| import Mathlib |
| import OAI.Probability.BinarySweep.YoungTheory.GenericHookAux |
| import OAI.Probability.BinarySweep.YoungTheory.PhysicalHookInduced |
| import OAI.Probability.BinarySweep.Conditional.CycleAugmentation |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators Classical |
| open Filter |
|
|
| namespace BinaryCoordinateSweeps |
| open Irrep Representation Young GridSplit Signed |
|
|
| def EnumeratedMomentBound (q : ℕ) {b : ℕ} (bits : Fin b → ℕ) (z : ℝ) : Prop := |
| ∀ h : ℕ, ∀ H : PathFamily bits h, ∀ α : (Fintype.card (FreeSlot H 0)).Partition, |
| evenMoment q (groupAverage (partitionHilbertRep α) |
| (fun a => (conditionalGroupLaw H z ((Fintype.equivFin _).symm.permCongr a):ℂ))) ≤ |
| Real.exp (-cExponent (gridSize bits)*typeF α+ |
| eExponent (gridSize bits)*h*Real.log (gridSize bits)-pathCost H) |
|
|
| def diagramF (μ : YoungDiagram) : ℝ := Real.log (Module.finrank ℂ (SpechtSpace μ)) |
|
|
| lemma moment_divide_dimension {D T B : ℝ} (hD : 0<D) (h : D*T≤Real.exp B) : |
| T≤Real.exp (B-Real.log D) := by |
| have hh : T*D≤Real.exp B := by rwa [mul_comm] |
| have he := (le_div_iff₀ hD).mpr hh |
| rwa [←Real.exp_log hD,←Real.exp_sub] at he |
|
|
| lemma dense_of_auxiliary_hook {m n h r : ℕ} (s q t : ℕ) (hq : 0<q) |
| (bits : Fin (m+n) → ℕ) (H : PathFamily bits h) |
| (hd : ∀j, bits j≤2*r) (μ : YoungDiagram) (e : Cell μ ≃ FreeSlot H 0) |
| (z : ℝ) (hz : 0≤z) (hzr : z≤linePerturbationRadius r) |
| (hhook : ∀ h' : ℕ, ∀ HH : PathFamily bits h', |
| ∀ ee : Cell (hookPart μ t) ≃ FreeSlot HH 0, |
| evenMoment q (groupAverage ((hilbertSpecht (hookPart μ t)).comp |
| ee.symm.permCongrHom.toMonoidHom) (fun a => (conditionalGroupLaw HH z a:ℂ)))≤ |
| Real.exp (-(cExponent s+inductionGap s)*diagramF (hookPart μ t)+ |
| (eExponent s-inductionGap s)*h'*Real.log s-pathCost HH+(s:ℝ)^(9/10:ℝ))) : |
| evenMoment q (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) |
| (fun a => (conditionalGroupLaw H z a:ℂ)))≤ |
| Real.exp (-(cExponent s+inductionGap s)*diagramF (hookPart μ t)+ |
| (eExponent s-inductionGap s)*(h+Fintype.card (Cell (southeast μ t)))*Real.log s+ |
| (s:ℝ)^(9/10:ℝ)-pathCost H+ |
| Real.log 4*Fintype.card (Cell (southeast μ t))*(m+n)-diagramF (southeast μ t)) := by |
| let k := Fintype.card (Cell (southeast μ t)) |
| let f := hookPlacementBasis H μ t e |
| let σ := hilbertSpecht (hookPart μ t) |
| let B := -(cExponent s+inductionGap s)*diagramF (hookPart μ t)+ |
| (eExponent s-inductionGap s)*(h+k)*Real.log s+(s:ℝ)^(9/10:ℝ) |
| have hf (x y : Placement H k 0) (hp : 0<placementKernel H z x y) : |
| placementFiberMoment H f σ z q x y ≤ |
| Real.exp (B-placementTransitionCost H z x (placementIdentification H k y)) := by |
| let g := realizePlacement H z x (placementIdentification H k y) hp |
| have hb := hhook (h+k) (augmentByChoice H x g) (augmentedInputBasis H x g f) |
| unfold placementFiberMoment |
| rw [dite_eq_left hp] |
| change evenMoment q (groupAverage (augmentedRepresentation H x g f σ) _)≤_ |
| change evenMoment q (groupAverage ((hilbertSpecht (hookPart μ t)).comp |
| (augmentedInputBasis H x g f).symm.permCongrHom.toMonoidHom) _)≤_ at hb ⊢ |
| refine hb.trans_eq ?_ |
| unfold placementTransitionCost |
| rw [dite_eq_left (show 0<placementProbability H z x (placementIdentification H k y) from hp)] |
| congr 1 |
| dsimp [B,diagramF] |
| push_cast |
| ring |
| have hi := induced_placement_of_hook_bound H f σ hd hz hzr |
| (hilbertSpecht_unitary (hookPart μ t)) hq B hf |
| have hm := (conditional_hook_induced_moment_le H μ t e z q).trans hi |
| have hD : (0:ℝ)<Module.finrank ℂ (SpechtSpace (southeast μ t)) := by |
| rw [←hilbertSpecht_finrank] |
| exact_mod_cast irreducible_finrank_pos (hilbertSpecht (southeast μ t)) |
| refine (moment_divide_dimension hD hm).trans_eq ?_ |
| congr 1 |
| dsimp [B,k,diagramF] |
| push_cast |
| ring |
|
|
| theorem dense_branch_operator_eventually (q : ℕ) (hq : 2≤q) : ∀ᶠ s : ℕ in atTop, |
| ∀ m n h r : ℕ, ∀ bits : Fin (m+n) → ℕ, ∀ H : PathFamily bits h, |
| gridSize bits=s → |
| 1≤Real.log (gridSize (leftBits bits)) → 1≤Real.log (gridSize (rightBits bits)) → |
| Real.log (gridSize (leftBits bits))≤(4/5:ℝ)*Real.log s → |
| Real.log (gridSize (rightBits bits))≤(4/5:ℝ)*Real.log s → |
| (∀j, bits j≤2*r) → |
| ∀ t : ℕ, 0<t → (t:ℝ)≤(s:ℝ)^(1/100:ℝ) → |
| ∀ μ : YoungDiagram, ∀ e : Cell μ ≃ FreeSlot H 0, ∀z : ℝ, |
| 0≤z → z≤linePerturbationRadius r → |
| EnumeratedMomentBound q (leftBits bits) z → EnumeratedMomentBound q (rightBits bits) z → |
| evenMoment q (groupAverage ((hilbertSpecht μ).comp e.symm.permCongrHom.toMonoidHom) |
| (fun a => (conditionalGroupLaw H z a:ℂ)))≤ |
| Real.exp (-(cExponent s+inductionGap s)*diagramF (hookPart μ t)+ |
| (eExponent s-inductionGap s)*(h+Fintype.card (Cell (southeast μ t)))*Real.log s+ |
| (s:ℝ)^(9/10:ℝ)-pathCost H+ |
| Real.log 4*Fintype.card (Cell (southeast μ t))*(m+n)-diagramF (southeast μ t)) := by |
| filter_upwards [specht_hook_auxiliary_eventually q hq] with s hs |
| intro m n h r bits H hsize hu hv hub hvb hd t ht hts μ e z hz hzr hR hC |
| apply dense_of_auxiliary_hook s q t (by omega) bits H hd μ e z hz hzr |
| intro h' HH ee |
| exact hs m n h' bits HH hsize hu hv hub hvb t ht hts |
| (hookPart μ t) (hookPart_inHook μ t) ee z |
| (fun y a => hR _ (rowFamily bits HH y) a.val) |
| (fun x a => hC _ (columnFamily bits HH x) a.val) |
|
|
| end BinaryCoordinateSweeps |
|
|
| end |
|
|
| end OAI |
|
|
