| import OAI.Probability.DilutedSpin.CompoundRate |
| import OAI.Probability.DilutedSpin.RandomInsertion |
|
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| namespace OAI |
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| 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} |
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| 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 |
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| 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 |
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|
| 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 |
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| end |
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| end OAI |
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