| import OAI.Probability.ClassicalON.PoleCrossing |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory Set |
| open scoped BigOperators Classical |
| namespace ClassicalON.LatticeGraph |
|
|
| def innerSet (G : LatticeGraph) (k : ℕ) : Set G.vertices := |
| {x | |(scaledSite k x.val).1|≤1 ∧ |(scaledSite k x.val).2|≤1} |
|
|
| def outerSet (G : LatticeGraph) (k : ℕ) : Set G.vertices := |
| {x | 2≤|(scaledSite k x.val).1| ∨ 2≤|(scaledSite k x.val).2|} |
|
|
| theorem inner_outer_disjoint (G : LatticeGraph) (k : ℕ) : Disjoint (G.innerSet k) (G.outerSet k) := by |
| apply Set.disjoint_left.mpr |
| intro x hx ho |
| rcases ho with h | h <;> linarith [hx.1,hx.2] |
|
|
| def positiveLayerPins (G : LatticeGraph) (I O : Set G.vertices) : G.vertices → Option (Spin 3) := |
| (poleSystem (fun e : G.edges => e.val.1) (fun e => e.val.2) (fun _ => 0) (I∪O) (fun _ => true)).pin |
|
|
| theorem positiveLayerPins_boundary (G : LatticeGraph) (k : ℕ) (I O : Set G.vertices) |
| (hI : I⊆G.innerSet k) (hO : O⊆G.outerSet k) : |
| G.BoundaryPins k 3 (G.positiveLayerPins I O) := by |
| intro x s hs |
| by_cases h : x∈I∪O |
| · rcases h with h | h |
| · exact Or.inl (hI h) |
| · exact Or.inr (hO h) |
| · simp only [positiveLayerPins,poleSystem,h,↓reduceDIte] at hs |
| cases hs |
|
|
| theorem rotate_positiveLayerPins_pi (G : LatticeGraph) (k : ℕ) (I O : Set G.vertices) |
| (hI : I⊆G.innerSet k) (hO : O⊆G.outerSet k) : |
| G.rotatedBoundaryPins k (G.positiveLayerPins I O) Real.pi= |
| (poleSystem (fun e : G.edges => e.val.1) (fun e => e.val.2) (fun _ => 0) (I∪O) |
| (fun v => layerSigns I v)).pin := by |
| funext x |
| by_cases hx : x∈I∪O |
| · have htheta : G.thetaPotential k x=if x∈I then 1 else 0 := by |
| by_cases h : x∈I |
| · simp only [h,ite_true,thetaPotential] |
| exact squareTheta_one _ (hI h).1 (hI h).2 |
| · simp only [h,ite_false,thetaPotential] |
| exact squareTheta_zero _ (hO (hx.resolve_left h)) |
| simp only [rotatedBoundaryPins,positiveLayerPins,poleSystem,hx,↓reduceDIte,Option.map_some] |
| by_cases hi : x∈I |
| · rw [show layerSigns I x=false by simp [layerSigns,hi]] |
| congr 1 |
| exact rotate_pole _ _ (by simpa [hi] using htheta) |
| · rw [show layerSigns I x=true by simp [layerSigns,hi]] |
| congr 1 |
| exact rotation_zero_potential _ _ _ (by simpa [hi] using htheta) _ |
| · simp only [rotatedBoundaryPins,positiveLayerPins,poleSystem,hx,↓reduceDIte,Option.map_none] |
|
|
| theorem rotate_positiveLayerPins_zero (G : LatticeGraph) (k : ℕ) (I O : Set G.vertices) : |
| G.rotatedBoundaryPins k (G.positiveLayerPins I O) 0=G.positiveLayerPins I O := by |
| funext x |
| simp only [rotatedBoundaryPins] |
| cases hp : G.positiveLayerPins I O x with |
| | none => rfl |
| | some s => |
| simp only [Option.map_some] |
| congr 1 |
| apply Subtype.ext |
| change NormedSpace.exp ((0*G.thetaPotential k x) • axisC) s.val=s.val |
| simp [zero_smul ℝ axisC] |
|
|
| theorem annulus_crossing_small (β : ℝ) (hβ : 0≤β) (ε : ℝ) (hε : 0<ε) : |
| ∃ k : ℕ,4≤k ∧ ∀ (G : LatticeGraph),G.WithinDyadicBox k → |
| ∀ (b : G.edges → ℝ),(∀ e,0≤b e ∧ b e≤β) → |
| ∀ (I O : Set G.vertices),I⊆G.innerSet k → O⊆G.outerSet k → |
| poleBondMean (fun e : G.edges => e.val.1) (fun e => e.val.2) b (I∪O) (fun _ => true) |
| (crossingIndicator (fun e : G.edges => e.val.1) (fun e => e.val.2) I O)≤ε := by |
| obtain ⟨k,hk,hr⟩ := annulus_partition_ratio β hβ ε hε |
| refine ⟨k,hk,?_⟩ |
| intro G hG b hb I O hI hO |
| have hd : Disjoint I O := (G.inner_outer_disjoint k).mono hI hO |
| rw [pole_crossing_ratio _ _ _ (fun e => (hb e).1) _ _ hd] |
| have h := hr G hG b hb (G.positiveLayerPins I O) |
| (G.positiveLayerPins_boundary k I O hI hO) Real.pi 0 |
| rw [G.rotate_positiveLayerPins_pi k I O hI hO,G.rotate_positiveLayerPins_zero k I O] at h |
| change 1-ε≤(poleSystem _ _ b (I∪O) (fun v => layerSigns I v)).Z (fun _ => 1)/ |
| (poleSystem _ _ b (I∪O) (fun _ => true)).Z (fun _ => 1) at h |
| linarith |
|
|
| end ClassicalON.LatticeGraph |
|
|
| end |
|
|
| end OAI |
|
|
