| import Mathlib |
| import OAI.Probability.SKGap.Localization.Magnetization |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators |
| namespace SKGapCutoff |
| open Filter Topology |
|
|
| noncomputable def dimensionDecay (a : ℝ) (n : ℕ) : ℝ := |
| Real.exp (-a * Real.log (n : ℝ)) |
|
|
| lemma dimensionDecay_tendsto (a : ℝ) (ha : 0 < a) : |
| Tendsto (dimensionDecay a) atTop (nhds 0) := by |
| have hlog : Tendsto (fun n : ℕ => Real.log (n : ℝ)) atTop atTop := |
| Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop |
| have hneg := tendsto_neg_atTop_atBot.comp (hlog.const_mul_atTop ha) |
| change Tendsto (fun n : ℕ => Real.exp (-a * Real.log (n : ℝ))) atTop (nhds 0) |
| simpa only [Function.comp_def, neg_mul] using Real.tendsto_exp_atBot.comp hneg |
|
|
| lemma dimensionDecay_upper_tendsto (a : ℝ) (ha : 0 < a) : |
| Tendsto (fun n => Real.sqrt (Real.exp (dimensionDecay a n) - 1) / 2) |
| atTop (nhds 0) := by |
| have hexp := (Real.continuous_exp.tendsto 0).comp (dimensionDecay_tendsto a ha) |
| have h := ((hexp.sub (tendsto_const_nhds (x := (1 : ℝ)))).sqrt).div_const (2 : ℝ) |
| simpa using h |
|
|
| lemma mul_exp_log_identity {n : ℕ} (hn : 0 < n) (a : ℝ) : |
| (n : ℝ) * Real.exp (-(1 + a) * Real.log (n : ℝ)) = dimensionDecay a n := by |
| have hn' : (0 : ℝ) < n := by exact_mod_cast hn |
| calc |
| _ = Real.exp (Real.log (n : ℝ)) * Real.exp (-(1 + a) * Real.log (n : ℝ)) := by |
| rw [Real.exp_log hn'] |
| _ = _ := by rw [← Real.exp_add]; unfold dimensionDecay; congr 1; ring |
|
|
| lemma exp_log_div_identity {n : ℕ} (hn : 0 < n) (a : ℝ) : |
| Real.exp ((1 - a) * Real.log (n : ℝ)) / (n : ℝ) = dimensionDecay a n := by |
| have hn' : (0 : ℝ) < n := by exact_mod_cast hn |
| calc |
| _ = Real.exp ((1 - a) * Real.log (n : ℝ)) / Real.exp (Real.log (n : ℝ)) := by |
| rw [Real.exp_log hn'] |
| _ = _ := by rw [← Real.exp_sub]; unfold dimensionDecay; congr 1; ring |
|
|
| lemma cutoffTime_one (n : ℕ) : cutoffTime 1 n = Real.log (n : ℝ) / 2 := by |
| simp [cutoffTime] |
|
|
| lemma cutoffTime_one_nonneg {n : ℕ} (hn : 0 < n) : 0 ≤ cutoffTime 1 n := by |
| rw [cutoffTime_one] |
| exact div_nonneg (Real.log_nonneg (by exact_mod_cast hn)) (by norm_num) |
|
|
| lemma tendsto_one_of_error (f e : ℕ → ℝ) (he : Tendsto e atTop (nhds 0)) |
| (hup : ∀ᶠ n in atTop, f n ≤ 1) (hlo : ∀ᶠ n in atTop, 1 - e n ≤ f n) : |
| Tendsto f atTop (nhds 1) := by |
| have hz : Tendsto (fun n => 1 - f n) atTop (nhds 0) := |
| squeeze_zero' (hup.mono (fun _ h => by linarith)) |
| (hlo.mono (fun _ h => by linarith)) he |
| have h := (tendsto_const_nhds (x := (1 : ℝ))).sub hz |
| simpa using h |
| |
| lemma continuous_zero_cutoff_upper (ε : ℝ) (hε : 0 < ε) : |
| Tendsto (fun n => worstContinuous (0 : Interaction n) ((1 + ε) * cutoffTime 1 n)) |
| atTop (nhds 0) := by |
| apply squeeze_zero (fun n => worstContinuous_nonneg _ _) |
| (fun n => ?_) (dimensionDecay_upper_tendsto ε hε) |
| by_cases hn : 0 < n |
| · have h := worstContinuous_zero_upper (n := n) ((1 + ε) * cutoffTime 1 n) |
| have he : (n : ℝ) * Real.exp (-2 * ((1 + ε) * cutoffTime 1 n)) = |
| dimensionDecay ε n := by |
| rw [cutoffTime_one] |
| convert mul_exp_log_identity hn ε using 2; congr 1; ring |
| simpa only [he] using h |
| · have hn0 : n = 0 := by omega |
| subst n |
| have hup := worstContinuous_zero_upper (n := 0) ((1 + ε) * cutoffTime 1 0) |
| simp only [Nat.cast_zero, zero_mul, Real.exp_zero, sub_self, Real.sqrt_zero, |
| zero_div] at hup |
| exact hup.trans (by positivity) |
| |
| lemma continuous_zero_cutoff_lower (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1) : |
| Tendsto (fun n => worstContinuous (0 : Interaction n) ((1 - ε) * cutoffTime 1 n)) |
| atTop (nhds 1) := by |
| apply tendsto_one_of_error _ (fun n => 8 * dimensionDecay ε n) |
| (by simpa using (dimensionDecay_tendsto ε hε).const_mul 8) |
| · filter_upwards [eventually_gt_atTop 0] with n hn |
| exact worstContinuous_le_one _ _ |
| (mul_nonneg (by linarith) (cutoffTime_one_nonneg hn)) |
| · filter_upwards [eventually_gt_atTop 0] with n hn |
| have h := worstContinuous_zero_lower hn ((1 - ε) * cutoffTime 1 n) |
| (mul_nonneg (by linarith) (cutoffTime_one_nonneg hn)) |
| have he : 8 * Real.exp (2 * ((1 - ε) * cutoffTime 1 n)) / (n : ℝ) = |
| 8 * dimensionDecay ε n := by |
| rw [mul_div_assoc, ← exp_log_div_identity hn ε, cutoffTime_one] |
| congr 2 |
| ring_nf |
| simpa only [he] using h |
| |
| @[simp] lemma sampledInteraction_zero (n : ℕ) : |
| sampledInteraction (0 : GaussianCoordinates n) = (0 : Interaction n) := by |
| funext i j |
| simp [sampledInteraction] |
| |
| lemma disorderLaw_zero (n : ℕ) : disorderLaw 0 n = |
| MeasureTheory.Measure.dirac (0 : GaussianCoordinates n) := by |
| unfold disorderLaw |
| simp only [zero_pow (by norm_num : 2 ≠ 0), zero_div, Real.toNNReal_zero, |
| ProbabilityTheory.gaussianReal_zero_var] |
| apply MeasureTheory.Measure.pi_eq |
| intro s _ |
| simp only [MeasureTheory.Measure.dirac_apply, Set.indicator, Pi.one_apply, |
| Pi.zero_apply, Set.mem_univ_pi] |
| have hm : (0 : GaussianCoordinates n) ∈ Set.univ.pi s ↔ |
| ∀ i : Fin n × Fin n, (0 : ℝ) ∈ s i := Set.mem_univ_pi |
| by_cases h : (0 : GaussianCoordinates n) ∈ Set.univ.pi s |
| · simp only [ite_eq_left h] |
| symm |
| exact Finset.prod_eq_one (fun i _ => ite_eq_left ((hm.mp h) i)) |
| · simp only [ite_eq_right h] |
| symm |
| obtain ⟨i, hi⟩ := not_forall.mp (fun hforall => h (hm.mpr hforall)) |
| exact Finset.prod_eq_zero (Finset.mem_univ i) (by simp [hi]) |
| |
| lemma disorderLimit_zero_of_tendsto (F : (n : ℕ) → Interaction n → ℝ) (a : ℝ) |
| (h : Tendsto (fun n => F n 0) atTop (nhds a)) : DisorderLimit 0 F a := by |
| intro δ hδ |
| have habs : Tendsto (fun n => |F n 0 - a|) atTop (nhds 0) := by |
| simpa using (h.sub (tendsto_const_nhds (x := a))).abs |
| have hlt := habs.eventually_lt_const hδ |
| have he : (fun n => (disorderLaw 0 n) |
| {g | δ < |F n (sampledInteraction g) - a|}) =ᶠ[atTop] (fun _ => 0) := by |
| filter_upwards [hlt] with n hn |
| rw [disorderLaw_zero, MeasureTheory.Measure.dirac_apply] |
| simp [Set.indicator, not_lt.mpr hn.le] |
| exact (tendsto_congr' he).mpr tendsto_const_nhds |
|
|
| lemma discrete_zero_cutoff_upper (ε : ℝ) (hε : 0 < ε) : |
| Tendsto (fun n => worstDiscrete (0 : Interaction n) |
| ⌈(1 + ε) * (n : ℝ) * cutoffTime 1 n⌉₊) atTop (nhds 0) := by |
| apply squeeze_zero' (Eventually.of_forall (fun n => worstDiscrete_nonneg _ _)) |
| _ (dimensionDecay_upper_tendsto ε hε) |
| filter_upwards [eventually_gt_atTop 0] with n hn |
| have hn' : (0 : ℝ) < n := by exact_mod_cast hn |
| let k := ⌈(1 + ε) * (n : ℝ) * cutoffTime 1 n⌉₊ |
| have hceil : (1 + ε) * (n : ℝ) * (Real.log (n : ℝ) / 2) ≤ (k : ℝ) := by |
| simpa only [k, cutoffTime_one] using Nat.le_ceil ((1 + ε) * (n : ℝ) * cutoffTime 1 n) |
| have hr : -2 * (k : ℝ) / (n : ℝ) ≤ -(1 + ε) * Real.log (n : ℝ) := by |
| apply (div_le_iff₀ hn').mpr |
| nlinarith only [hceil] |
| have hq : (n : ℝ) * Real.exp (-2 * (k : ℝ) / n) ≤ dimensionDecay ε n := by |
| rw [← mul_exp_log_identity hn ε] |
| exact mul_le_mul_of_nonneg_left (Real.exp_le_exp.mpr hr) hn'.le |
| apply (worstDiscrete_zero_upper hn k).trans |
| gcongr |
|
|
| lemma discrete_zero_cutoff_lower (ε : ℝ) (hε : 0 < ε) (hε1 : ε < 1) : |
| Tendsto (fun n => worstDiscrete (0 : Interaction n) |
| ⌊(1 - ε) * (n : ℝ) * cutoffTime 1 n⌋₊) atTop (nhds 1) := by |
| apply tendsto_one_of_error _ (fun n => 8 * dimensionDecay (ε / 2) n) |
| (by simpa using (dimensionDecay_tendsto (ε / 2) (by positivity)).const_mul 8) |
| (Eventually.of_forall (fun n => worstDiscrete_le_one _ _)) |
| have hlarge : ∀ᶠ n : ℕ in atTop, 1 ≤ (ε / 2) * (n : ℝ) := |
| ((tendsto_natCast_atTop_atTop : Tendsto (fun n : ℕ => (n : ℝ)) atTop atTop).const_mul_atTop |
| (by positivity : 0 < ε / 2)).eventually (eventually_ge_atTop 1) |
| filter_upwards [eventually_ge_atTop 2, hlarge] with n hn hlargeN |
| have hn0 : 0 < n := by omega |
| have hn' : (1 : ℝ) < n := by exact_mod_cast (show 1 < n by omega) |
| have hs : 0 < (n : ℝ) - 1 := by linarith |
| have hlog : 0 ≤ Real.log (n : ℝ) := Real.log_nonneg (by linarith) |
| let k := ⌊(1 - ε) * (n : ℝ) * cutoffTime 1 n⌋₊ |
| have hfloor : (k : ℝ) ≤ (1 - ε) * (n : ℝ) * (Real.log (n : ℝ) / 2) := by |
| rw [← cutoffTime_one] |
| exact Nat.floor_le (mul_nonneg (mul_nonneg (by linarith) (Nat.cast_nonneg n)) |
| (cutoffTime_one_nonneg hn0)) |
| have hc : (1 - ε) * (n : ℝ) ≤ (1 - ε / 2) * ((n : ℝ) - 1) := by |
| nlinarith only [hlargeN, hε] |
| have hr : 2 * (k : ℝ) / ((n : ℝ) - 1) ≤ (1 - ε / 2) * Real.log (n : ℝ) := by |
| apply (div_le_iff₀ hs).mpr |
| nlinarith only [hfloor, mul_le_mul_of_nonneg_right hc hlog] |
| have he : 8 * Real.exp (2 * (k : ℝ) / ((n : ℝ) - 1)) / (n : ℝ) ≤ |
| 8 * dimensionDecay (ε / 2) n := by |
| rw [← exp_log_div_identity hn0 (ε / 2), ← mul_div_assoc] |
| gcongr |
| exact (sub_le_sub_left he 1).trans (worstDiscrete_zero_lower hn k) |
| |
| theorem beta_zero_cutoff : CutoffAtRate 0 1 := by |
| intro ε hε hε1 |
| exact ⟨disorderLimit_zero_of_tendsto _ _ (continuous_zero_cutoff_lower ε hε hε1), |
| disorderLimit_zero_of_tendsto _ _ (continuous_zero_cutoff_upper ε hε), |
| disorderLimit_zero_of_tendsto _ _ (discrete_zero_cutoff_lower ε hε hε1), |
| disorderLimit_zero_of_tendsto _ _ (discrete_zero_cutoff_upper ε hε)⟩ |
| |
| end SKGapCutoff |
| |
| end |
| |
| end OAI |
| |
