| import Mathlib |
| import OAI.Probability.SKGap.Matrix.WordDiag |
| import OAI.Probability.SKGap.Matrix.RecipeWordEvent |
|
|
| namespace OAI |
|
|
| section |
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|
| noncomputable section |
| open scoped BigOperators |
| namespace SKGapCutoff.Recipe |
| open SKGap SKGap.Noncrossing SKGap.Noncrossing.Primary MeasureTheory ProbabilityTheory Real Set |
| variable {n : ℕ} |
|
|
| lemma wordBounded_lift (A : ℝ) (w : OrdinaryWord n) : |
| wordBounded A w ↔ ∀l∈liftWord w,l.bounded A := by |
| constructor |
| · intro h l hl |
| obtain ⟨a,ha,rfl⟩:=List.mem_map.mp hl |
| cases a <;> exact h _ ha |
| · intro h l hl |
| have hh:=h (Letter.toWord l) (List.mem_map.mpr ⟨l,hl,rfl⟩) |
| cases l <;> exact hh |
| |
| def WordSeminormBound (J : Interaction n) (j A C : ℝ) (L : ℕ) : Prop := |
| ∀(p : Bool) (w : OrdinaryWord n),w.length≤L→wordBounded A w→ |
| matrixWordSeminorm p (matrixWord J w-Matrix.diagonal (Diagram.prediction j w))≤C |
|
|
| lemma wordSeminormBound_diagram {J : Interaction n} {j A C : ℝ} {L : ℕ} |
| (h : WordSeminormBound J j A C L) : WordDiagramBound J j A C L := h false |
|
|
| lemma offDiagonalSeminorm_sub_diagonal (M : Interaction n) (a : Fin n→ℝ) : |
| SKGap.offDiagonalSeminorm (M-Matrix.diagonal a)=SKGap.offDiagonalSeminorm M := by |
| unfold SKGap.offDiagonalSeminorm |
| congr 1 |
| ext ⟨i,k⟩ |
| by_cases hik:i=k <;> simp [SKGap.offDiagonalVector,Matrix.diagonal_apply,hik] |
| |
| lemma matrixWord_opNorm (J : Interaction n) (hn : 0<n) {R : ℝ} |
| (hR : 1≤R) (hJ : SKGap.opNorm J≤R) (w : OrdinaryWord n) (hw : wordBounded R w) : |
| SKGap.opNorm (matrixWord J w)≤R^w.length := by |
| let : Nonempty (Fin n):=Fin.pos_iff_nonempty.mp hn |
| induction w with |
| | nil=>simpa [matrixWord] using (SKGap.opNorm_one_le (ι:=Fin n)) |
| | cons l w ih=> |
| have hl : SKGap.opNorm (matrixLetter J l)≤R := by |
| cases l with |
| | noise=>exact hJ |
| | diag a=>exact SKGap.opNorm_diagonal_le (by linarith) (hw (.diag a) (List.mem_cons_self)) |
| have hh:=ih (fun a ha=>hw a (List.mem_cons_of_mem _ ha)) |
| simp only [matrixWord,List.map_cons,List.prod_cons,List.length_cons,pow_succ] |
| exact (SKGap.opNorm_mul _ _).trans ((mul_le_mul hl hh (norm_nonneg _) (by linarith)).trans_eq (mul_comm _ _)) |
|
|
| lemma wordSeminormBound_test {J : Interaction n} {j A C R : ℝ} {L : ℕ} |
| (h : WordSeminormBound J j A C L) (hn : 0<n) (hR : 1≤R) (hA : A≤R) (hJ : SKGap.opNorm J≤R) : |
| WordTestBound J A (R^L+C) L := by |
| intro w hw hb |
| have hs:=h true w hw hb |
| simp only [matrixWordSeminorm,↓reduceIte,offDiagonalSeminorm_sub_diagonal] at hs |
| exact add_le_add ((matrixWord_opNorm J hn hR hJ w (wordBounded_mono hb hA)).trans |
| (pow_le_pow_right₀ hR hw)) hs |
|
|
| theorem ordinary_word_event {j A : ℝ} (hj : 0<j) (hj1 : j<1) (hA : 1≤A) (L : ℕ) : |
| ∃(C : ℝ) (N : ℕ),0<C ∧ 0<N ∧ ∀n,N≤n→ |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n)=>gaussianReal 0 1)) |
| {g | ¬WordSeminormBound (removeDiagonal (goeMatrix (j/n) g)) j A C L}≤ |
| 2*((wordPatternSet L L).card*ENNReal.ofReal (3*exp (-(n:ℝ)))+ |
| ENNReal.ofReal (exp (-2*(n:ℝ)/(π^2*(sqrt (2*j))^2)))+ |
| (ENNReal.ofReal (2*exp (-(n:ℝ)/(π^2*j)))+ |
| ENNReal.ofReal (2*(n:ℝ)*exp (-1/(8*(j/n)))))) := by |
| have hs : sqrt j*(1:ℝ)<1 := by nlinarith [sq_sqrt hj.le,sqrt_nonneg j] |
| obtain ⟨C,N,hC,hN,hb⟩:=zeroDiag_exact_words_two_seminorms hj (by norm_num : (0:ℝ)<1) |
| hs hA hA (by norm_num : (0:ℝ)<1) L |
| refine ⟨C,N,hC,hN,fun n hn=>?_⟩ |
| apply (measure_mono ?_).trans (hb n hn) |
| intro g hg |
| simp only [WordSeminormBound,not_forall,not_le] at hg |
| obtain ⟨p,w,hw,hwb,hfail⟩:=hg |
| refine ⟨p,(fun _=>0),liftWord w,?_,?_,(wordBounded_lift A w).mp hwb,?_,?_,?_⟩ |
| · intro i; norm_num |
| · simpa [liftWord] using hw |
| · simp |
| · exact Or.inl (inverseCount_lift w) |
| · have he : wordPrediction j (fun _ : Fin n=>0) 1 (liftWord w)=Diagram.prediction j w:= |
| funext (wordPrediction_lift j _ _ _) |
| simpa only [exactWord_lift,he] using hfail |
|
|
| end SKGapCutoff.Recipe |
|
|
| end |
| end |
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| end OAI |
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|
