| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.PressureGapProbability |
| import OAI.Probability.SKBarriers.Dynamics.ClockEndpoint |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Classical MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| theorem continuous_coupling {n : ℕ} (i j : Fin n) : Continuous (fun J : Disorder n => coupling J i j) := by |
| unfold coupling |
| split_ifs <;> first | exact continuous_apply _ | exact continuous_const |
|
|
| theorem continuous_localField {n : ℕ} (x : Config n) (i : Fin n) : Continuous (fun J : Disorder n => localField J x i) := by |
| unfold localField |
| apply Continuous.div_const |
| exact continuous_finsetSum _ (fun j _ => (continuous_coupling i j).mul continuous_const) |
|
|
| theorem continuous_heatBath {n : ℕ} (β : ℝ) (x y : Config n) : Continuous (fun J : Disorder n => heatBath β J x y) := by |
| unfold heatBath |
| apply Continuous.div_const |
| apply continuous_finsetSum |
| intro i _ |
| split_ifs |
| · apply Continuous.div |
| · exact (continuous_const.mul (continuous_localField x i)).rexp |
| · exact continuous_const.mul (Real.continuous_cosh.comp (continuous_const.mul (continuous_localField x i))) |
| · intro J; exact mul_ne_zero (by norm_num) (Real.cosh_pos _).ne' |
| · exact continuous_const |
|
|
| theorem continuous_discreteKernel {n : ℕ} (β : ℝ) (k : ℕ) (x y : Config n) : |
| Continuous (fun J : Disorder n => discreteKernel β J k x y) := by |
| induction k generalizing x y with |
| | zero => exact continuous_const |
| | succ k ih => |
| exact continuous_finsetSum _ (fun z _ => (continuous_heatBath β x z).mul (ih z y)) |
|
|
| theorem measurable_continuousKernel {n : ℕ} (β : ℝ) (t : ℝ) (x y : Config n) : |
| Measurable (fun J : Disorder n => continuousKernel β J t x y) := by |
| exact Measurable.tsum (fun k => measurable_const.mul (continuous_discreteKernel β k x y).measurable) |
|
|
| theorem continuous_discreteDistance {n : ℕ} (β : ℝ) (k : ℕ) (x : Config n) : |
| Continuous (fun J : Disorder n => discreteDistance β J x k) := by |
| exact continuous_const.mul (continuous_finsetSum _ (fun y _ => ((continuous_discreteKernel β k x y).sub (continuous_gibbs β y)).abs)) |
|
|
| theorem measurable_continuousDistance {n : ℕ} (β : ℝ) (t : ℝ) (x : Config n) : |
| Measurable (fun J : Disorder n => continuousDistance β J x t) := by |
| exact measurable_const.mul (Finset.measurable_sum _ (fun y _ => ((measurable_continuousKernel β t x y).sub (continuous_gibbs β y).measurable).abs)) |
|
|
| theorem measurable_gibbs_predicate_mass {n : ℕ} (β : ℝ) (E : Disorder n → Config n → Prop) |
| (hE : ∀x,MeasurableSet {J | E J x}) : |
| Measurable (fun J => ∑x,if E J x then gibbs β J x else 0) := by |
| exact Finset.measurable_sum _ (fun x _ => Measurable.ite (hE x) (continuous_gibbs β x).measurable measurable_const) |
|
|
| theorem gibbs_predicate_mass_nonneg {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop) : |
| 0≤∑x,if E x then gibbs β J x else 0 := by |
| apply Finset.sum_nonneg |
| intro x _ |
| split_ifs <;> first | exact (gibbs_pos β J x).le | exact le_rfl |
|
|
| theorem gibbs_predicate_mass_le_one {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop) : |
| (∑x,if E x then gibbs β J x else 0)≤1 := by |
| rw [← gibbs_sum β J] |
| apply Finset.sum_le_sum |
| intro x _ |
| split_ifs <;> first | exact le_rfl | exact (gibbs_pos β J x).le |
|
|
| theorem integrable_gibbs_predicate_mass {n : ℕ} (β : ℝ) (E : Disorder n → Config n → Prop) |
| (hE : ∀x,MeasurableSet {J | E J x}) : |
| Integrable (fun J => ∑x,if E J x then gibbs β J x else 0) (disorderLaw n) := by |
| apply (integrable_const (1:ℝ)).mono' (measurable_gibbs_predicate_mass β E hE).aestronglyMeasurable |
| filter_upwards [] with J |
| simpa only [Real.norm_eq_abs,abs_of_nonneg (gibbs_predicate_mass_nonneg β J _)] using gibbs_predicate_mass_le_one β J (E J) |
|
|
| theorem discreteMixedMass_integrable (n : ℕ) (β : ℝ) (k : ℕ) : |
| Integrable (fun J : Disorder n => discreteMixedMass β J k) (disorderLaw n) := |
| integrable_gibbs_predicate_mass β _ (fun x => measurableSet_le (continuous_discreteDistance β k x).measurable measurable_const) |
|
|
| theorem continuousMixedMass_integrable (n : ℕ) (β : ℝ) (t : ℝ) : |
| Integrable (fun J : Disorder n => continuousMixedMass β J t) (disorderLaw n) := |
| integrable_gibbs_predicate_mass β _ (fun x => measurableSet_le (measurable_continuousDistance β t x) measurable_const) |
|
|
| theorem gibbs_predicate_mass_compl {n : ℕ} (β : ℝ) (J : Disorder n) (E : Config n → Prop) : |
| (∑x,if ¬E x then gibbs β J x else 0)=1-∑x,if E x then gibbs β J x else 0 := by |
| rw [← gibbs_sum β J,← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro x _ |
| split_ifs <;> simp_all |
|
|
| theorem discreteBadMass_eq_one_sub (β : ℝ) (n : ℕ) : |
| discreteBadMass β n=1-∫J,discreteMixedMass β J ⌊timeScale n⌋₊ ∂disorderLaw n := by |
| have he (J : Disorder n) : (∑x,if (1/4:ℝ)<discreteDistance β J x ⌊timeScale n⌋₊ then gibbs β J x else 0)= |
| 1-discreteMixedMass β J ⌊timeScale n⌋₊ := by |
| simpa only [not_le,discreteMixedMass] using gibbs_predicate_mass_compl β J (fun x => discreteDistance β J x ⌊timeScale n⌋₊≤(1/4:ℝ)) |
| unfold discreteBadMass |
| simp_rw [he] |
| rw [integral_sub (integrable_const 1) (discreteMixedMass_integrable n β _)] |
| simp |
|
|
| theorem continuousBadMass_eq_one_sub (β : ℝ) (n : ℕ) : |
| continuousBadMass β n=1-∫J,continuousMixedMass β J (timeScale n) ∂disorderLaw n := by |
| have he (J : Disorder n) : (∑x,if (1/4:ℝ)<continuousDistance β J x (timeScale n) then gibbs β J x else 0)= |
| 1-continuousMixedMass β J (timeScale n) := by |
| simpa only [not_le,continuousMixedMass] using gibbs_predicate_mass_compl β J (fun x => continuousDistance β J x (timeScale n)≤(1/4:ℝ)) |
| unfold continuousBadMass |
| simp_rw [he] |
| rw [integral_sub (integrable_const 1) (continuousMixedMass_integrable n β _)] |
| simp |
|
|
| theorem main_of_quenched_mixed_bounds (β : ℝ) (C : (n : ℕ) → Set (Disorder n)) |
| (hC : ∀n,MeasurableSet (C n)) (hC1 : Tendsto (fun n => (disorderLaw n).real (C n)) atTop (𝓝 1)) |
| (e : ℕ → ℝ) (he : ∀n,0≤e n) (he0 : Tendsto e atTop (𝓝 0)) |
| (hbound : ∀ᶠ n : ℕ in atTop,∀J∈C n,continuousMixedMass β J (timeScale n)≤e n ∧ |
| discreteMixedMass β J ⌊timeScale n⌋₊≤e n) : |
| Tendsto (continuousBadMass β) atTop (𝓝 1) ∧ Tendsto (discreteBadMass β) atTop (𝓝 1) := by |
| have hcomp : Tendsto (fun n => (disorderLaw n).real (C n)ᶜ) atTop (𝓝 0) := by |
| have H := (tendsto_const_nhds (x:=(1:ℝ))).sub hC1 |
| simpa only [measureReal_compl (hC _),probReal_univ,sub_self] using H |
| have hlim := he0.add hcomp |
| simp only [add_zero] at hlim |
| have HC : Tendsto (fun n => ∫J,continuousMixedMass β J (timeScale n) ∂disorderLaw n) atTop (𝓝 0) := by |
| apply squeeze_zero' (Eventually.of_forall (fun n => integral_nonneg (fun J => gibbs_predicate_mass_nonneg β J _))) _ hlim |
| filter_upwards [hbound] with n hn |
| apply integral_le_off_exception (continuousMixedMass_integrable n β _) (hC n).compl (he n) |
| (fun J => gibbs_predicate_mass_le_one β J _) |
| intro J hJ |
| exact (hn J (by simpa only [mem_compl_iff,not_not] using hJ)).1 |
| have HD : Tendsto (fun n => ∫J,discreteMixedMass β J ⌊timeScale n⌋₊ ∂disorderLaw n) atTop (𝓝 0) := by |
| apply squeeze_zero' (Eventually.of_forall (fun n => integral_nonneg (fun J => gibbs_predicate_mass_nonneg β J _))) _ hlim |
| filter_upwards [hbound] with n hn |
| apply integral_le_off_exception (discreteMixedMass_integrable n β _) (hC n).compl (he n) |
| (fun J => gibbs_predicate_mass_le_one β J _) |
| intro J hJ |
| exact (hn J (by simpa only [mem_compl_iff,not_not] using hJ)).2 |
| constructor |
| · have H := (tendsto_const_nhds (x:=(1:ℝ))).sub HC |
| simpa only [← continuousBadMass_eq_one_sub,sub_zero] using H |
| · have H := (tendsto_const_nhds (x:=(1:ℝ))).sub HD |
| simpa only [← discreteBadMass_eq_one_sub,sub_zero] using H |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
