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f205175 | 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 | import OAI.Probability.DilutedSpin.CompoundRate
import OAI.Probability.DilutedSpin.RandomInsertion
namespace OAI
section
namespace DilutedSpinGlass.PhysicalRoot
open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory HeterogeneousMarks KernelTower
open scoped NNReal ENNReal BigOperators
variable {X Y I : Type} [MeasurableSpace X] [MeasurableSpace Y]
[MeasurableSpace I] [Countable I] [MeasurableSingletonClass I]
{A : I → Type} [∀ i,Fintype (A i)] {N L : ℕ}
noncomputable def energyRoot (Q : (i : I) → Fin (L+1) → FiniteLaw (A i))
(m : Fin (L+1) → ℝ) (field : Y → ℝ)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
(h : RootPath Y N) (a : Sigma (RootPath I)) (E : (Fin N → Spin) → ℝ) : ℝ :=
root (terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L) Q m
(fun y => E (terminalState L y)+∑ i,field (rootArray N h i)*spin (terminalState L y i))
(rootArray a.1 a.2) factor
omit [MeasurableSpace Y] [MeasurableSpace I] [Countable I] [MeasurableSingletonClass I] in
lemma energyRoot_lipschitz (Q : (i : I) → Fin (L+1) → FiniteLaw (A i))
(m : Fin (L+1) → ℝ) (hm : ∀ i,0 < m i) (field : Y → ℝ)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
(h : RootPath Y N) (a : Sigma (RootPath I)) :
LipschitzWith 1 (energyRoot Q m field factor h a) := by
apply LipschitzWith.of_dist_le_mul
intro E F
simp only [NNReal.coe_one,one_mul,dist_eq_norm,Real.norm_eq_abs]
apply root_base_stability _ Q m hm
intro y
simp only [add_sub_add_right_eq_sub]
simpa only [Pi.sub_apply,Real.norm_eq_abs] using norm_le_pi_norm (E-F) (terminalState L y)
noncomputable def rootSample (V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
(Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
(h : RootPath Y N) (k : ℕ) (x : RootPath X k) (n : ℕ) (a : RootPath I n) : ℝ :=
root (terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L) Q m
(fun y => energy V field h k x (terminalState L y)) (rootArray n a) factor
omit [MeasurableSpace I] [Countable I] [MeasurableSingletonClass I] in
lemma measurable_rootSample_fixed
(V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
(hV : Measurable V) (hh : Measurable field)
(Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
(k n : ℕ) (a : RootPath I n) :
Measurable (fun z : RootPath Y N × RootPath X k =>
rootSample V field Q m factor z.1 k z.2 n a) := by
let b := fun z : RootPath Y N × RootPath X k =>
fun y : FinitePath (Fin N → Spin) (L+1) => energy V field z.1 k z.2 (terminalState L y)
have hb : ∀ y,Measurable (fun z => b z y) := fun y =>
measurable_energy V field (fun σ => (measurable_pi_apply σ).comp hV) hh k _
exact measurable_root (terminalTower (fun _ : Fin N => false) FiniteLaw.uniform L) Q m b
(rootArray n a) (fun _ => factor) hb (fun _ _ _ => measurable_const)
lemma measurable_rootSample
(V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
(hV : Measurable V) (hh : Measurable field)
(Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
(k n : ℕ) :
Measurable (fun z : (RootPath Y N × RootPath X k) × RootPath I n =>
rootSample V field Q m factor z.1.1 k z.1.2 n z.2) := by
apply measurable_from_prod_countable_left
intro a
exact measurable_rootSample_fixed V field hV hh Q m factor k n a
omit [MeasurableSpace X] [MeasurableSpace Y] [MeasurableSpace I] [Countable I]
[MeasurableSingletonClass I] in
lemma rootSample_bound (V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
(Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ) (hm : ∀ i,0 < m i)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
{C H D : ℝ} (hVb : ∀ x σ,|V x σ|≤C) (hhb : ∀ y,|field y|≤H)
(hfb : ∀ i y a,|Real.log (factor i y a)|≤D)
(h : RootPath Y N) (k : ℕ) (x : RootPath X k) (n : ℕ) (a : RootPath I n) :
|rootSample V field Q m factor h k x n a|≤H*N+C*k+D*n := by
exact root_uniform_bound _ Q m hm _ _ factor
(fun y => energy_bound V field hVb hhb h k x _) hfb
omit [MeasurableSpace Y] [MeasurableSpace I] [Countable I] [MeasurableSingletonClass I] in
lemma integral_compound_energyRoot (μ : Measure X) [IsProbabilityMeasure μ] (r : ℝ≥0)
(V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ) (hV : Measurable V)
(Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ) (hm : ∀ i,0 < m i)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
(h : RootPath Y N) (a : Sigma (RootPath I)) :
(∫ z : Sigma (RootPath X),rootSample V field Q m factor h z.1 z.2 a.1 a.2 ∂compoundRootLaw μ r)=
∫ E,energyRoot Q m field factor h a E ∂compoundPoisson r (Measure.map V μ) := by
have he : Measurable (fun z : Sigma (RootPath X) => ∑ i,V (rootArray z.1 z.2 i)) := by
apply measurable_sigmaUncurry (f := fun n (x : RootPath X n) => ∑ i,V (rootArray n x i))
intro k
simpa only [rootArrayEquiv_apply] using
(show Measurable (fun x : RootPath X k => ∑ i,V (rootArrayEquiv (X := X) k x i)) by fun_prop)
rw [← map_compoundRoot_energy μ V hV r,
integral_map he.aemeasurable (energyRoot_lipschitz Q m hm field factor h a).continuous.aestronglyMeasurable]
apply integral_congr_ae
filter_upwards [] with z
simp only [rootSample,energyRoot,energy,Finset.sum_apply]
lemma fullRoot_energy_integral
(ξ : Measure Y) [IsProbabilityMeasure ξ] (μ : Measure X) [IsProbabilityMeasure μ]
(ν : Measure I) [IsProbabilityMeasure ν] (r s : ℝ≥0)
(V : X → (Fin N → Spin) → ℝ) (field : Y → ℝ)
(hV : Measurable V) (hh : Measurable field)
(Q : (i : I) → Fin (L+1) → FiniteLaw (A i)) (m : Fin (L+1) → ℝ) (hm : ∀ i,0 < m i)
(factor : (i : I) → FinitePath (Fin N → Spin) (L+1) → FinitePath (A i) (L+1) → ℝ)
{C H D : ℝ} (hC : 0≤C) (hD : 0≤D)
(hVb : ∀ x σ,|V x σ|≤C) (hhb : ∀ y,|field y|≤H)
(hfb : ∀ i y a,|Real.log (factor i y a)|≤D) :
(∫ z,packRoot (rootSample V field Q m factor) z ∂fullRootLaw (fun _ => ξ) μ ν r s) =
∫ h,∫ a,∫ E,energyRoot Q m field factor h a E
∂compoundPoisson r (Measure.map V μ) ∂compoundRootLaw ν s ∂rootLaw N (fun _ => ξ) := by
rw [integral_fullRootLaw_reordered (fun _ => ξ) μ ν r s
(measurable_rootSample V field hV hh Q m factor)
(fun h k x n a => (rootSample_bound V field Q m hm factor hVb hhb hfb h k x n a).trans
(count_linear_le_quadratic hC hD k n))]
apply integral_congr_ae
filter_upwards [] with h
apply integral_congr_ae
filter_upwards [] with a
exact integral_compound_energyRoot μ r V field hV Q m hm factor h a
end DilutedSpinGlass.PhysicalRoot
end
end OAI
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