| import Mathlib |
| import OAI.Probability.SKGap.Localization.PlantedTilt |
| import OAI.Probability.SKGap.Localization.PlantedMarginalLaw |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace SKGap.ObservationBridge |
| open Matrix Real Set MeasureTheory ProbabilityTheory |
| open scoped BigOperators ENNReal NNReal |
| variable {n : ℕ} |
| local instance annealedMatrixMeasurable : MeasurableSpace (Matrix (Fin n) (Fin n) ℝ) := borel _ |
| local instance annealedMatrixBorel : BorelSpace (Matrix (Fin n) (Fin n) ℝ) := ⟨rfl⟩ |
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| def gibbsObservationBad (j A K ε c ρ t : ℝ) (s : Spin n) : Set (Disorder n × Field n) := |
| {p | (coupling p.1,(fun i=>t*spinValue (s i)+p.2 i))∈literalBadPair j A K ε c ρ} |
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| lemma measurableSet_gibbsObservationBad (j A K ε c ρ t : ℝ) (s : Spin n) : |
| MeasurableSet (gibbsObservationBad j A K ε c ρ t s) := by |
| have hmap : Continuous (fun p : Disorder n × Field n=> |
| (coupling p.1,(fun i=>t*spinValue (s i)+p.2 i)) : |
| Disorder n × Field n→Matrix (Fin n) (Fin n) ℝ × Field n) := |
| (continuous_coupling.comp continuous_fst).prodMk (by fun_prop) |
| have hmmap : Measurable (fun p : Disorder n × Field n=> |
| (coupling p.1,(fun i=>t*spinValue (s i)+p.2 i)) : |
| Disorder n × Field n→Matrix (Fin n) (Fin n) ℝ × Field n) := hmap.measurable |
| exact (measurableSet_literalBadPair j A K ε c ρ).preimage hmmap |
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| def conditionalSpinFailure (j A K ε c ρ t : ℝ) (s : Spin n) (J : Disorder n) : ℝ≥0∞ := |
| (Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal)) |
| ((Prod.mk J) ⁻¹' gibbsObservationBad j A K ε c ρ t s) |
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| lemma measurable_conditionalSpinFailure (j A K ε c ρ t : ℝ) (s : Spin n) : |
| Measurable (conditionalSpinFailure j A K ε c ρ t s) := |
| measurable_measure_prodMk_left (measurableSet_gibbsObservationBad j A K ε c ρ t s) |
| lemma conditionalSpinFailure_le_one (j A K ε c ρ t : ℝ) (s : Spin n) (J : Disorder n) : |
| conditionalSpinFailure j A K ε c ρ t s J ≤ 1 := prob_le_one |
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| def fixedTimeFailureENN (j A K ε c ρ t : ℝ) (J : Disorder n) : ℝ≥0∞ := |
| ∑ s : Spin n, ENNReal.ofReal (mass J 0 s)*conditionalSpinFailure j A K ε c ρ t s J |
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| lemma measurable_fixedTimeFailureENN (j A K ε c ρ t : ℝ) : |
| Measurable (fixedTimeFailureENN (n:=n) j A K ε c ρ t) := by |
| apply Finset.measurable_sum |
| intro s _ |
| have hm : Measurable (fun J : Disorder n=>mass J 0 s) := by |
| simpa only [sqrt_one,one_mul] using scaled_spin_mass_measurable 1 s |
| exact hm.ennreal_ofReal.mul (measurable_conditionalSpinFailure j A K ε c ρ t s) |
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| lemma fixedTimeFailureENN_le_one (j A K ε c ρ t : ℝ) (J : Disorder n) : |
| fixedTimeFailureENN j A K ε c ρ t J ≤ 1 := by |
| calc |
| _ ≤ ∑ s : Spin n, ENNReal.ofReal (mass J 0 s)*1 := |
| Finset.sum_le_sum (fun s _=>mul_le_mul_right (conditionalSpinFailure_le_one j A K ε c ρ t s J) _) |
| _ = 1 := by |
| simp only [mul_one] |
| rw [← ENNReal.ofReal_sum_of_nonneg (fun s _=>mass_nonneg J 0 s),sum_mass,ENNReal.ofReal_one] |
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| lemma planted_spin_failure_integral (s : Spin n) (r j A K ε c ρ t : ℝ) : |
| (∫⁻ g : Disorder n,conditionalSpinFailure j A K ε c ρ t s (plantedEdges r s g) |
| ∂gaussianCoordinates (Edge n)) = |
| ((gaussianCoordinates (Edge n)).prod (Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))) |
| {p | (coupling (plusPlantedEdges r p.1),(fun i=>t+p.2 i))∈literalBadPair j A K ε c ρ} := by |
| let E := (Prod.map (plantedEdges r s) id) ⁻¹' gibbsObservationBad j A K ε c ρ t s |
| have hE : MeasurableSet E := (measurableSet_gibbsObservationBad j A K ε c ρ t s).preimage |
| ((show Measurable (plantedEdges r s) by unfold plantedEdges;fun_prop).prodMap measurable_id) |
| have he := Measure.prod_apply (μ:=gaussianCoordinates (Edge n)) |
| (ν:=Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal)) hE |
| exact he.symm.trans (gauged_fixed_time_probability s r j A K ε c ρ t) |
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|
| theorem fixed_time_size_bias_identity (β A K ε c ρ t : ℝ) : |
| (∫⁻ J : Disorder n, |
| ENNReal.ofReal (partition J 0/(∫ J,partition J 0 ∂disorderLaw β n))* |
| fixedTimeFailureENN (β^2) A K ε c ρ t J ∂disorderLaw β n) = |
| ((gaussianCoordinates (MatrixCoordinates (Fin n))).prod |
| (Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))) |
| (erasedPlantedRootBad n (β^2) A K ε c ρ t) := by |
| rw [show fixedTimeFailureENN (n:=n) (β^2) A K ε c ρ t = |
| (fun J=>∑ s,ENNReal.ofReal (mass J 0 s)*conditionalSpinFailure (β^2) A K ε c ρ t s J) from rfl] |
| rw [planted_disorder_joint_tilt β _ (fun s=>measurable_conditionalSpinFailure (β^2) A K ε c ρ t s)] |
| have he (s : Spin n) : |
| (∫⁻ g : Disorder n,conditionalSpinFailure (β^2) A K ε c ρ t s |
| (fun e=>sqrt (β^2/(n:ℝ))*g e+β^2/(n:ℝ)*spinValue (s e.1.1)*spinValue (s e.1.2)) |
| ∂gaussianCoordinates (Edge n)) = |
| ((gaussianCoordinates (MatrixCoordinates (Fin n))).prod |
| (Measure.pi (fun _ : Fin n=>gaussianReal 0 t.toNNReal))) |
| (erasedPlantedRootBad n (β^2) A K ε c ρ t) := |
| (planted_spin_failure_integral s _ _ _ _ _ _ _ _).trans |
| (plus_planted_marginal_probability (sq_nonneg β) A K ε c ρ t) |
| simp_rw [he] |
| rw [Finset.sum_const,Finset.card_univ,nsmul_eq_mul,← mul_assoc] |
| have hc : (Fintype.card (Spin n):ℝ) ≠ 0 := by positivity |
| have hh : ENNReal.ofReal (1/(Fintype.card (Spin n):ℝ))*(Fintype.card (Spin n):ℝ≥0∞)=1 := by |
| rw [← ENNReal.ofReal_natCast,← ENNReal.ofReal_mul (by positivity : (0:ℝ)≤1/(Fintype.card (Spin n):ℝ))] |
| simp |
| rw [hh,one_mul] |
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| theorem annealed_fixed_time_root_stability {β T : ℝ} (hβ : 0<β) (hβ1 : β<1) (hT : 0<T) : |
| ∃ A K ε c ρ a : ℝ, 1<A ∧ 2*β<K ∧ β*A<1 ∧ |
| 0<ε ∧ 0<c ∧ 0<ρ ∧ 0<a ∧ ∃ N : ℕ, 0<N ∧ ∀ n≥N, ∀ t∈Icc 0 T, |
| (∫⁻ J : Disorder n, |
| ENNReal.ofReal (partition J 0/(∫ J,partition J 0 ∂disorderLaw β n))* |
| fixedTimeFailureENN (β^2) A K ε c ρ t J ∂disorderLaw β n) ≤ |
| ENNReal.ofReal (exp (-a*(n:ℝ))) := by |
| obtain ⟨A,K,ε,c,ρ,a,hA,hK,hsub,hε,hc,hρ,ha,N,hN,hp⟩ := |
| zero_diagonal_fixed_time_root_stability (sq_pos_of_pos hβ) (by nlinarith : β^2<1) hT |
| rw [sqrt_sq hβ.le] at hK hsub |
| refine ⟨A,K,ε,c,ρ,a,hA,hK,hsub,hε,hc,hρ,ha,N,hN,?_⟩ |
| intro n hn t ht |
| rw [fixed_time_size_bias_identity,← ofReal_measureReal] |
| exact ENNReal.ofReal_le_ofReal (hp n hn t ht) |
| end SKGap.ObservationBridge |
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| end |
| end |
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| end OAI |
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