| import Mathlib |
| import OAI.Probability.SKBarriers.SpinGlass.Basic |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators InnerProductSpace |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem finite_unit_packing {E I : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] |
| [DecidableEq I] (S : Finset I) (u : I → E) (w : E) (b e : ℝ) |
| (hb : 0<b) (he : 0≤e) (heb : e<b^2/2) |
| (hw : ⟪w,w⟫_ℝ=1) (hu : ∀ i∈S,⟪u i,u i⟫_ℝ=1) |
| (hpair : ∀ i∈S,∀ j∈S,i≠j → ⟪u i,u j⟫_ℝ≤e) |
| (hlarge : ∀ i∈S,b<⟪u i,w⟫_ℝ) : (S.card:ℝ)<2/b^2 := by |
| by_cases hS : S.Nonempty |
| · let d : ℝ := S.card |
| have hd : 0<d := by dsimp [d]; exact_mod_cast hS.card_pos |
| have hl : d*b<⟪∑ i∈S,u i,w⟫_ℝ := by |
| rw [sum_inner] |
| simpa [d] using Finset.sum_lt_sum_of_nonempty hS hlarge |
| have hu' : ⟪∑ i∈S,u i,∑ i∈S,u i⟫_ℝ≤d+d^2*e := by |
| rw [sum_inner] |
| simp_rw [inner_sum] |
| calc |
| _ ≤ ∑ i∈S,∑ j∈S,((if i=j then (1:ℝ) else 0)+e) := by |
| apply Finset.sum_le_sum |
| intro i hi |
| apply Finset.sum_le_sum |
| intro j hj |
| by_cases hij : i=j |
| · subst j; rw [ite_eq_left rfl,hu i hi]; linarith only [he] |
| · simpa only [ite_eq_right hij,zero_add] using hpair i hi j hj hij |
| _ = _ := by |
| simp only [Finset.sum_add_distrib,Finset.sum_ite_eq,Finset.sum_const,nsmul_eq_mul] |
| simp only [Finset.sum_ite_mem,Finset.inter_self] |
| simp [d] |
| ring |
| have hcs := real_inner_mul_inner_self_le (∑ i∈S,u i) w |
| rw [hw,mul_one] at hcs |
| have hls : (d*b)^2<⟪∑ i∈S,u i,w⟫_ℝ^2 := by |
| exact pow_lt_pow_left₀ hl (mul_nonneg hd.le hb.le) (by decide) |
| have hquad : d^2*b^2<d+d^2*e := by nlinarith only [hls,hcs,hu'] |
| have hdb : d*b^2<2 := by |
| by_contra h |
| have hd2 : 2≤d*b^2 := le_of_not_gt h |
| have hpos : 0<d^2 := sq_pos_of_pos hd |
| have hs := mul_lt_mul_of_pos_left heb hpos |
| have hm := mul_le_mul_of_nonneg_left hd2 hd.le |
| nlinarith only [hquad,hs,hm] |
| exact (lt_div_iff₀ (sq_pos_of_pos hb)).mpr (by simpa [d] using hdb) |
| · have hz : S=∅ := Finset.not_nonempty_iff_eq_empty.mp hS |
| simp only [hz,Finset.card_empty,Nat.cast_zero] |
| positivity |
|
|
| def spinUnitVector {n : ℕ} (x : Config n) : EuclideanSpace ℝ (Fin n) := |
| WithLp.toLp 2 (fun i => spin (x i)/Real.sqrt n) |
|
|
| theorem spinUnitVector_inner {n : ℕ} (x y : Config n) : |
| ⟪spinUnitVector x,spinUnitVector y⟫_ℝ=overlap x y := by |
| change (∑ i : Fin n,(spin (y i)/Real.sqrt n)*(spin (x i)/Real.sqrt n))=overlap x y |
| simp only [div_mul_div_comm,← sq,Real.sq_sqrt (Nat.cast_nonneg n),← Finset.sum_div] |
| exact congrArg (fun z : ℝ => z/(n:ℝ)) (Finset.sum_congr rfl (fun i _ => mul_comm _ _)) |
|
|
| @[simp] theorem overlap_flip_first {n : ℕ} (x y : Config n) : overlap (flip x) y = -overlap x y := by |
| rw [overlap_comm,overlap_flip_right,overlap_comm x y] |
|
|
| theorem overlap_conflict_card {n : ℕ} (hn : 0<n) {I : Type*} [DecidableEq I] |
| (S : Finset I) (v : I → Config n) (w : Config n) (b e : ℝ) |
| (hb : 0<b) (he : 0≤e) (heb : e<b^2/2) |
| (hpair : ∀ i∈S,∀ j∈S,i≠j → |overlap (v i) (v j)|≤e) |
| (hlarge : ∀ i∈S,b< |overlap (v i) w|) : (S.card:ℝ)<2/b^2 := by |
| let u (i : I) : Config n := if 0≤overlap (v i) w then v i else flip (v i) |
| apply finite_unit_packing S (fun i => spinUnitVector (u i)) (spinUnitVector w) b e hb he heb |
| · rw [spinUnitVector_inner,overlap_self hn] |
| · intro i hi; rw [spinUnitVector_inner,overlap_self hn] |
| · intro i hi j hj hij |
| rw [spinUnitVector_inner] |
| have h := hpair i hi j hj hij |
| dsimp only [u] |
| split_ifs <;> (try simp only [overlap_flip_first,overlap_flip_right,neg_neg]) <;> |
| linarith only [(abs_le.mp h).1,(abs_le.mp h).2] |
| · intro i hi |
| rw [spinUnitVector_inner] |
| dsimp only [u] |
| split_ifs with h |
| · simpa only [abs_of_nonneg h] using hlarge i hi |
| · simpa only [overlap_flip_first,abs_of_neg (lt_of_not_ge h)] using hlarge i hi |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
