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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 | import OAI.Probability.FreeIsing.Trees.CutContinuations
import OAI.Probability.FreeIsing.Trees.CutLevelConcentration
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367.CutContinuationChannels
open RootedBroadcast
variable {d : ℕ} {A : Type*} [MeasurableSpace A]
abbrev ConeSample (A : Type*) (v : TreeVertex d) :=
(DescendantVertex v → IsingSpin) × (DescendantVertex v → A)
def globalCone (v : TreeVertex d)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) : ConeSample A v :=
((fun u => z.1 u.1), fun u => z.2 u.1)
def restrictCone {u v : TreeVertex d} (h : u.1 <+: v.1)
(z : ConeSample A u) : ConeSample A v :=
((fun w => z.1 ⟨w.1, h.trans w.2⟩), fun w => z.2 ⟨w.1, h.trans w.2⟩)
theorem measurable_globalCone (v : TreeVertex d) : Measurable (globalCone (A := A) v) := by
unfold globalCone
fun_prop
theorem measurable_restrictCone {u v : TreeVertex d} (h : u.1 <+: v.1) :
Measurable (restrictCone (A := A) h) := by
unfold restrictCone
fun_prop
def globalFactors (F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) (v : TreeVertex d) : ℝ :=
F v (globalCone v z)
def continuationChannel (F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
{k L : ℕ} (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L)
(z : ConeSample A p.1) : ℝ :=
∏ j ∈ Finset.range L,
F (rootPathVertex (appendCutContinuation p w).1 (k + 1 + (j + 1)))
(restrictCone (cutContinuation_vertex_descendant p w j) z)
theorem measurable_continuationChannel
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ) (hF : ∀ v, Measurable (F v))
{k L : ℕ} (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L) :
Measurable (continuationChannel F p w) := by
apply Finset.measurable_fun_prod
intro j hj
exact (hF _).comp (measurable_restrictCone _)
omit [MeasurableSpace A] in
theorem continuationChannel_mem_Icc
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
(hb : ∀ v z, 0 ≤ F v z ∧ F v z ≤ 1)
{k L : ℕ} (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L)
(z : ConeSample A p.1) : continuationChannel F p w z ∈ Set.Icc (0 : ℝ) 1 :=
⟨Finset.prod_nonneg (fun _ _ => (hb _ _).1),
Finset.prod_le_one₀ (fun _ _ => (hb _ _).1) (fun _ _ => (hb _ _).2)⟩
omit [MeasurableSpace A] in
@[simp] theorem continuationChannel_globalCone
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
{k L : ℕ} (p : TreeShell d (k + 1)) (w : ReducedTail (cutLastColor p) L)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) :
continuationChannel F p w (globalCone p.1 z) =
cutContinuationProduct (globalFactors F z) p w := rfl
def averageChannel (F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
{k : ℕ} (p : TreeShell d (k + 1)) (L : ℕ) (z : ConeSample A p.1) : ℝ :=
(∑ w : ReducedTail (cutLastColor p) L, continuationChannel F p w z) /
(Fintype.card (ReducedTail (cutLastColor p) L) : ℝ)
theorem measurable_averageChannel
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ) (hF : ∀ v, Measurable (F v))
{k : ℕ} (p : TreeShell d (k + 1)) (L : ℕ) : Measurable (averageChannel F p L) :=
(Finset.measurable_fun_sum _ (fun w _ => measurable_continuationChannel F hF p w)).div_const _
omit [MeasurableSpace A] in
theorem averageChannel_mem_Icc (hd : 2 ≤ d)
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
(hb : ∀ v z, 0 ≤ F v z ∧ F v z ≤ 1)
{k : ℕ} (p : TreeShell d (k + 1)) (L : ℕ) (z : ConeSample A p.1) :
averageChannel F p L z ∈ Set.Icc (0 : ℝ) 1 := by
have hc : (0 : ℝ) < Fintype.card (ReducedTail (cutLastColor p) L) := by
rw [card_reducedTail]
exact_mod_cast pow_pos (by omega : 0 < d - 1) L
constructor
· exact div_nonneg (Finset.sum_nonneg
(fun w _ => (continuationChannel_mem_Icc F hb p w z).1)) hc.le
· apply (div_le_iff₀ hc).mpr
simpa using Finset.sum_le_sum
(fun w (_ : w ∈ (Finset.univ : Finset (ReducedTail (cutLastColor p) L))) =>
(continuationChannel_mem_Icc F hb p w z).2)
omit [MeasurableSpace A] in
@[simp] theorem averageChannel_globalCone
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
{k : ℕ} (p : TreeShell d (k + 1)) (L : ℕ)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) :
averageChannel F p L (globalCone p.1 z) =
(∑ w : ReducedTail (cutLastColor p) L,
cutContinuationProduct (globalFactors F z) p w) / (((d - 1) ^ L : ℕ) : ℝ) := by
simp [averageChannel]
omit [MeasurableSpace A] in
theorem path_average_le_averageChannel
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ)
(hb : ∀ v z, 0 ≤ F v z ∧ F v z ≤ 1)
{k : ℕ} (I : Finset (TreeShell d (k + 1))) (L : ℕ)
(z : (TreeVertex d → IsingSpin) × (TreeVertex d → A)) :
(∑ v ∈ cutDescendants I L, rootPathProduct (globalFactors F z) v.1) /
((cutDescendants I L).card : ℝ) ≤
(∑ p ∈ I, averageChannel F p L (globalCone p.1 z)) / (I.card : ℝ) := by
simpa only [averageChannel_globalCone] using
average_rootPathProduct_le_continuation_average (globalFactors F z)
(fun v => hb v _) I L
theorem integral_averageChannel_le (hd : 2 ≤ d)
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ) (hF : ∀ v, Measurable (F v))
(hb : ∀ v z, 0 ≤ F v z ∧ F v z ≤ 1)
{k : ℕ} (p : TreeShell d (k + 1)) (L : ℕ)
(P : Measure ((TreeVertex d → IsingSpin) × (TreeVertex d → A)))
[IsProbabilityMeasure P] {m : ℝ}
(hmean : ∀ w : ReducedTail (cutLastColor p) L,
(∫ z, cutContinuationProduct (globalFactors F z) p w ∂P) ≤ m) :
(∫ z, averageChannel F p L (globalCone p.1 z) ∂P) ≤ m := by
have hc : (0 : ℝ) < Fintype.card (ReducedTail (cutLastColor p) L) := by
rw [card_reducedTail]
exact_mod_cast pow_pos (by omega : 0 < d - 1) L
have hi (w : ReducedTail (cutLastColor p) L) :
Integrable (fun z => continuationChannel F p w (globalCone p.1 z)) P := by
apply Integrable.of_bound
((measurable_continuationChannel F hF p w).comp (measurable_globalCone p.1)).aestronglyMeasurable 1
apply Filter.Eventually.of_forall
intro z
change ‖continuationChannel F p w (globalCone p.1 z)‖ ≤ 1
rw [Real.norm_eq_abs, abs_of_nonneg (continuationChannel_mem_Icc F hb p w _).1]
exact (continuationChannel_mem_Icc F hb p w _).2
calc
_ = (∑ w : ReducedTail (cutLastColor p) L,
∫ z, continuationChannel F p w (globalCone p.1 z) ∂P) /
(Fintype.card (ReducedTail (cutLastColor p) L) : ℝ) := by
unfold averageChannel
rw [integral_div, integral_finsetSum _ (fun w _ => hi w)]
_ ≤ (∑ _w : ReducedTail (cutLastColor p) L, m) /
(Fintype.card (ReducedTail (cutLastColor p) L) : ℝ) := by
apply div_le_div_of_nonneg_right _ hc.le
exact Finset.sum_le_sum (fun w _ => hmean w)
_ = m := by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
exact mul_div_cancel_left₀ m hc.ne'
theorem measure_path_average_ge_le [StandardBorelSpace A]
{theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (hd : 2 ≤ d)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ) (hF : ∀ v, Measurable (F v))
(hb : ∀ v z, 0 ≤ F v z ∧ F v z ≤ 1)
{k : ℕ} (I : Finset (TreeShell d (k + 1))) (hI : I.Nonempty) (L : ℕ)
{τ : ℝ} (hτ : 0 ≤ τ)
(hmean : ∀ p ∈ I, ∀ w : ReducedTail (cutLastColor p) L,
(∫ z, cutContinuationProduct (globalFactors F z) p w
∂μ.prod (Measure.infinitePi ν)) ≤ τ / 4) :
(μ.prod (Measure.infinitePi ν))
{z | τ ≤ (∑ v ∈ cutDescendants I L, rootPathProduct (globalFactors F z) v.1) /
((cutDescendants I L).card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(I.card : ℝ) * τ ^ 2 / 2)) := by
calc
_ ≤ (μ.prod (Measure.infinitePi ν))
{z | τ ≤ (∑ p ∈ I, averageChannel F p L (globalCone p.1 z)) / (I.card : ℝ)} := by
apply measure_mono
intro z hz
exact hz.trans (path_average_le_averageChannel F hb I L z)
_ ≤ _ := hμ.measure_cutLevel_average_ge_le h0 h1 (k + 1) ν
(fun p => averageChannel F p L) (fun p => measurable_averageChannel F hF p L)
(fun p z => averageChannel_mem_Icc hd F hb p L z) I hI hτ
(fun p hp => integral_averageChannel_le hd F hF hb p L _ (hmean p hp))
theorem measure_restricted_path_average_ge_le_double_exp [StandardBorelSpace A]
{theta : ℝ} {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
(hμ : HasFreeIsingLaw theta μ) (h0 : -1 ≤ theta) (h1 : theta ≤ 1) (hd : 2 ≤ d)
(ν : TreeVertex d → Measure A) [∀ v, IsProbabilityMeasure (ν v)]
(F : ∀ v : TreeVertex d, ConeSample A v → ℝ) (hF : ∀ v, Measurable (F v))
(hb : ∀ v z, 0 ≤ F v z ∧ F v z ≤ 1)
(forbidden : Finset (Fin d)) (hforbidden : forbidden.card ≤ 1) (k L : ℕ)
{q ℓ : ℝ} (hℓ : 0 ≤ ℓ) (hq : q ≤ Real.log ((d : ℝ) - 1) / 8)
(hlevel : ℓ / 2 - 1 ≤ (k + 1 : ℕ))
(hmean : ∀ p ∈ avoidingTreeCut forbidden k, ∀ w : ReducedTail (cutLastColor p) L,
(∫ z, cutContinuationProduct (globalFactors F z) p w
∂μ.prod (Measure.infinitePi ν)) ≤ Real.exp (-q * ℓ) / 4) :
(μ.prod (Measure.infinitePi ν))
{z | Real.exp (-q * ℓ) ≤
(∑ v ∈ cutDescendants (avoidingTreeCut forbidden k) L,
rootPathProduct (globalFactors F z) v.1) /
((cutDescendants (avoidingTreeCut forbidden k) L).card : ℝ)} ≤
ENNReal.ofReal (Real.exp (-(((d : ℝ) - 1) ^ (ℓ / 4)) / (2 * ((d : ℝ) - 1)))) := by
have hdR : (2 : ℝ) ≤ d := by exact_mod_cast hd
exact (measure_path_average_ge_le hμ h0 h1 hd ν F hF hb
(avoidingTreeCut forbidden k) (avoidingTreeCut_nonempty hd forbidden hforbidden k)
L (Real.exp_pos _).le hmean).trans
(ENNReal.ofReal_le_ofReal (BoundedAverageConcentration.exponential_card_bound
(by linarith) hℓ hq (rpow_le_card_avoidingTreeCut hd forbidden hforbidden k hlevel)))
end Problem367.CutContinuationChannels
end
end OAI
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