File size: 2,838 Bytes
b317718 | 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 | import Mathlib
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
abbrev TreeVertex (d : ℕ) :=
{word : List (Fin d) // List.IsChain (fun first second : Fin d => first ≠ second) word}
def treeAdj {d : ℕ} (u v : TreeVertex d) : Prop :=
(∃ c : Fin d, v.1 = u.1 ++ [c]) ∨
(∃ c : Fin d, u.1 = v.1 ++ [c])
structure TreeAut (d : ℕ) where
toEquiv : TreeVertex d ≃ TreeVertex d
map_adj : ∀ u v, treeAdj (toEquiv u) (toEquiv v) ↔ treeAdj u v
def relabel {d : ℕ} {α : Type*} (g : TreeAut d)
(x : TreeVertex d → α) : TreeVertex d → α :=
fun v => x (g.toEquiv.symm v)
inductive IsingSpin where
| neg
| pos
deriving DecidableEq
protected abbrev IsingSpin.enumList : List IsingSpin := [.neg, .pos]
protected theorem IsingSpin.enumList_getElem?_ctorIdx_eq (x : IsingSpin) :
IsingSpin.enumList[x.ctorIdx]? = some x := by
cases x <;> rfl
protected theorem IsingSpin.enumList_nodup : IsingSpin.enumList.Nodup := by decide
instance : Fintype IsingSpin where
elems := ⟨IsingSpin.enumList, IsingSpin.enumList_nodup⟩
complete x := by cases x <;> decide
instance isingSpinMeasurable : MeasurableSpace IsingSpin := ⊤
def uniform01 : Measure ℝ :=
volume.restrict (Set.Icc (0 : ℝ) 1)
instance uniform01Probability : IsProbabilityMeasure uniform01 where
measure_univ := by simp [uniform01]
def labelMeasure (d : ℕ) : Measure (TreeVertex d → ℝ) := by
classical
exact Measure.infinitePi (fun _ : TreeVertex d => uniform01)
def ConnectedFinset {d : ℕ} (D : Finset (TreeVertex d)) : Prop :=
D.Nonempty ∧
∀ u ∈ D, ∀ v ∈ D,
Relation.ReflTransGen
(fun x y : TreeVertex d => x ∈ D ∧ y ∈ D ∧ treeAdj x y) u v
def inducedEdges {d : ℕ} (D : Finset (TreeVertex d)) :
Finset (TreeVertex d × TreeVertex d) := by
classical
exact (D.product D).filter (fun e =>
treeAdj e.1 e.2 ∧ e.1.1.length < e.2.1.length)
def freeCylinderWeight {d : ℕ} (theta : ℝ)
(D : Finset (TreeVertex d)) (eta : TreeVertex d → IsingSpin) : ENNReal :=
ENNReal.ofReal
((1 / 2 : ℝ) *
Finset.prod (inducedEdges D) (fun e =>
(1 + (if eta e.1 = eta e.2 then theta else -theta)) / 2))
def HasFreeIsingLaw {d : ℕ} (theta : ℝ)
(mu : Measure (TreeVertex d → IsingSpin)) : Prop :=
∀ (D : Finset (TreeVertex d)), ConnectedFinset D →
∀ eta : TreeVertex d → IsingSpin,
mu {x | ∀ v ∈ D, x v = eta v} = freeCylinderWeight theta D eta
def IsFactorOfIID (d : ℕ) (theta : ℝ) : Prop :=
∃ Phi : (TreeVertex d → ℝ) → (TreeVertex d → IsingSpin),
Measurable Phi ∧
HasFreeIsingLaw theta (Measure.map Phi (labelMeasure d)) ∧
∀ g : TreeAut d,
∀ᵐ U ∂labelMeasure d, Phi (relabel g U) = relabel g (Phi U)
end Problem367
end
end OAI
|