| import OAI.Probability.ClassicalON.AnnulusCut |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped Classical |
| namespace ClassicalON.LatticeGraph |
|
|
| def cellVisited (G : LatticeGraph) (η : G.edges → Bool) (x : G.vertices) (N : ℤ) (z : Site) : Prop := |
| ∃ v : G.vertices,bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x v ∧ coarseSite N v.val=z |
|
|
| def coarseCluster (G : LatticeGraph) (η : G.edges → Bool) (x : G.vertices) (N : ℤ) : SimpleGraph Site where |
| Adj z w := z≠w ∧ siteRadius z w≤1 ∧ G.cellVisited η x N z ∧ G.cellVisited η x N w |
| symm := ⟨by intro z w ⟨h,hd,hz,hw⟩; exact ⟨h.symm,by simpa only [siteRadius_symm] using hd,hw,hz⟩⟩ |
| loopless := ⟨by intro z; simp⟩ |
|
|
| theorem fine_edge_radius (G : LatticeGraph) (η : G.edges → Bool) {v w : G.vertices} |
| (h : (bondGraph (fun e : G.edges => e.val.1) (fun e => e.val.2) η).Adj v w) : |
| siteRadius v.val w.val=1 := by |
| obtain ⟨_,e,_,he⟩ := h |
| rcases he with ⟨rfl,rfl⟩ | ⟨rfl,rfl⟩ |
| · exact siteRadius_neighbor (G.nearest e.val e.property) |
| · rw [siteRadius_symm]; exact siteRadius_neighbor (G.nearest e.val e.property) |
|
|
| theorem coarse_reachable_of_walk (G : LatticeGraph) (η : G.edges → Bool) (x : G.vertices) |
| (N : ℤ) (hN : 0<N) {v w : G.vertices} |
| (p : (bondGraph (fun e : G.edges => e.val.1) (fun e => e.val.2) η).Walk v w) |
| (hv : bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x v) : |
| (G.coarseCluster η x N).Reachable (coarseSite N v.val) (coarseSite N w.val) := by |
| induction p with |
| | nil => exact SimpleGraph.Reachable.refl _ |
| | @cons v w u hvw p ih => |
| have hw := bondConnected_trans _ _ _ hv ((bondConnected_iff_reachable _ _ _ _ _).mpr hvw.reachable) |
| apply SimpleGraph.Reachable.trans (v := coarseSite N w.val) ?_ (ih hw) |
| by_cases he : coarseSite N v.val=coarseSite N w.val |
| · rw [he] |
| · apply SimpleGraph.Adj.reachable |
| refine ⟨he,coarseSite_step N hN (le_of_eq (G.fine_edge_radius η hvw)),⟨v,hv,rfl⟩,⟨w,hw,rfl⟩⟩ |
|
|
| theorem coarse_walk_visited (G : LatticeGraph) (η : G.edges → Bool) (x : G.vertices) |
| (N : ℤ) {z w : Site} (p : (G.coarseCluster η x N).Walk z w) |
| (hz : G.cellVisited η x N z) (i : ℕ) : G.cellVisited η x N (p.getVert i) := by |
| induction p generalizing i with |
| | nil => exact hz |
| | @cons z v w h p ih => |
| cases i with |
| | zero => exact hz |
| | succ i => exact ih h.2.2.2 i |
|
|
| theorem coarse_walk_radius (G : LatticeGraph) (η : G.edges → Bool) (x : G.vertices) |
| (N : ℤ) {z w : Site} (p : (G.coarseCluster η x N).Walk z w) : siteRadius z w≤p.length := by |
| induction p with |
| | nil => simp |
| | @cons z v w h p ih => |
| have ht := siteRadius_triangle z v w |
| have hs := h.2.1 |
| simpa only [SimpleGraph.Walk.length_cons,Nat.cast_add,Nat.cast_one] using (show siteRadius z w≤(p.length:ℤ)+1 by omega) |
|
|
| theorem connected_has_coarse_code (G : LatticeGraph) (η : G.edges → Bool) (x y : G.vertices) |
| (N : ℤ) (hN : 0<N) (L : ℕ) (hfar : N*((L:ℤ)+2)≤ siteRadius x.val y.val) |
| (hxy : bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x y) : |
| ∃ c : SimpleCoarseCode (coarseSite N x.val) L, |
| ∀ i : Fin (L+1),G.cellVisited η x N (codePosition (coarseSite N x.val) c.val i) := by |
| obtain ⟨p⟩ := (bondConnected_iff_reachable _ _ _ _ _).mp hxy |
| obtain ⟨q⟩ := G.coarse_reachable_of_walk η x N hN p (fun _ _ => rfl) |
| let r := q.toPath.val |
| have hr : r.IsPath := q.toPath.property |
| have hl : L≤r.length := by |
| have hd := coarseSite_distance N hN x.val y.val |
| have hq := G.coarse_walk_radius η x N r |
| by_contra hn |
| have hln : (r.length:ℤ)<(L:ℤ) := by exact_mod_cast (Nat.lt_of_not_ge hn) |
| nlinarith |
| let a : Fin (L+1) → Site := fun i => r.getVert i.val |
| have ha : Function.Injective a := by |
| intro i j hij |
| apply Fin.ext |
| exact hr.getVert_injOn (by simp only [Set.mem_ofPred_eq]; omega) (by simp only [Set.mem_ofPred_eq]; omega) hij |
| have hs (i : Fin L) : siteRadius (a i.castSucc) (a i.succ)≤1 := |
| (r.adj_getVert_succ (by omega : i.val<r.length)).2.1 |
| obtain ⟨c,hc⟩ := codePosition_exists a hs |
| have h0 : a 0=coarseSite N x.val := r.getVert_zero |
| rw [h0] at hc |
| refine ⟨⟨c,hc ▸ ha⟩,?_⟩ |
| intro i |
| rw [congrFun hc i] |
| exact G.coarse_walk_visited η x N r ⟨x,(fun _ _ => rfl),rfl⟩ i.val |
|
|
| theorem connected_code_crossings (G : LatticeGraph) (η : G.edges → Bool) (x y : G.vertices) |
| (N : ℤ) (hN : 0<N) (L : ℕ) (hfar : N*((L:ℤ)+4)≤ siteRadius x.val y.val) |
| (hxy : bondConnected (fun e : G.edges => e.val.1) (fun e => e.val.2) η x y) : |
| ∃ c : SimpleCoarseCode (coarseSite N x.val) L, |
| ∀ i : Fin (L+1),G.annulusIndicator N (coarseCenter N (codePosition (coarseSite N x.val) c.val i)) η=1 := by |
| obtain ⟨c,hc⟩ := G.connected_has_coarse_code η x y N hN L (by nlinarith) hxy |
| refine ⟨c,?_⟩ |
| intro i |
| obtain ⟨w,hw,he⟩ := hc i |
| apply G.annulus_crossing_of_component N hN η x y w (by nlinarith) hxy hw |
| rw [← he] |
| exact le_of_lt (coarseSite_close N hN w.val) |
|
|
| end ClassicalON.LatticeGraph |
|
|
| end |
|
|
| end OAI |
|
|
