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c4cdbbd | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 | import OAI.Probability.InvariantIsing.Spectral.SpectralGroupIdentification
import OAI.Probability.InvariantIsing.Spectral.SpectralReconstruction
import OAI.Probability.InvariantIsing.Arrays.ReplicaProductRoots
namespace OAI
/-! Root reconstruction for a finite family of canonical overlap paths.
The Ward equations and reconstructed symbols determine the roots, whose
sum follows from the almost-everywhere path sum. -/
noncomputable section
open MeasureTheory Set Filter Function
open scoped BigOperators Topology
namespace InvariantIsing
variable {ι : Type*} [Fintype ι]
lemma projectedResolvent_deficit_path_root_tendsto (ρ eig : ι → ℝ)
(hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) (p : OverlapPath) (a : ι) :
Tendsto (fun s => projectedResolvent ρ eig hρ hρsum a (deficit p (p s)))
(𝓝[>] (0 : ℝ))
(𝓝 (projectedResolvent ρ eig hρ hρsum a (deficit p (Function.rightLim p.val 0)))) := by
have hc : ContinuousOn (fun r => projectedResolvent ρ eig hρ hρsum a (deficit p r))
(Icc (0 : ℝ) 1) :=
(continuousOn_projectedResolvent ρ eig hρ hρsum a).comp
(continuous_deficit p).continuousOn (fun _ hr => deficit_mem_unit p hr)
have hp : Tendsto p.val (𝓝[>] (0 : ℝ)) (𝓝[Icc (0 : ℝ) 1] (Function.rightLim p.val 0)) :=
tendsto_nhdsWithin_iff.mpr ⟨p.monotone.tendsto_rightLim 0,
Filter.Eventually.of_forall fun s => ⟨p.nonneg s, p.le_one s⟩⟩
exact (hc _ p.rightLim_mem_unit).tendsto.comp hp
lemma canonical_symbol_root_eq (ρ eig : ι → ℝ)
(hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) (p q : OverlapPath) (d : ℝ) (a : ι)
(hsymbol : ∀ᵐ s ∂pathMeasure, pathSymbol d q.val s =
projectedResolvent ρ eig hρ hρsum a (deficit p (p s))) :
d - (∫ u in 0..1, q u) =
projectedResolvent ρ eig hρ hρsum a (deficit p (Function.rightLim p.val 0)) := by
have hqB (u : ℝ) : |q u| ≤ 1 := by
rw [abs_of_nonneg (q.nonneg u)]
exact q.le_one u
have he : ∀ᵐ s ∂pathMeasure, pathSymbol d q.val s -
projectedResolvent ρ eig hρ hρsum a (deficit p (p s)) = 0 := by
filter_upwards [hsymbol] with s hs
exact sub_eq_zero.mpr hs
have ht := (pathSymbol_root_tendsto d q.val q.measurable hqB
(q.monotone.tendsto_rightLim 0)).sub
(projectedResolvent_deficit_path_root_tendsto ρ eig hρ hρsum p a)
have hz := ae_constant_right_root he ht
linarith
lemma canonical_group_roots_sum (p : OverlapPath) (q : ι → OverlapPath)
(hsum : ∀ᵐ s ∂pathMeasure, ∑ a, q a s = p s) :
(∑ a, Function.rightLim (q a).val 0) = Function.rightLim p.val 0 := by
classical
have hs : Tendsto (fun s => ∑ a, q a s) (𝓝[>] (0 : ℝ))
(𝓝 (∑ a, Function.rightLim (q a).val 0)) := by
exact tendsto_finsetSum Finset.univ (fun a _ => (q a).monotone.tendsto_rightLim 0)
have he : ∀ᵐ s ∂pathMeasure, (∑ a, q a s) - p s = 0 := by
filter_upwards [hsum] with s hs
exact sub_eq_zero.mpr hs
have hz := ae_constant_right_root he (hs.sub (p.monotone.tendsto_rightLim 0))
linarith
/-- The actual root Ward equations fix every canonical group root once
the symbols and the total path have been identified. -/
theorem canonical_group_root_reconstruction (ρ eig : ι → ℝ)
(hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) (p : OverlapPath)
(q : ι → OverlapPath) (d : ι → ℝ)
(hsymbol : ∀ a, ∀ᵐ s ∂pathMeasure, pathSymbol (d a) (q a).val s =
projectedResolvent ρ eig hρ hρsum a (deficit p (p s)))
(hsum : ∀ᵐ s ∂pathMeasure, ∑ a, q a s = p s)
(hward : ∀ a b, ρ b * Function.rightLim (q a).val 0 -
ρ a * Function.rightLim (q b).val 0 = (eig a - eig b) *
((d a - ∫ u in 0..1, q a u) * Function.rightLim (q b).val 0 +
Function.rightLim (q a).val 0 * (d b - ∫ u in 0..1, q b u))) (a : ι) :
Function.rightLim (q a).val 0 = Function.rightLim p.val 0 *
projectedResolventDerivative ρ eig hρ hρsum a (deficit p (Function.rightLim p.val 0)) := by
have hsym (b : ι) := canonical_symbol_root_eq ρ eig hρ hρsum p (q b) (d b) b (hsymbol b)
have hw (b c : ι) : ρ c * Function.rightLim (q b).val 0 -
ρ b * Function.rightLim (q c).val 0 = (eig b - eig c) *
(projectedResolvent ρ eig hρ hρsum b (deficit p (Function.rightLim p.val 0)) *
Function.rightLim (q c).val 0 +
Function.rightLim (q b).val 0 *
projectedResolvent ρ eig hρ hρsum c (deficit p (Function.rightLim p.val 0))) := by
simpa only [hsym b, hsym c] using hward b c
have he := spectral_root_reconstruction ρ eig hρ hρsum
(deficit_mem_unit p p.rightLim_mem_unit).1 (fun b => Function.rightLim (q b).val 0) hw a
rwa [canonical_group_roots_sum p q hsum] at he
/-- The symbols, path sum and root Ward equations determine the complete
canonical spectral family, including its diagonal. -/
theorem canonical_groups_identification_of_symbols_rootWard (ρ eig : ι → ℝ)
(hρ : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) (p : OverlapPath)
(q : ι → OverlapPath) (d : ι → ℝ)
(hsymbol : ∀ a, ∀ᵐ s ∂pathMeasure, pathSymbol (d a) (q a).val s =
projectedResolvent ρ eig hρ hρsum a (deficit p (p s)))
(hsum : ∀ᵐ s ∂pathMeasure, ∑ a, q a s = p s)
(hward : ∀ a b, ρ b * Function.rightLim (q a).val 0 -
ρ a * Function.rightLim (q b).val 0 = (eig a - eig b) *
((d a - ∫ u in 0..1, q a u) * Function.rightLim (q b).val 0 +
Function.rightLim (q a).val 0 * (d b - ∫ u in 0..1, q b u))) :
(∀ a, d a = spectralGroupDiagonal ρ eig hρ hρsum p a) ∧
∀ᵐ s ∂pathMeasure, ∀ a, q a s = spectralGroupPath ρ eig hρ hρsum p a s :=
spectralGroups_identification_of_symbols_roots ρ eig hρ hρsum p q d hsymbol
(canonical_group_root_reconstruction ρ eig hρ hρsum p q d hsymbol hsum hward)
end InvariantIsing
end
end OAI
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