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9b6e014 | 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 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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