : ((G.eraseBond f a).openGraph ω).Reachable a v) : |
| G.connects a t ω ↔ (G.eraseBond f a).wiredEvent a z t ω := by |
| dsimp only [connects,wiredEvent] |
| rw [G.openGraph_eq_eraseBond_sup f a v z ω hf he] |
| exact Wiring.reachable_replace_tail _ a v z a t hv |
|
|
| lemma pivot_eq_deleted_joined (f : E) (a v z t : V) (ω : Set E) |
| (hf : f ∈ ω) (he : G.ends f = s(v,z)) |
| (hv : ((G.eraseBond f a).openGraph ω).Reachable a v) : |
| openPivot (G.connects a t) f ω ↔ (G.eraseBond f a).joinedPair a z t ω := by |
| unfold openPivot |
| rw [G.connects_eq_wired_deleted f a v z t ω hf he hv] |
| have hclose : G.connects a t (Function.update ω f False) ↔ |
| ((G.eraseBond f a).openGraph ω).Reachable a t := by |
| dsimp only [connects] |
| erw [← G.eraseBond_openGraph_force_false] |
| erw [hclose] |
| exact (and_iff_right hf).trans ((G.eraseBond f a).earlier_pivot_iff a z t ω) |
|
|
| variable [Fintype E] |
|
|
| lemma deleted_joined_pin_eq (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) |
| (f : E) (h0 : w f False ≠ 0) (h1 : w f True ≠ 0) (a z t : V) : |
| prob (pin w f True) ((G.eraseBond f a).joinedPair a z t) = |
| prob (pin w f False) ((G.eraseBond f a).joinedPair a z t) := by |
| rw [prob_pin_transport w hw f False True h0 h1] |
| apply prob_congr |
| intro ω |
| dsimp [joinedPair,isolateThree] |
| erw [G.eraseBond_force] |
|
|
| lemma delta_eq_deleted_joined (w : E → Prop → ℝ) (hw : ∀ e, ∑ b, w e b = 1) |
| (f : E) (h0 : w f False ≠ 0) (h1 : w f True ≠ 0) (a v z t : V) |
| (he : G.ends f = s(v,z)) |
| (hv : ∀ ω, density (pin w f False) ω ≠ 0 → ((G.eraseBond f a).openGraph ω).Reachable a v) : |
| delta w (G.connects a t) f = prob (pin w f True) ((G.eraseBond f a).joinedPair a z t) := by |
| rw [G.deleted_joined_pin_eq w hw f h0 h1 a z t, |
| delta_eq_pivotal w hw f h0 h1 _ (G.connects_mono a t)] |
| apply prob_congr_support |
| intro ω hω |
| have hf0 := density_pin_support w f False ω hω |
| have hf1 : Function.update ω f True f := by |
| erw [Function.update_self] |
| trivial |
| have hv' : ((G.eraseBond f a).openGraph (Function.update ω f True)).Reachable a v := by |
| erw [G.eraseBond_force] |
| exact hv ω hω |
| have hopen := G.connects_eq_wired_deleted f a v z t (Function.update ω f True) hf1 he hv' |
| have hgraph : G.openGraph ω = (G.eraseBond f a).openGraph ω := by |
| erw [G.eraseBond_openGraph_force_false] |
| have hu : Function.update ω f False = ω := by |
| erw [← hf0,Function.update_eq_self] |
| erw [hu] |
| erw [hopen] |
| dsimp only [wiredEvent,connects] |
| erw [G.eraseBond_force,hgraph] |
| exact (G.eraseBond f a).earlier_pivot_iff a z t ω |
|
|
| end CriticalPercolation.BondGraph |
|
|
| namespace CriticalPercolation.Wiring |
| variable {V : Type*} |
|
|
| lemma pivot_component_avoids (H : SimpleGraph V) (A B : Set V) (w a z : V) |
| (ha : a ∈ A) (hA : ∀ u ∈ A, H.Reachable w u) |
| (hopen : ∃ t ∈ B, (H ⊔ SimpleGraph.edge a z).Reachable w t) |
| (hclosed : ¬∃ t ∈ B, H.Reachable w t) : |
| (∃ t ∈ B, H.Reachable z t) ∧ ∀ u ∈ A, ¬H.Reachable z u := by |
| obtain ⟨t,ht,hwt⟩ := hopen |
| have hzt : H.Reachable z t := by |
| rcases (reachable_sup_edge H a z w t).mp hwt with h | ⟨_,h⟩ | ⟨_,h⟩ |
| · exact (hclosed ⟨t,ht,h⟩).elim |
| · exact h |
| · exact (hclosed ⟨t,ht,(hA a ha).trans h⟩).elim |
| refine ⟨⟨t,ht,hzt⟩,?_⟩ |
| intro u hu hzu |
| exact hclosed ⟨t,ht,(hA u hu).trans (hzu.symm.trans hzt)⟩ |
| end CriticalPercolation.Wiring |
|
|
| end |
| end |
|
|
| end OAI |
|
|
