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f205175 | 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 99 100 101 102 103 104 105 106 107 108 109 110 | import Mathlib
import OAI.Probability.CriticalPercolation.Modular.ActionOrbitTransport
namespace OAI
section
noncomputable section
open Set
open scoped Classical BigOperators
namespace ActionOrbitTransport
variable (Λ X : Type*) [Group Λ] [MulAction Λ X]
namespace Representatives
variable (R : Representatives Λ X)
def weightedRow (w : X → ℝ) (F : X → X → ℝ) (p : R.RootedPair) : ℝ :=
w p.1 * F p.1 p.2
lemma weightedRow_finite (w : X → ℝ) (F : X → X → ℝ)
(hF : ∀ x, (Function.support (F x)).Finite) :
(Function.support (weightedRow Λ X R w F)).Finite := by
apply (Set.finite_iUnion (fun x : {x // x∈R.roots} =>
(hF x).image (Prod.mk x))).subset
intro p hp
refine Set.mem_iUnion.mpr ⟨p.1,p.2,?_,rfl⟩
intro hz
exact hp (by simp only [weightedRow,hz,mul_zero])
lemma fiber_sum_root (hfin : ∀ x y, (stabilizerOrbit Λ X x y).Finite)
(w : X → ℝ) (F : X → X → ℝ)
(hF : ∀ (φ : Λ) x y, F (φ • x) (φ • y) = F x y)
(x : {x // x∈R.roots}) (y : X) :
(∑' p : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y}, weightedRow Λ X R w F p) =
w x * (suborbitCard Λ X x y : ℝ) * F x y := by
let := (hfin x y).fintype
rw [← (fiberEquiv Λ X R x y).symm.tsum_eq]
have hv (z : stabilizerOrbit Λ X x y) :
weightedRow Λ X R w F ((fiberEquiv Λ X R x y).symm z) = w x * F x y := by
change w x * F x z = w x * F x y
obtain ⟨φ,hx,hy⟩ := (mem_stabilizerOrbit Λ X).mp z.property
have hh := hF φ x y
rw [hx,hy] at hh
rw [hh]
simp only [hv,tsum_fintype,Finset.sum_const,Finset.card_univ,nsmul_eq_mul]
have hcard : Fintype.card (stabilizerOrbit Λ X x y) = suborbitCard Λ X x y := by
rw [← Nat.card_eq_fintype_card,Nat.card_coe_set_eq]
rfl
rw [hcard]
ring
lemma fiber_sum (hfin : ∀ x y, (stabilizerOrbit Λ X x y).Finite)
(w : X → ℝ) (hw : ∀ (φ : Λ) x, w (φ • x) = w x)
(F : X → X → ℝ) (hF : ∀ (φ : Λ) x y, F (φ • x) (φ • y) = F x y)
(x y : X) :
(∑' p : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X x y}, weightedRow Λ X R w F p) =
w x * (suborbitCard Λ X x y : ℝ) * F x y := by
obtain ⟨u,hu,φ,rfl⟩ := R.cover x
let v := φ⁻¹ • y
have hy : φ • v = y := smul_inv_smul φ y
have hq : pairOrbit Λ X (φ • u) y = pairOrbit Λ X u v := by
rw [←hy,smul_pairOrbit]
rw [hq,fiber_sum_root Λ X R hfin w F hF ⟨u,hu⟩ v]
change w u * (suborbitCard Λ X u v : ℝ) * F u v = _
rw [←hy,hw,smul_suborbitCard,hF]
lemma tsum_weightedRow (w : X → ℝ) (F : X → X → ℝ)
(hF : ∀ x, (Function.support (F x)).Finite) :
(∑' p, weightedRow Λ X R w F p) = ∑ x ∈ R.roots, w x * ∑' y, F x y := by
have hs : Summable (weightedRow Λ X R w F) := summable_of_hasFiniteSupport (weightedRow_finite Λ X R w F hF)
rw [hs.tsum_prod]
simp only [weightedRow,tsum_mul_left,tsum_fintype]
exact Finset.sum_coe_sort R.roots (fun x => w x * ∑' y, F x y)
lemma massTransport (hfin : ∀ x y, (stabilizerOrbit Λ X x y).Finite)
(w : X → ℝ) (hw : ∀ (φ : Λ) x, w (φ • x) = w x)
(hbal : ∀ x y, w x * (suborbitCard Λ X x y : ℝ) = w y * (suborbitCard Λ X y x : ℝ))
(F : X → X → ℝ) (hF : ∀ (φ : Λ) x y, F (φ • x) (φ • y) = F x y)
(hrow : ∀ x, (Function.support (F x)).Finite)
(hcol : ∀ y, (Function.support (fun x => F x y)).Finite) :
(∑ x ∈ R.roots, w x * ∑' y, F x y) = ∑ y ∈ R.roots, w y * ∑' x, F x y := by
let F' := fun x y => F y x
have hF' (φ : Λ) (x y : X) : F' (φ • x) (φ • y) = F' x y := hF φ y x
have hs : Summable (weightedRow Λ X R w F) := summable_of_hasFiniteSupport (weightedRow_finite Λ X R w F hrow)
have ht : Summable (weightedRow Λ X R w F') := summable_of_hasFiniteSupport (weightedRow_finite Λ X R w F' hcol)
rw [←tsum_weightedRow Λ X R w F hrow,←tsum_weightedRow Λ X R w F' hcol]
calc
_ = ∑' q : PairOrbit Λ X, ∑' p : (pairMap Λ X R) ⁻¹' {q}, weightedRow Λ X R w F p :=
(hs.hasSum.tsum_fiberwise (pairMap Λ X R)).tsum_eq.symm
_ = ∑' q : PairOrbit Λ X, ∑' p : (pairMap Λ X R) ⁻¹' {swapPairOrbit Λ X q},
weightedRow Λ X R w F' p := by
apply tsum_congr
intro q
induction q using Quotient.inductionOn' with
| h p =>
change (∑' z : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X p.1 p.2}, weightedRow Λ X R w F z) =
∑' z : (pairMap Λ X R) ⁻¹' {pairOrbit Λ X p.2 p.1}, weightedRow Λ X R w F' z
rw [fiber_sum Λ X R hfin w hw F hF,fiber_sum Λ X R hfin w hw F' hF',hbal]
_ = ∑' p, weightedRow Λ X R w F' p := by
exact ((swapPairOrbit Λ X).tsum_eq (fun q =>
∑' p : (pairMap Λ X R) ⁻¹' {q}, weightedRow Λ X R w F' p)).trans
(ht.hasSum.tsum_fiberwise (pairMap Λ X R)).tsum_eq
end Representatives
end ActionOrbitTransport
end
end
end OAI
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