| 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 |
|
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| end OAI |
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