| import OAI.Probability.InvariantIsing.Haar.HaarKernelConvolution |
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| namespace OAI |
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| /-! Quantitative concentration of the positive polynomial kernels near the identity. -/ |
| noncomputable section |
| open Matrix MvPolynomial MeasureTheory Set Filter |
| open scoped Topology |
| namespace InvariantIsing |
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| lemma haarKernelMass_lower {N : ℕ} (μ : Measure (SpecialOrthogonal N)) |
| [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] (b : ℝ) (hb : 0 ≤ b) (n : ℕ) : |
| μ.real {V | b < haarKernelBase 1 V} * b^n ≤ haarKernelMass μ n := by |
| let S : Set (SpecialOrthogonal N) := {V | b < haarKernelBase 1 V} |
| have hS : MeasurableSet S := measurableSet_lt measurable_const (continuous_haarKernelBase 1).measurable |
| have hi : Integrable (S.indicator (fun _ => b^n)) μ := (integrable_const _).indicator hS |
| have hk := continuous_haar_integrable μ _ ((continuous_haarKernelBase 1).pow n) |
| have hle : (∫ V, S.indicator (fun _ => b^n) V ∂μ) ≤ haarKernelMass μ n := by |
| apply integral_mono hi hk |
| intro V |
| by_cases hV : V ∈ S |
| · rw [Set.indicator_of_mem hV] |
| exact pow_le_pow_left₀ hb hV.le n |
| · rw [Set.indicator_of_notMem hV] |
| exact pow_nonneg (haarKernelBase_nonneg 1 V) n |
| simpa only [integral_indicator_const _ hS,smul_eq_mul] using hle |
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| lemma haarKernel_superlevel_pos {N : ℕ} (μ : Measure (SpecialOrthogonal N)) |
| [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] {b : ℝ} (hb : b < 2*N) : |
| 0 < μ.real {V | b < haarKernelBase 1 V} := by |
| have hm : 0 < μ {V | b < haarKernelBase 1 V} := IsOpen.measure_pos μ |
| (isOpen_lt continuous_const (continuous_haarKernelBase 1)) |
| ⟨1,by simpa only [mem_ofPred_eq,haarKernelBase_self] using hb⟩ |
| exact ENNReal.toReal_pos hm.ne' (measure_ne_top μ _) |
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| theorem haarKernelMass_ratio_tendsto_zero {N : ℕ} (hN : 0 < N) |
| (μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] [μ.IsMulLeftInvariant] |
| (a : ℝ) (ha : 0 ≤ a) (haN : a < 2*N) : |
| Tendsto (fun n : ℕ => a^n/haarKernelMass μ n) atTop (𝓝 0) := by |
| let b := (a+2*N)/2 |
| have hab : a < b := by dsimp [b]; linarith |
| have hbN : b < 2*N := by dsimp [b]; linarith |
| have hb : 0 < b := ha.trans_lt hab |
| let m := μ.real {V | b < haarKernelBase 1 V} |
| have hm : 0 < m := haarKernel_superlevel_pos μ hbN |
| have hr : a/b < 1 := (div_lt_one hb).mpr hab |
| have hlim : Tendsto (fun n : ℕ => (a/b)^n/m) atTop (𝓝 0) := by |
| simpa using (tendsto_pow_atTop_nhds_zero_of_lt_one (div_nonneg ha hb.le) hr).div_const m |
| apply squeeze_zero (fun n => div_nonneg (pow_nonneg ha n) (haarKernelMass_pos hN μ n).le) _ hlim |
| intro n |
| have hl := haarKernelMass_lower μ b hb.le n |
| have hZ := haarKernelMass_pos hN μ n |
| have hp := pow_pos hb n |
| rw [div_pow] |
| apply (div_le_div_iff₀ hZ hm).mpr |
| rw [div_mul_eq_mul_div] |
| apply (le_div_iff₀ hp).mpr |
| calc |
| (a^n*m)*b^n = a^n*(m*b^n) := by ring |
| _ ≤ a^n*haarKernelMass μ n := mul_le_mul_of_nonneg_left hl (pow_nonneg ha n) |
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| end InvariantIsing |
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| end |
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| end OAI |
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