| import Mathlib |
| import OAI.Probability.SKRatio.Gaussian.GaussianEvents |
| import OAI.Probability.SKRatio.Variational.UniformDrift |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators Topology Matrix |
| open MeasureTheory Filter |
| namespace SKRatio.Calculus |
| attribute [local instance] Classical.propDecidable |
|
|
| lemma rowControl_le {n : ℕ} (J : Interaction n) (k : ℕ) {B : ℝ} |
| (hB : 0 ≤ B) (h : ∀ i, ∑ j, |J i j|^k ≤ B) : rowControl J k ≤ B := by |
| change ((Finset.univ.sup (fun i => ∑ j, ‖J i j‖₊^k) : NNReal) : ℝ) ≤ B |
| apply (NNReal.coe_le_coe (r₂ := ⟨B,hB⟩)).mpr |
| apply Finset.sup_le |
| intro i _ |
| have hh : ((∑ j, ‖J i j‖₊^k : NNReal) : ℝ) ≤ B := by |
| simpa only [NNReal.coe_sum,NNReal.coe_pow,coe_nnnorm,Real.norm_eq_abs] using h i |
| exact hh |
|
|
| lemma R3_le_entry_times_row {n : ℕ} (J : Interaction n) {δ B : ℝ} |
| (hδ : 0 ≤ δ) (hB : 0 ≤ B) |
| (he : ∀ i j, |J i j| ≤ δ) (hrow : ∀ i, ∑ j, J i j^2 ≤ B) : |
| R3 J ≤ δ*B := by |
| apply rowControl_le J 3 (mul_nonneg hδ hB) |
| intro i |
| calc |
| _ ≤ ∑ j, δ*(J i j)^2 := by |
| apply Finset.sum_le_sum |
| intro j _ |
| calc |
| _ = |J i j| *(J i j)^2 := by rw [pow_succ,sq_abs];ring |
| _ ≤ _ := mul_le_mul_of_nonneg_right (he i j) (sq_nonneg _) |
| _ = δ*(∑ j, J i j^2) := (Finset.mul_sum ..).symm |
| _ ≤ _ := mul_le_mul_of_nonneg_left (hrow i) hδ |
|
|
| theorem uniform_gradient_small_temperature {β δ : ℝ} |
| (hβ : 0 ≤ β) (hβsmall : β ≤ 7/20) (hδ : 0 ≤ δ) (hδsmall : δ ≤ 1/10000) |
| {n : ℕ} (g : Disorder n) (hg : g ∈ Gaussian.matrixGood β δ n) |
| (f : Observables n) (r : ℝ) (y : Spin n) : |
| deriv (fun v => unweightedGradient (semigroup (coupling g) v f) y) r - |
| generator (coupling g) (unweightedGradient (semigroup (coupling g) r f)) y ≤ |
| -(1/10:ℝ)*unweightedGradient (semigroup (coupling g) r f) y := by |
| have hb2 : β^2 ≤ 49/400 := by |
| nlinarith [mul_nonneg hβ (sub_nonneg.mpr hβsmall)] |
| have hrow (i : Fin n) : ∑ j, coupling g i j^2 ≤ 1 := (hg.2.2 i).trans (by linarith) |
| apply gradient_dissipation_small_norm (coupling g) (coupling_symm g) (coupling_diag g) |
| · intro i j |
| exact (hg.2.1 i j).trans (by linarith) |
| · exact hg.1.trans (by linarith) |
| · apply rowControl_le _ 2 (by norm_num) |
| intro i |
| simpa only [sq_abs] using (hg.2.2 i).trans (by linarith : β^2+δ ≤ 49/400+1/10000) |
| · exact (R3_le_entry_times_row (coupling g) hδ (by norm_num : (0:ℝ) ≤ 1) |
| hg.2.1 hrow).trans (by simpa only [mul_one] using hδsmall) |
|
|
| end SKRatio.Calculus |
|
|
| namespace SKRatio |
| open Calculus |
|
|
| theorem ratio_cutoff_small_temperature (β ε η : ℝ) |
| (hβ0 : 0 ≤ β) (hβsmall : β ≤ 7/20) |
| (hε0 : 0 < ε) (hεhalf : ε < 1/2) (hη : 0 < η) : |
| Tendsto |
| (fun n : ℕ => disorderLaw β n |
| {g : Disorder n | 1+η < |
| (mixingTime g ε : ℝ)/(mixingTime g (1-ε) : ℝ)}) |
| atTop (𝓝 0) ∧ |
| (∀ᶠ n : ℕ in atTop, ∀ g : Disorder n, 0 < mixingTime g (1-ε)) := by |
| obtain ⟨δ,γ,hd,hγ,hgap⟩ := Fields.exists_uniform_gap_tolerance hβ0 (by linarith : β < 1/2) |
| let d := min δ (1/10000:ℝ) |
| have hdp : 0 < d := lt_min hd (by norm_num) |
| have hdδ : d ≤ δ := min_le_left _ _ |
| have hds : d ≤ 1/10000 := min_le_right _ _ |
| let G := Gaussian.matrixGood β d |
| have hnorm : ∀ n, ∀ g ∈ G n, euclideanOpNorm (coupling g) ≤ 1 := by |
| intro n g hg |
| exact hg.1.trans (by linarith) |
| have hgp (n : ℕ) (g : Disorder n) (hg : g ∈ G n) (f : Observables n) : |
| γ*FiniteLaw.variance (mass g 0) f ≤ stationaryEnergy g f := by |
| have h := (hgap n g (hg.1.trans (by linarith)) |
| (fun i j => (hg.2.1 i j).trans hdδ) |
| (fun i => (hg.2.2 i).trans (by linarith)) 1 ⟨by norm_num,le_rfl⟩ 0).1 f |
| simpa only [one_smul,Fields.variance_eq_finiteLaw,Fields.dirichlet_zero_eq_stationaryEnergy] using h |
| apply ratio_cutoff_of_uniform_gradient β ε η hε0 (by linarith) hη G |
| (by norm_num : (0:ℝ) < 1/10) hγ (by norm_num : (0:ℝ) < 1) |
| (Gaussian.matrixGood_probability hβ0 hdp) hnorm hgp |
| intro n g hg f r y |
| exact uniform_gradient_small_temperature hβ0 hβsmall hdp.le hds g hg f r y |
|
|
| end SKRatio |
|
|
| end |
| end |
|
|
| end OAI |
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|
