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135052a | 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 | import Mathlib
import OAI.Probability.SKBarriers.SpinGlass.LogPartition
namespace OAI
section
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Filter Set
namespace SK.Analytic
theorem rotation_logPartition_tail {n : ℕ} (hn : 0 < n) (β θ u : ℝ) (hu : 0 ≤ u) :
(disorderPairLaw n).real {p | u ≤ logPartition β (rotateDisorder θ p).1-
logPartition β p.1} ≤ Real.exp (-u^2/(θ^2*β^2*(n:ℝ))) := by
let v : NNReal := ⟨θ^2*β^2*(n:ℝ)/2,by positivity⟩
have HS : HasSubgaussianMGF (fun p : Disorder n × Disorder n =>
logPartition β (rotateDisorder θ p).1-logPartition β p.1) v (disorderPairLaw n) := by
refine ⟨rotation_exp_integrable hn β θ,fun t => ?_⟩
apply (rotation_exp_integral_le hn β θ t).trans
apply Real.exp_le_exp.mpr
change t^2*θ^2*β^2*((n:ℝ)-1)/4 ≤ (θ^2*β^2*(n:ℝ)/2)*t^2/2
have H : 0 ≤ t^2*θ^2*β^2 := by positivity
nlinarith
convert HS.measure_ge_le hu using 1
congr 2
change θ^2*β^2*(n:ℝ) = 2*(θ^2*β^2*(n:ℝ)/2)
ring
theorem product_event_lower_of_sections {X Y : Type*} [MeasurableSpace X] [MeasurableSpace Y]
(μ : Measure X) (ν : Measure Y) [IsFiniteMeasure μ] [IsFiniteMeasure ν]
{A : Set X} {E : Set (X × Y)} (hA : MeasurableSet A) (hE : MeasurableSet E)
{c : ℝ} (hc : 0 ≤ c) (hsec : ∀ x ∈ A, c ≤ ν.real {y | (x,y) ∈ E}) :
c*μ.real A ≤ (μ.prod ν).real E := by
have H : ENNReal.ofReal c*μ A ≤ (μ.prod ν) E := by
rw [Measure.prod_apply hE,← lintegral_indicator_const hA (ENNReal.ofReal c)]
apply lintegral_mono
intro x
by_cases hx : x ∈ A
· rw [Set.indicator_of_mem hx]
exact (ENNReal.ofReal_le_iff_le_toReal (measure_ne_top ν _)).mpr (hsec x hx)
· rw [Set.indicator_of_notMem hx]
exact bot_le
have HT := ENNReal.toReal_mono (measure_ne_top (μ.prod ν) E) H
simpa only [ENNReal.toReal_mul,ENNReal.toReal_ofReal hc,measureReal_def] using HT
end SK.Analytic
end
end
end
end OAI
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