File size: 2,740 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 | import OAI.Probability.InvariantIsing.Pressure.RandomOrbitTail
import OAI.Probability.InvariantIsing.Pressure.RandomSpectralBound
import OAI.Probability.InvariantIsing.Pressure.LocalizedAlmostSure
namespace OAI
/-! Almost-sure conditional Haar concentration for random spectra and fields. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology ENNReal
namespace InvariantIsing
theorem ae_random_orbit_pressure_sub_mean (hhaar : HaarConcentrationInput)
{Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P]
(d : (N : ℕ) → Ω → FieldSpectralData N) (Y : ℕ → Ω → ℝ)
(hd : ∀ N, Measurable (d N)) (hY : ∀ N, Measurable (Y N))
(H : (N : ℕ) → Measure (Orthogonal N)) [∀ N, IsProbabilityMeasure (H N)]
[∀ N, (H N).IsMulRightInvariant]
(hlaw : ∀ N, ConditionalFieldOrbitLaw P (d N) (Y N) (H N))
(hbound : ∀ᵐ ω ∂P, ∃ K : ℝ, ∀ᶠ k in atTop, spectralRadius ((d (k+1) ω).1) ≤ K) :
∀ᵐ ω ∂P, Tendsto (fun k => Y (k+1) ω-
∫ U, dataPhysicalPressure (d (k+1) ω) U ∂H (k+1)) atTop (𝓝 0) := by
apply ae_tendsto_zero_of_localized_exponential_tail P
(fun k ω => Y (k+1) ω-∫ U, dataPhysicalPressure (d (k+1) ω) U ∂H (k+1))
(fun k ω => spectralRadius ((d (k+1) ω).1)) _ hbound
intro K ε hK hε
obtain ⟨C,a,hC,ha,ht⟩ := uniform_physical_pressure_mean_tail hhaar
obtain ⟨N₀,hN₀⟩ := ht K ε hK hε
refine ⟨C,a*(ε/2)^2/K^2,hC.le,by positivity,max N₀ 3,?_⟩
intro n hn
have hb := conditional_orbit_tail_bound (Nat.succ_pos n) P (d (n+1)) (Y (n+1))
(hd (n+1)) (hY (n+1)) (H (n+1)) (hlaw (n+1)) K ε
(C*Real.exp (-a*(n+1 : ℝ)*(ε/2)^2/K^2)) (fun data hdata => by
simpa only [dataPhysicalPressure,Nat.cast_add,Nat.cast_one] using
hN₀ (n+1) (by omega) (by omega) (H (n+1)) inferInstance inferInstance
data.1 data.2 hdata)
have he : {ω | (∀ i, |(d (n+1) ω).1 i| ≤ K) ∧
ε ≤ |Y (n+1) ω-∫ U, dataPhysicalPressure (d (n+1) ω) U ∂H (n+1)|}=
{ω | ε ≤ |Y (n+1) ω-∫ U, dataPhysicalPressure (d (n+1) ω) U ∂H (n+1)| ∧
spectralRadius ((d (n+1) ω).1) ≤ K} := by
ext ω
simp only [mem_ofPred_eq,spectralRadius_le_iff,and_comm]
rw [he] at hb
have hreal := ENNReal.toReal_mono (ENNReal.ofReal_ne_top) hb
rw [ENNReal.toReal_ofReal (by positivity)] at hreal
refine hreal.trans ?_
apply mul_le_mul_of_nonneg_left _ hC.le
apply Real.exp_le_exp.mpr
have hc : 0 ≤ a*(ε/2)^2/K^2 := by positivity
calc
-a*(n+1 : ℝ)*(ε/2)^2/K^2= -(a*(ε/2)^2/K^2)*((n : ℝ)+1) := by ring
_ ≤ -(a*(ε/2)^2/K^2)*(n : ℝ) := by nlinarith
end InvariantIsing
end
end OAI
|