File size: 2,886 Bytes
c4cdbbd | 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 68 69 70 71 72 | import OAI.Probability.InvariantIsing.Fields.FieldSpinTransition
namespace OAI
/-! Even bounded tests of the tilted scalar position chain. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set
open scoped NNReal
namespace InvariantIsing
lemma fieldSpinTransition_even (ζ : ℝ) (v : ℝ≥0) {F a : ℝ → ℝ}
(hF : Measurable F) (hFe : Function.Even F)
(ha : Measurable a) (hae : Function.Even a) :
Function.Even (fieldSpinTransition ζ v F a) := by
intro z
rw [fieldSpinTransition, fieldSpinTransition, integral_tilted_eq_div, integral_tilted_eq_div]
have he : Function.Even (fun u => Real.exp (ζ * F u) * a u) := by
intro u
dsimp only
rw [hFe u, hae u]
have hm : Measurable (fun u => Real.exp (ζ * F u) * a u) :=
(hF.const_mul ζ).exp.mul ha
rw [field_gaussian_integral_reflect v z hm he,
field_gaussian_integral_reflect v z (hF.const_mul ζ).exp
(fun u => by rw [hFe u])]
lemma fieldSpinTransition_even_monotone (ζ : ℝ) (v : ℝ≥0) {F a : ℝ → ℝ}
(hF : Measurable F) (hFe : Function.Even F) (hFg : HasLinearGrowth F)
(ha : Measurable a) (hae : Function.Even a) (ham : MonotoneOn a (Ici 0))
{K : ℝ} (haB : ∀ u, |a u| ≤ K) :
MonotoneOn (fieldSpinTransition ζ v F a) (Ici 0) := by
by_cases hv : v = 0
· subst v
intro x hx y hy hxy
rw [fieldSpinTransition_zero_variance, fieldSpinTransition_zero_variance]
exact ham hx hy hxy
intro x hx y _hy hxy
have h := field_gaussian_tilt_abs_parent_order v hv ζ F a hF hFe hFg
ha ham haB hx hxy
simpa only [fieldSpinTransition, field_even_abs hae] using h
lemma fieldSpinTransition_even_order (v : ℝ≥0) {ζ : ℝ} (hζ : 0 ≤ ζ)
{F G a b : ℝ → ℝ} (hF : Measurable F) (hG : Measurable G)
(hFe : Function.Even F) (hGe : Function.Even G)
(hFg : HasLinearGrowth F) (hGg : HasLinearGrowth G)
(hFG : MonotoneOn (fun u => G u - F u) (Ici 0))
(ha : Measurable a) (hb : Measurable b)
(hae : Function.Even a) (hbe : Function.Even b)
(ham : MonotoneOn a (Ici 0)) {K : ℝ}
(haB : ∀ u, |a u| ≤ K) (hbB : ∀ u, |b u| ≤ K)
(hab : ∀ u ∈ Ici (0 : ℝ), a u ≤ b u) (z : ℝ) :
fieldSpinTransition ζ v F a z ≤ fieldSpinTransition ζ v G b z := by
have hab' (u : ℝ) : a u ≤ b u := by
rw [← field_even_abs hae u, ← field_even_abs hbe u]
exact hab |u| (abs_nonneg u)
by_cases hv : v = 0
· subst v
simpa only [fieldSpinTransition_zero_variance] using hab' z
have h := field_gaussian_tilt_abs_payoff_order v hv hζ F G a hF hG hFe hGe
hFg hGg hFG ha ham haB z
have h' : fieldSpinTransition ζ v F a z ≤ fieldSpinTransition ζ v G a z := by
simpa only [fieldSpinTransition, field_even_abs hae] using h
exact h'.trans (fieldSpinTransition_mono_test ζ v G ha hb haB hbB hab' z)
end InvariantIsing
end
end OAI
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