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f61e547 | 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 | import OAI.Probability.FreeIsing.Ising.ObservationResponseEnergy
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped NNReal ENNReal Topology
namespace Problem367.IsingObservationResponseEnergy
open IsingObservationInnovation IsingObservationFactor CriticalCanonicalRowEnergy
variable {d : ℕ} {beta : ℝ}
variable {μ : Measure (TreeVertex d → IsingSpin)} [IsProbabilityMeasure μ]
def timeEnergyConstant (d : ℕ) (beta : ℝ) : ℝ :=
1 + (d : ℝ) * Real.tanh beta ^ 2 /
ComponentSignalAttenuation.lossCoefficient (Real.tanh beta)
theorem row_bound_le_time_majorant
(hb : 0 ≤ beta) (hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
{t : ℝ} (ht : 0 < t) :
1 + (d : ℝ) * Real.tanh beta ^ 2 / rate beta t.toNNReal ≤
timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2)) := by
have hC : 0 < ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) := by
simpa [rate] using (rate_pos hb hc (by norm_num : (0 : ℝ≥0) < 1))
have hK : 0 ≤ (d : ℝ) * Real.tanh beta ^ 2 /
ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) := by positivity
have hs : 0 < Real.sqrt t := Real.sqrt_pos.mpr ht
have hp : t ^ (-(1 : ℝ) / 2) = (Real.sqrt t)⁻¹ := by
rw [show -(1 : ℝ) / 2 = -((1 : ℝ) / 2) by ring, Real.rpow_neg ht.le,
← Real.sqrt_eq_rpow]
rw [hp]
unfold rate timeEnergyConstant
rw [Real.coe_toNNReal t ht.le]
by_cases hs1 : Real.sqrt t ≤ 1
· rw [min_eq_left hs1, div_mul_eq_div_mul_one_div, one_div]
nlinarith [inv_nonneg.mpr hs.le]
· rw [min_eq_right (le_of_not_ge hs1), mul_one]
nlinarith [inv_nonneg.mpr hs.le]
theorem integrable_time_majorant (d : ℕ) (beta : ℝ) {T : ℝ} (hT : 0 ≤ T) :
Integrable (fun t : ℝ => timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2)))
(volume.restrict (Ioc 0 T)) := by
have hp : IntervalIntegrable (fun t : ℝ => t ^ (-(1 : ℝ) / 2)) volume 0 T :=
intervalIntegral.intervalIntegrable_rpow' (by norm_num)
exact ((integrable_const (1 : ℝ)).add
((intervalIntegrable_iff_integrableOn_Ioc_of_le hT).mp hp)).const_mul _
theorem integrable_response_row_time
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
(u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
Integrable (fun p : ℝ × Paths d => ∑' v, response beta u v p.1.toNNReal p.2 ^ 2)
((volume.restrict (Ioc 0 T)).prod (observationLaw μ)) := by
have hm : Measurable (fun p : ℝ × Paths d => ∑' v,
response beta u v p.1.toNNReal p.2 ^ 2) := by
apply StronglyMeasurable.measurable
apply StronglyMeasurable.tsum
intro v
exact (((measurable_response_joint beta u v).comp
(measurable_real_toNNReal.prodMap measurable_id)).pow_const 2).stronglyMeasurable
apply (integrable_prod_iff hm.aestronglyMeasurable).mpr
constructor
· filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
exact (response_row_energy hμ hb hd hc (Real.toNNReal_pos.mpr ht.1) u).2.1
· apply (integrable_time_majorant d beta hT).mono'
hm.aestronglyMeasurable.norm.integral_prod_right'
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
have hn (x : Paths d) : 0 ≤ ∑' v, response beta u v t.toNNReal x ^ 2 :=
tsum_nonneg (fun v => sq_nonneg _)
simp_rw [Real.norm_eq_abs, abs_of_nonneg (hn _)]
rw [abs_of_nonneg (integral_nonneg hn)]
exact (response_row_energy hμ hb hd hc (Real.toNNReal_pos.mpr ht.1) u).2.2.trans
(row_bound_le_time_majorant hb hc ht.1)
theorem integrable_copy_response_row_time
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
(u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) (j : Bool) :
Integrable (fun p : ℝ × (Paths d × (Paths d × Paths d)) => ∑' v,
response beta u v p.1.toNNReal (copyObservation j p.2) ^ 2)
((volume.restrict (Ioc 0 T)).prod (copies beta μ)) :=
((MeasurePreserving.id (volume.restrict (Ioc 0 T))).prod
(measurePreserving_copyObservation (beta := beta) (μ := μ) j)).integrable_comp_of_integrable
(integrable_response_row_time hμ hb hd hc u hT)
theorem integral_time_majorant (d : ℕ) (beta : ℝ) {T : ℝ} (hT : 0 ≤ T) :
(∫ t in Ioc 0 T, timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) =
timeEnergyConstant d beta * (T + 2 * Real.sqrt T) := by
have hp : IntervalIntegrable (fun t : ℝ => t ^ (-(1 : ℝ) / 2)) volume 0 T :=
intervalIntegral.intervalIntegrable_rpow' (by norm_num)
rw [integral_const_mul]
congr 1
rw [← intervalIntegral.integral_of_le hT,
intervalIntegral.integral_add intervalIntegrable_const hp,
intervalIntegral.integral_const, integral_rpow (Or.inl (by norm_num))]
norm_num
rw [← Real.sqrt_eq_rpow]
ring
theorem lintegral_response_row_time_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
(u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) :
(∫⁻ p : ℝ × Paths d, ∑' v, ENNReal.ofReal
(response beta u v p.1.toNNReal p.2 ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (observationLaw μ)) ≤
ENNReal.ofReal (timeEnergyConstant d beta * (T + 2 * Real.sqrt T)) := by
have hm : Measurable (fun p : ℝ × Paths d => ∑' v, ENNReal.ofReal
(response beta u v p.1.toNNReal p.2 ^ 2)) :=
Measurable.tsum fun v => (((measurable_response_joint beta u v).comp
(measurable_real_toNNReal.prodMap measurable_id)).pow_const 2).ennreal_ofReal
have hC : 0 < ComponentSignalAttenuation.lossCoefficient (Real.tanh beta) := by
simpa [rate] using (rate_pos hb hc (by norm_num : (0 : ℝ≥0) < 1))
have hK : 0 ≤ timeEnergyConstant d beta := by
unfold timeEnergyConstant
positivity
calc
_ = ∫⁻ t in Ioc 0 T, ∫⁻ x, ∑' v, ENNReal.ofReal
(response beta u v t.toNNReal x ^ 2) ∂observationLaw μ :=
lintegral_prod _ hm.aemeasurable
_ ≤ ∫⁻ t in Ioc 0 T,
ENNReal.ofReal (timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) := by
apply lintegral_mono_ae
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
exact (lintegral_response_row_le hμ hb hd hc (Real.toNNReal_pos.mpr ht.1) u).trans
(ENNReal.ofReal_le_ofReal (row_bound_le_time_majorant hb hc ht.1))
_ = ENNReal.ofReal (∫ t in Ioc 0 T,
timeEnergyConstant d beta * (1 + t ^ (-(1 : ℝ) / 2))) := by
symm
apply ofReal_integral_eq_lintegral_ofReal (integrable_time_majorant d beta hT)
filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht
exact mul_nonneg hK (add_nonneg zero_le_one (Real.rpow_nonneg ht.1.le _))
_ = _ := congrArg ENNReal.ofReal (integral_time_majorant d beta hT)
theorem lintegral_copy_response_row_time_le
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
(u : TreeVertex d) {T : ℝ} (hT : 0 ≤ T) (j : Bool) :
(∫⁻ p : ℝ × (Paths d × (Paths d × Paths d)), ∑' v, ENNReal.ofReal
(response beta u v p.1.toNNReal (copyObservation j p.2) ^ 2)
∂(volume.restrict (Ioc 0 T)).prod (copies beta μ)) ≤
ENNReal.ofReal (timeEnergyConstant d beta * (T + 2 * Real.sqrt T)) := by
have hp := (MeasurePreserving.id (volume.restrict (Ioc 0 T))).prod
(measurePreserving_copyObservation (beta := beta) (μ := μ) j)
have hm : Measurable (fun p : ℝ × Paths d => ∑' v, ENNReal.ofReal
(response beta u v p.1.toNNReal p.2 ^ 2)) :=
Measurable.tsum fun v => (((measurable_response_joint beta u v).comp
(measurable_real_toNNReal.prodMap measurable_id)).pow_const 2).ennreal_ofReal
exact (hp.hasLaw.lintegral_comp hm.aemeasurable).trans_le
(lintegral_response_row_time_le hμ hb hd hc u hT)
theorem tendsto_copy_response_row_time_zero
(hμ : HasFreeIsingLaw (Real.tanh beta) μ) (hb : 0 ≤ beta) (hd : 3 ≤ d)
(hc : ((d - 1 : ℕ) : ℝ) * Real.tanh beta ^ 2 = 1)
(u : TreeVertex d) (j : Bool) :
Tendsto (fun T : ℝ≥0 => ∫⁻ p : ℝ × (Paths d × (Paths d × Paths d)),
∑' v, ENNReal.ofReal (response beta u v p.1.toNNReal (copyObservation j p.2) ^ 2)
∂(volume.restrict (Ioc 0 (T : ℝ))).prod (copies beta μ)) (𝓝 0) (𝓝 0) := by
have hcont : Continuous (fun T : ℝ≥0 => ENNReal.ofReal
(timeEnergyConstant d beta * ((T : ℝ) + 2 * Real.sqrt (T : ℝ)))) :=
ENNReal.continuous_ofReal.comp
(continuous_const.mul (NNReal.continuous_coe.add
(continuous_const.mul (Real.continuous_sqrt.comp NNReal.continuous_coe))))
have hlim : Tendsto (fun T : ℝ≥0 => ENNReal.ofReal
(timeEnergyConstant d beta * ((T : ℝ) + 2 * Real.sqrt (T : ℝ)))) (𝓝 0) (𝓝 0) := by
simpa only [NNReal.coe_zero, Real.sqrt_zero, mul_zero, add_zero, ENNReal.ofReal_zero]
using hcont.tendsto 0
exact tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hlim
(Eventually.of_forall fun _ => bot_le)
(Eventually.of_forall fun T => lintegral_copy_response_row_time_le hμ hb hd hc u T.property j)
end Problem367.IsingObservationResponseEnergy
end
end OAI
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