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7d9f027 | 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 | import Mathlib
import OAI.Probability.SKGap.Localization.ConditionalWellDefined
import OAI.Probability.SKGap.Localization.ConditionalHessianLaw
import OAI.Probability.SKGap.Gaussian.GaussianTiltMass
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real Set MeasureTheory ProbabilityTheory
open scoped BigOperators Matrix.Norms.Frobenius ENNReal
theorem conditional_field_stability {j A R t0 T : ℝ}
(hj : 0 < j) (hj1 : j < 1) (hA : 1 ≤ A) (hsub : sqrt j*A < 1)
(hR : 0 ≤ R) (ht0 : 0 < t0) :
∃ m ν σ0 : ℝ, 0 < m ∧ 0 < ν ∧ 0 < σ0 ∧
∃ U : Set (ScalarTimeDomain R t0 T), IsOpen U ∧ scalarEqualitySet j R t0 T ⊆ U ∧
∀ R1 : ℝ, 1 ≤ R1 → ∃ ε c : ℝ, ∃ N : ℕ, 0 < ε ∧ 0 < c ∧ 0 < N ∧
∀ (n : ℕ) [NeZero n], N ≤ n → ∀ x ∈ U, ∀ y : Fin n → ℝ,
empiricalLaw y=(x.1 : ProbabilityMeasure ℝ) →
(∑ i,if R1 < |y i| then y i^2 else 0) ≤ ν*(n:ℝ) →
∀ σ : ℝ, |σ| ≤ σ0 →
(conditionalGoeLaw (j/(n:ℝ)) ((x.2:ℝ)+σ^2) (magnetization y)
(empiricalObservation j x.2 σ y)).real
{W | ((fun i k=>W (i,k)),y) ∈ badFieldSet n j A ε m} ≤ exp (-c*(n:ℝ)) := by
obtain ⟨m,ν,σ0,hm,hν,hσ0,U,hU,hKU,hbound⟩ :=
conditional_empirical_simultaneous (T := T) hj hj1 hA hsub hR ht0
let V : Set (ScalarTimeDomain R t0 T) := {x | 0 < scalarQMoment (x.1:ProbabilityMeasure ℝ)}
have hV : IsOpen V := isOpen_lt continuous_const
(continuous_scalarQMoment.comp (continuous_subtype_val.comp continuous_fst))
have hKV : scalarEqualitySet j R t0 T ⊆ V := by
intro x hx
change 0 < scalarQMoment (x.1:ProbabilityMeasure ℝ)
rw [hx,scalarQMoment_curve hj hj1 (ht0.le.trans x.2.2.1)]
exact (scalar_fixed_pos hj hj1 (ht0.trans_le x.2.2.1)).2
refine ⟨m/2,ν,σ0,half_pos hm,hν,hσ0,U∩V,hU.inter hV,fun x hx=>⟨hKU hx,hKV hx⟩,?_⟩
intro R1 hR1
obtain ⟨ε,c,N,hε,hc,hN,hb⟩ := hbound R1 hR1
obtain ⟨M,hM⟩ := absorb_finite_exponential_factor 103 (half_pos hc)
refine ⟨ε,c/2,max N M,hε,half_pos hc,lt_of_lt_of_le hN (le_max_left _ _),?_⟩
intro n _ hn x hx y hP hy σ hσ
have hq : 0 < scalarQMoment (empiricalLaw y) := by rw [hP]; exact hx.2
apply (conditional_bad_le_freshbad hj.le (ht0.trans_le x.2.2.1) (by linarith : m/2 < m) y hq).trans
apply (hb n ((le_max_left N M).trans hn) x hx.1 y hP hy σ hσ).trans
simpa only [Nat.cast_ofNat,show (n:ℝ)*(-c)= -c*(n:ℝ) from by ring,
show (n:ℝ)*(-c+c/2)= -(c/2)*(n:ℝ) from by ring] using
hM n ((le_max_right N M).trans hn) (-c)
end SKGap
end
end
end OAI
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