| import OAI.Probability.DilutedSpin.CompoundSuperposition |
| import OAI.Probability.DilutedSpin.InsertionStability |
|
|
| namespace OAI |
|
|
| section |
| namespace DilutedSpinGlass |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory |
| open scoped NNReal ENNReal BigOperators |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] |
|
|
| omit [NormedSpace ℝ E] in |
| lemma iid_energy_integrable (μ : Measure E) [IsProbabilityMeasure μ] |
| (hi : Integrable id μ) (n : ℕ) : |
| Integrable id (Measure.map (fun x : Fin n → E => ∑ i,x i) (Measure.pi (fun _ : Fin n => μ))) := by |
| rw [integrable_map_measure aestronglyMeasurable_id (by fun_prop)] |
| change Integrable (fun x : Fin n → E => ∑ i,x i) (Measure.pi (fun _ : Fin n => μ)) |
| exact integrable_finsetSum _ (fun i _ => integrable_eval (μ := fun _ : Fin n => μ) (i := i) hi) |
|
|
| omit [NormedSpace ℝ E] in |
| lemma iid_energy_norm_le (μ : Measure E) [IsProbabilityMeasure μ] |
| (hi : Integrable id μ) (n : ℕ) : |
| (∫ x, ‖x‖ ∂Measure.map (fun x : Fin n → E => ∑ i,x i) (Measure.pi (fun _ : Fin n => μ))) ≤ |
| n*(∫ x, ‖x‖ ∂μ) := by |
| rw [integral_map (by fun_prop) (by fun_prop)] |
| calc |
| _ ≤ ∫ x : Fin n → E, ∑ i, ‖x i‖ ∂Measure.pi (fun _ : Fin n => μ) := |
| integral_mono (integrable_finsetSum _ (fun i _ => integrable_eval hi)).norm |
| (integrable_finsetSum _ (fun i _ => integrable_comp_eval hi.norm)) |
| (fun x => norm_sum_le _ _) |
| _ = _ := by |
| rw [integral_finsetSum _ (fun i _ => integrable_comp_eval hi.norm)] |
| have he (i : Fin n) : (∫ x : Fin n → E, ‖x i‖ ∂Measure.pi (fun _ : Fin n => μ)) = ∫ x,‖x‖ ∂μ := |
| integral_comp_eval (μ := fun _ : Fin n => μ) (i := i) (by fun_prop) |
| simp only [he, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul] |
|
|
| omit [NormedSpace ℝ E] in |
| lemma compoundPoisson_integrable_id (r : ℝ≥0) (μ : Measure E) [IsProbabilityMeasure μ] |
| (hi : Integrable id μ) : Integrable id (compoundPoisson r μ) := by |
| unfold compoundPoisson |
| apply integrable_sum_measure |
| · intro n |
| exact (iid_energy_integrable μ hi n).smul_measure ENNReal.ofReal_ne_top |
| · have hs := ((hasSum_integral_poissonMeasure (poisson_integrable_count r)).mul_right |
| (∫ x, ‖x‖ ∂μ)).summable |
| apply Summable.of_nonneg_of_le (fun n => integral_nonneg (fun _ => norm_nonneg _)) _ hs |
| intro n |
| rw [integral_smul_measure,ENNReal.toReal_ofReal (by positivity)] |
| change (Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*_ ≤ |
| ((Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*(n:ℝ))*(∫ x,‖x‖ ∂μ) |
| rw [mul_assoc] |
| exact mul_le_mul_of_nonneg_left (iid_energy_norm_le μ hi n) (by positivity) |
|
|
| omit [NormedSpace ℝ E] in |
| lemma compoundPoisson_norm_le (r : ℝ≥0) (μ : Measure E) [IsProbabilityMeasure μ] |
| (hi : Integrable id μ) : |
| (∫ x, ‖x‖ ∂compoundPoisson r μ) ≤ r*(∫ x, ‖x‖ ∂μ) := by |
| have hc : Integrable (fun x : E => ‖x‖) (compoundPoisson r μ) := |
| (compoundPoisson_integrable_id r μ hi).norm |
| rw [compoundPoisson,integral_sum_measure hc] |
| have hs := (hasSum_integral_poissonMeasure (poisson_integrable_count r)).mul_right (∫ x,‖x‖ ∂μ) |
| rw [poisson_mean] at hs |
| rw [← hs.tsum_eq] |
| apply Summable.tsum_le_tsum |
| · intro n |
| rw [integral_smul_measure,ENNReal.toReal_ofReal (by positivity)] |
| change (Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*_ ≤ |
| ((Real.exp (-(r:ℝ))*(r:ℝ)^n/n.factorial)*(n:ℝ))*(∫ x,‖x‖ ∂μ) |
| rw [mul_assoc] |
| exact mul_le_mul_of_nonneg_left (iid_energy_norm_le μ hi n) (by positivity) |
| · exact (hasSum_integral_measure hc).summable |
| · exact hs.summable |
|
|
| end DilutedSpinGlass |
|
|
| end |
|
|
| end OAI |
|
|
