File size: 2,873 Bytes
180bc03 | 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 | import OAI.NumberTheory.TwoPoint.Bounds.MatrixSpectralTail
import Mathlib.Topology.Algebra.Module.Spaces.ContinuousLinearMap
namespace OAI
/-! Identify the concrete nonbacktracking matrix with the operator in
the spectral transfer lemma. The change of norm is by conjugacy. -/
namespace TwoPointCorrelations
open Finset
open scoped Classical
variable {D V : Type*} [Fintype D] [DecidableEq D] [Fintype V] [DecidableEq V]
noncomputable def blockEuclideanEquiv :
(D → EuclideanSpace ℂ V) ≃L[ℂ] EuclideanSpace ℂ (D × V) :=
LinearEquiv.toContinuousLinearEquiv
{ toFun := fun v => WithLp.toLp 2 (fun x => v x.1 x.2)
invFun := fun v d => WithLp.toLp 2 (fun i => v (d, i))
left_inv := by intro v; ext d i; rfl
right_inv := by intro v; ext x; rfl
map_add' := by intro v w; ext x; rfl
map_smul' := by intro c v; ext x; rfl }
omit [DecidableEq D] [DecidableEq V] in
lemma blockEuclideanEquiv_apply (v : D → EuclideanSpace ℂ V) (d : D) (i : V) :
blockEuclideanEquiv v (d, i) = v d i := rfl
omit [DecidableEq D] [DecidableEq V] in
lemma blockEuclideanEquiv_symm_apply (v : EuclideanSpace ℂ (D × V)) (d : D) (i : V) :
blockEuclideanEquiv.symm v d i = v (d, i) := rfl
lemma blockNonbacktracking_operator_conj (A : D → Matrix V V ℝ) :
blockEuclideanEquiv.conjContinuousAlgEquiv
(nonbacktrackingContinuous (fun d => matrixOperator (fun i j => (A d i j : ℂ)))) =
matrixOperator (fun i j => (blockNonbacktrackingMatrix A i j : ℂ)) := by
ext v x
rcases x with ⟨d, i⟩
rw [ContinuousLinearEquiv.conjContinuousAlgEquiv_apply_apply, blockEuclideanEquiv_apply]
change (∑ l ∈ univ.erase d, matrixOperator (fun i j => (A l i j : ℂ))
(blockEuclideanEquiv.symm v l)) i = _
simp only [WithLp.ofLp_sum, Finset.sum_apply, matrixOperator_apply,
blockEuclideanEquiv_symm_apply]
rw [Fintype.sum_prod_type]
calc
_ = ∑ l : D, if d ≠ l then ∑ j : V, (A l i j : ℂ) * v (l, j) else 0 := by
have he : univ.erase d = univ.filter (fun l : D => d ≠ l) := by
ext l
simp [ne_comm]
rw [he, sum_filter]
_ = _ := by
apply sum_congr rfl
intro l _
by_cases hd : d ≠ l <;> simp [blockNonbacktrackingMatrix, hd]
theorem blockNonbacktracking_spectralRadius (A : D → Matrix V V ℝ) :
spectralRadius ℂ
(nonbacktrackingContinuous (fun d => matrixOperator (fun i j => (A d i j : ℂ)))) =
spectralRadius ℂ (matrixOperator (fun i j => (blockNonbacktrackingMatrix A i j : ℂ))) := by
let e := (blockEuclideanEquiv (D := D) (V := V)).conjContinuousAlgEquiv
have hs := AlgEquiv.spectrum_eq e
(nonbacktrackingContinuous (fun d => matrixOperator (fun i j => (A d i j : ℂ))))
rw [blockNonbacktracking_operator_conj] at hs
simp only [spectralRadius_eq_of_unital, hs]
end TwoPointCorrelations
end OAI
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