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9b6e014 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | import Mathlib
import OAI.Probability.BinarySweep.SparseBounds.SparseAmplitude
import OAI.Probability.BinarySweep.SparseBounds.ConditionalCentered
import OAI.Probability.BinarySweep.SparseBounds.CenteredReindex
namespace OAI
noncomputable section
section
open scoped BigOperators Classical
namespace BinaryCoordinateSweeps
open Young Irrep
variable {b h k r : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h)
lemma groupAverage_comp_slots {X Y V : Type*} [Fintype X] [DecidableEq X]
[Fintype Y] [DecidableEq Y] [NormedAddCommGroup V] [InnerProductSpace ℂ V] [FiniteDimensional ℂ V]
(e : X ≃ Y) (ρ : Representation ℂ (Equiv.Perm X) V) (p : Equiv.Perm Y → ℂ) :
groupAverage (ρ.comp e.symm.permCongrHom.toMonoidHom) p =
groupAverage ρ (fun g => p (e.permCongr g)) := by
unfold groupAverage
rw [← Equiv.sum_comp e.permCongr]
apply Finset.sum_congr rfl
intro g _
congr 1
change ρ (e.permCongr.symm (e.permCongr g))=ρ g
rw [Equiv.symm_apply_apply]
theorem conditional_specht_hs (μ : YoungDiagram) (e : Cell μ ≃ FreeSlot H 0)
(a : Tail μ ≃ Fin k) (z : ℝ) :
evenMoment 1 (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom)
(fun g => (conditionalGroupLaw H z g:ℂ))) ≤
∑x : Placement H k 0, ∑y : Placement H k (Fin.last b),
(centeredEndpointKernel H z (fun i => ((x i).val,(y i).val)))^2 := by
rw [groupAverage_comp_slots]
have hs : (gridSize bits:ℝ)≠0 := by exact_mod_cast (gridSize_pos bits).ne'
have he := specht_moment_one_le_centered_matrix μ (gridSize bits) hs
(fun g => (conditionalGroupLaw H z (e.permCongr g):ℂ))
have hr := centeredMatrix_hs_reindex a e (gridSize bits)
(fun g => (conditionalGroupLaw H z (e.permCongr g):ℂ))
simp only [Equiv.apply_symm_apply] at hr
rw [← hr] at he
refine he.trans_eq ?_
change (∑x : Placement H k 0, ∑y : Placement H k 0,
‖centeredMatrix (gridSize bits) (fun g => (conditionalGroupLaw H z g:ℂ)) y x‖^2)=_
apply Finset.sum_congr rfl
intro x _
simp_rw [centeredMatrix_conditional H z,Complex.norm_real,Real.norm_eq_abs,sq_abs]
exact Equiv.sum_comp (placementIdentification H k) (fun y : Placement H k (Fin.last b) =>
(centeredEndpointKernel H z (fun i => ((x i).val,(y i).val)))^2)
theorem conditional_specht_many_bound (μ : YoungDiagram) (e : Cell μ ≃ FreeSlot H 0)
(a : Tail μ ≃ Fin k) (hd : ∀j, bits j≤2*r) {z : ℝ}
(hz : 0≤z) (hzr : z≤linePerturbationRadius r) (hz1 : z<1)
(hsmall : ((k+h:ℕ):ℝ) * (∑j : Fin b, (1:ℝ)/Fintype.card (GridOutside bits j)) ≤ 1) :
evenMoment 1 (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom)
(fun g => (conditionalGroupLaw H z g:ℂ))) ≤
((2:ℝ)^k * Real.exp ((b:ℝ)*(h+k)+Real.log 4*k*b))^2 *
((2:ℝ)^k * (((k+h:ℕ):ℝ) *
(∑j : Fin b, (1:ℝ)/Fintype.card (GridOutside bits j)))^((k+1)/2)) :=
(conditional_specht_hs H μ e a z).trans
(centeredEndpointKernel_hs_bound H hd hz hzr hz1 hsmall)
end BinaryCoordinateSweeps
end
open scoped BigOperators Classical
open Filter Topology
namespace BinaryCoordinateSweeps
variable {b h k : ℕ} {bits : Fin b → ℕ} (H : PathFamily bits h)
lemma continuous_lineLaw (d : ℕ) (g : Equiv.Perm (Slot d)) :
Continuous (fun z => lineLaw d z g) := by
unfold lineLaw
fun_prop
lemma continuous_gridWeight (g : GridChoices bits) : Continuous (fun z => gridWeight bits z g) := by
unfold gridWeight
exact continuous_finsetProd _ (fun j _ => continuous_finsetProd _ (fun y _ => continuous_lineLaw _ _))
lemma continuous_conditionalNormalizer : Continuous (conditionalNormalizer H) := by
unfold conditionalNormalizer
apply continuous_finsetSum
intro g _
split_ifs
· exact continuous_gridWeight g
· exact continuous_const
lemma continuousAt_conditionalChoiceWeight (g : ConditionalChoices H) :
ContinuousAt (fun z => conditionalChoiceWeight H z g) 0 :=
(continuous_gridWeight g.val).continuousAt.div (continuous_conditionalNormalizer H).continuousAt
(conditionalNormalizer_pos H (le_refl 0) (by norm_num)).ne'
lemma continuousAt_conditionalGroupLaw (a : Equiv.Perm (FreeSlot H 0)) :
ContinuousAt (fun z => conditionalGroupLaw H z a) 0 := by
unfold conditionalGroupLaw
apply tendsto_finsetSum
intro g _
split_ifs
· exact continuousAt_conditionalChoiceWeight H g
· exact continuousAt_const
lemma continuousAt_endpointProbability {I : Type*} [Fintype I] [DecidableEq I]
(A : Finset I) (e : I → GridSlot bits × GridSlot bits) :
ContinuousAt (fun z => endpointProbability H z A e) 0 := by
simp_rw [endpointProbability_conditional]
unfold conditionalEventWeight
apply tendsto_finsetSum
intro g _
split_ifs
· exact continuousAt_conditionalChoiceWeight H g
· exact continuousAt_const
lemma continuousAt_centeredEndpointKernel {I : Type*} [Fintype I] [DecidableEq I]
(e : I → GridSlot bits × GridSlot bits) :
ContinuousAt (fun z => centeredEndpointKernel H z e) 0 := by
unfold centeredEndpointKernel alternatingSubsetSum
apply ContinuousAt.mul continuousAt_const
apply tendsto_finsetSum
intro A _
exact continuousAt_const.mul (continuousAt_endpointProbability H A e)
lemma continuousAt_centeredEndpointHS (k : ℕ) :
ContinuousAt (fun z => ∑x : Placement H k 0, ∑y : Placement H k (Fin.last b),
(centeredEndpointKernel H z (fun i => ((x i).val,(y i).val)))^2) 0 := by
apply tendsto_finsetSum
intro x _
apply tendsto_finsetSum
intro y _
exact (continuousAt_centeredEndpointKernel H _).pow 2
end BinaryCoordinateSweeps
end
end OAI
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