File size: 2,985 Bytes
a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 | import OAI.Probability.InvariantIsing.Haar.HaarKernelConvolution
namespace OAI
/-! Quantitative concentration of the positive polynomial kernels near the identity. -/
noncomputable section
open Matrix MvPolynomial MeasureTheory Set Filter
open scoped Topology
namespace InvariantIsing
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
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 _)
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)
end InvariantIsing
end
end OAI
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