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b317718 | 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 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 | import OAI.Probability.InvariantIsing.Arrays.NSpinCascade
import OAI.Probability.InvariantIsing.Fields.SpinFactorization
import OAI.Probability.InvariantIsing.Fields.CascadeTilt
import OAI.Probability.InvariantIsing.Fields.FieldCascade
namespace OAI
/-! Site factorization through the finite Gaussian backward recursion. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
theorem vectorCascade_logCosh_sum (N n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0)
(hb : ∀ i < n, 0 < b i) (c z : Fin N → ℝ) :
IsingPerceptron.cascadeRecursion n b (fun i => vectorGaussianLaw N (v i))
(fun _ p => p.1 + p.2) (fun y => ∑ i, Real.log (Real.cosh (c i + y i))) z =
∑ i, IsingPerceptron.cascadeRecursion n b (fun j => gaussianCascadeMarks (v := v) j)
(fun _ p => p.1 + p.2) (fun y => Real.log (Real.cosh y)) (c i + z i) := by
induction n generalizing b v z with
| zero => rfl
| succ n ih =>
let bs := fun j => b (j + 1)
let vs := fun j => v (j + 1)
let F := IsingPerceptron.cascadeRecursion n bs (fun j => gaussianCascadeMarks vs j)
(fun _ p => p.1 + p.2) (fun y => Real.log (Real.cosh y))
have hbs : ∀ j < n, 0 < bs j := fun j hj => hb (j + 1) (by omega)
have hm : Measurable F := IsingPerceptron.measurable_cascadeRecursion n bs
(fun j => gaussianCascadeMarks vs j) (fun _ => measurable_fst.add measurable_snd)
IsingPerceptron.measurable_logCosh
have hg : IsingPerceptron.HasLinearGrowth F :=
IsingPerceptron.cascadeRecursion_linearGrowth n bs (fun j => gaussianCascadeMarks vs j)
(fun j _ => IsingPerceptron.gaussianReal_exponentialNormMoments 0 (vs j))
IsingPerceptron.measurable_logCosh IsingPerceptron.logCosh_linearGrowth hbs
have he : (fun g : Fin N → ℝ => IsingPerceptron.cascadeRecursion n bs
(fun j => vectorGaussianLaw N (vs j)) (fun _ p => p.1 + p.2)
(fun y => ∑ i, Real.log (Real.cosh (c i + y i))) (z + g)) =
(fun g => ∑ i, F ((c i + z i) + g i)) := by
funext g
rw [ih bs vs hbs]
simp only [F, Pi.add_apply, add_assoc]
change IsingPerceptron.logMean (b 0) (vectorGaussianLaw N (v 0) : Measure (Fin N → ℝ))
(fun g => IsingPerceptron.cascadeRecursion n bs
(fun j => vectorGaussianLaw N (vs j)) (fun _ p => p.1 + p.2)
(fun y => ∑ i, Real.log (Real.cosh (c i + y i))) (z + g)) =
∑ i, IsingPerceptron.logMean (b 0) (gaussianReal 0 (v 0))
(fun g => F ((c i + z i) + g))
rw [he]
exact IsingPerceptron.logMean_independent_sum (fun _ : Fin N => gaussianReal 0 (v 0))
(fun i g => F ((c i + z i) + g)) (hb 0 (by omega)).ne'
(fun _ => hm.comp (measurable_const.add measurable_id))
(fun i => IsingPerceptron.integrable_exp_of_linearGrowth _
(IsingPerceptron.gaussianReal_exponentialNormMoments 0 (v 0))
(hm.comp (measurable_const.add measurable_id)) (hg.add_left (c i + z i)) (b 0))
theorem rotatedCascadeValue_zero_interaction {N : ℕ} (n : ℕ) (b : ℕ → ℝ)
(v : ℕ → ℝ≥0) (hb : IsingPerceptron.CascadeExponents n b)
(U : Rotation N) (c z : Fin N → ℝ) :
rotatedCascadeValue n b v (fun _ => 0) U c z =
∑ i, ((List.ofFn (fun j : Fin n => (b j, (v j : ℝ)))).foldr
(fun av f => gaussianOperator av.1 av.2 f) (fun y => Real.log (Real.cosh y)))
(c i + z i) := by
have ht : rotatedFieldTerminal (fun _ : Fin N => 0) U c =
(fun y => ∑ i, Real.log (Real.cosh (c i + y i))) := by
funext y
simp only [rotatedFieldTerminal, rotatedEnergy_constant, zero_mul, zero_div, zero_add]
simpa only [Pi.add_apply] using logPartition_fieldEnergy (c + y)
unfold rotatedCascadeValue
rw [ht, vectorCascade_logCosh_sum N n b v (fun i hi => (hb.1 i hi).1)]
apply Finset.sum_congr rfl
intro i _
exact congrFun (gaussianCascadeRecursion_eq_fold n b v
(fun j hj => (hb.1 j hj).1.ne') IsingPerceptron.measurable_logCosh) (c i + z i)
lemma integrable_gaussianReal_of_linearGrowth (v : ℝ≥0) {F : ℝ → ℝ}
(hF : Measurable F) (hg : IsingPerceptron.HasLinearGrowth F) :
Integrable F (gaussianReal 0 v) := by
obtain ⟨C, L, _, hL, hbound⟩ := hg
have hi := (integrable_const C).add
((IsingPerceptron.gaussianReal_exponentialNormMoments 0 v 1).const_mul L)
apply hi.mono' hF.aestronglyMeasurable
exact ae_of_all _ fun z => by
change |F z| ≤ C + L * Real.exp (1 * ‖z‖)
rw [one_mul]
exact (hbound z).trans (add_le_add le_rfl (mul_le_mul_of_nonneg_left
((le_add_of_nonneg_right zero_le_one).trans (Real.add_one_le_exp ‖z‖)) hL))
theorem rotatedEnrichedPressure_zero_interaction {N : ℕ} (n : ℕ) (b : ℕ → ℝ)
(v : ℕ → ℝ≥0) (root : ℝ≥0) (hb : IsingPerceptron.CascadeExponents n b)
(U : Rotation N) (c : Fin N → ℝ) :
rotatedEnrichedPressure n b v root (fun _ => 0) U c =
(N : ℝ)⁻¹ * ∑ i, gaussianOperator 0 root
((List.ofFn (fun j : Fin n => (b j, (v j : ℝ)))).foldr
(fun av f => gaussianOperator av.1 av.2 f) (fun y => Real.log (Real.cosh y))) (c i) := by
let F := IsingPerceptron.cascadeRecursion n b (fun j => gaussianCascadeMarks v j)
(fun _ p => p.1 + p.2) (fun y => Real.log (Real.cosh y))
have hm : Measurable F := IsingPerceptron.measurable_cascadeRecursion n b
(fun j => gaussianCascadeMarks v j) (fun _ => measurable_fst.add measurable_snd)
IsingPerceptron.measurable_logCosh
have hg : IsingPerceptron.HasLinearGrowth F :=
IsingPerceptron.cascadeRecursion_linearGrowth n b (fun j => gaussianCascadeMarks v j)
(fun j _ => IsingPerceptron.gaussianReal_exponentialNormMoments 0 (v j))
IsingPerceptron.measurable_logCosh IsingPerceptron.logCosh_linearGrowth
(fun j hj => (hb.1 j hj).1)
have hfi (i : Fin N) : Integrable (fun g => F (c i + g)) (gaussianReal 0 root) :=
integrable_gaussianReal_of_linearGrowth root (hm.comp (measurable_const.add measurable_id))
(hg.add_left (c i))
have hF : F = (List.ofFn (fun j : Fin n => (b j, (v j : ℝ)))).foldr
(fun av f => gaussianOperator av.1 av.2 f) (fun y => Real.log (Real.cosh y)) :=
gaussianCascadeRecursion_eq_fold n b v (fun j hj => (hb.1 j hj).1.ne')
IsingPerceptron.measurable_logCosh
unfold rotatedEnrichedPressure
congr 1
calc
_ = ∫ z : Fin N → ℝ, (∑ i, F (c i + z i))
∂(vectorGaussianLaw N root : Measure (Fin N → ℝ)) := by
apply integral_congr_ae
exact ae_of_all _ fun z => by
rw [rotatedCascadeValue_zero_interaction n b v hb U c z]
simp only [hF]
_ = ∑ i, ∫ z : Fin N → ℝ, F (c i + z i)
∂(vectorGaussianLaw N root : Measure (Fin N → ℝ)) :=
integral_finsetSum _ (fun i _ =>
(measurePreserving_eval (fun _ : Fin N => gaussianReal 0 root) i).integrable_comp
(hfi i).aestronglyMeasurable |>.mpr (hfi i))
_ = ∑ i, ∫ g, F (c i + g) ∂gaussianReal 0 root := by
apply Finset.sum_congr rfl
intro i _
exact (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 root) i).hasLaw.integral_comp
((hm.comp (measurable_const.add measurable_id)).aestronglyMeasurable)
_ = _ := by
apply Finset.sum_congr rfl
intro i _
have hs : Measurable (fun g : ℝ => F (c i + g)) :=
hm.comp (measurable_const.add measurable_id)
rw [integral_gaussianReal_zero_eq_standard root hs, hF]
simp [gaussianOperator]
/-- The exact zero-time normalization in `up:zero-time`, before perturbations. -/
theorem rotatedEnrichedPressure_field_zero_time {N : ℕ} (hN : 0 < N)
(h : FieldStep) (U : Rotation N) :
rotatedEnrichedPressure h.depth (IsingPerceptron.chainExponent h.cut) (fieldCascadeVariance h)
(NNReal.mk (h.height 0) (h.nonneg 0)) (fun _ => 0) U (fun _ => 0) -
h.height (Fin.last h.depth) / 2 = fieldValue h 0 := by
rw [rotatedEnrichedPressure_zero_interaction h.depth (IsingPerceptron.chainExponent h.cut)
(fieldCascadeVariance h) _
(IsingPerceptron.chainExponent_admissible h.ordered_cut h.first h.last) U (fun _ => 0)]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
have hn : (N : ℝ) ≠ 0 := by exact_mod_cast Nat.ne_of_gt hN
rw [← mul_assoc, inv_mul_cancel₀ hn, one_mul, fieldCascadeFold_eq_magneticRecursion]
simp only [gaussianOperator, zero_add, NNReal.coe_mk]
rw [ite_true]
rw [fieldValue_eq_magneticFieldValue]
unfold IsingPerceptron.magneticFieldValue
have hroot : IsingPerceptron.pathAmplitude (heightSequence h) 0 = Real.sqrt (h.height 0) := by
simp [IsingPerceptron.pathAmplitude, IsingPerceptron.pathIncrement, heightSequence]
have hlast : heightSequence h h.depth = h.height (Fin.last h.depth) :=
heightSequence_eq h _ le_rfl
rw [hroot, hlast]
/-- The unperturbed contact objective cannot be negative at zero interaction. -/
theorem zero_time_contact_nonneg {N : ℕ} (hN : 0 < N) (h : FieldStep)
(U : Rotation N) (p : OverlapPath) {S : ℝ} (hS : entropyFunctional p ≤ (S : EReal)) :
0 ≤ -(rotatedEnrichedPressure h.depth (IsingPerceptron.chainExponent h.cut)
(fieldCascadeVariance h) (NNReal.mk (h.height 0) (h.nonneg 0))
(fun _ => 0) U (fun _ => 0) - h.height (Fin.last h.depth) / 2) -
fieldPairing p h / 2 + S := by
rw [rotatedEnrichedPressure_field_zero_time hN h U]
have he : ((fieldValue h 0 + fieldPairing p h / 2 : ℝ) : EReal) ≤ (S : EReal) :=
(le_iSup (fun g : FieldStep =>
((fieldValue g 0 + fieldPairing p g / 2 : ℝ) : EReal)) h).trans hS
have hr : fieldValue h 0 + fieldPairing p h / 2 ≤ S := by exact_mod_cast he
linarith
end InvariantIsing
end
end OAI
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