| import OAI.Probability.RadialTransition.FreeClusterCover |
|
|
| namespace OAI |
|
|
| universe uIndex |
|
|
| noncomputable section |
| open scoped BigOperators ENNReal NNReal |
| open MeasureTheory Set Filter |
| namespace RadialTransition |
| attribute [local instance] Classical.propDecidable |
|
|
| def badLabels {ι : Type uIndex} [Fintype ι] (t : ℝ) (x : ι → Position) : Finset ι := |
| Finset.univ.filter (fun i => ∃ j, i ≠ j ∧ dist (x i) (x j) < 1 ∧ |
| ¬ (microLength t ≤ dist (x i) (x j) ∧ dist (x i) (x j) ≤ (1+(1:ℝ)/1000)*microLength t)) |
|
|
| lemma after_removal_free {ι : Type uIndex} [Fintype ι] (t : ℝ) (x : ι → Position) : |
| CoreFree t (fun i : {i // i ∉ badLabels t x} => x i) := by |
| intro i j hij hd |
| by_contra hs |
| apply i.property |
| exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, j.val, fun he => hij (Subtype.ext he), hd, hs⟩ |
|
|
| lemma badLabels_energy {ι : Type uIndex} [Fintype ι] {t : ℝ} (ht : 0 ≤ t) (x : ι → Position) : |
| t^2 * (badLabels t x).card / 2 ≤ orderedEnergy (corePotential t) x := by |
| classical |
| have hrow (i : ι) : (if i ∈ badLabels t x then t^2 else 0) ≤ |
| ∑ j, if i ≠ j then corePotential t (nndist (x i) (x j)) else 0 := by |
| have hnon (j : ι) : (0:ℝ) ≤ if i ≠ j then corePotential t (nndist (x i) (x j)) else 0 := by |
| split_ifs |
| · exact coreReal_nonneg _ _ |
| · rfl |
| by_cases hi : i ∈ badLabels t x |
| · obtain ⟨j,hij,hd,hs⟩ := (Finset.mem_filter.mp hi).2 |
| have hfull : corePotential t (nndist (x i) (x j)) = t^2 := coreReal_full_of_not_shell ht hd.le hs |
| rw [ite_eq_left hi] |
| have hh := Finset.single_le_sum (fun k _ => hnon k) (Finset.mem_univ j) |
| simpa only [ite_eq_left hij, hfull] using hh |
| · rw [ite_eq_right hi] |
| exact Finset.sum_nonneg (fun j _ => hnon j) |
| have hh := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset ι)) => hrow i) |
| have hsum : (∑ i : ι, if i ∈ badLabels t x then t^2 else 0) = t^2*(badLabels t x).card := by |
| rw [← Finset.sum_filter] |
| simp [mul_comm] |
| rw [hsum] at hh |
| exact div_le_div_of_nonneg_right hh (by norm_num) |
|
|
| def removalWeight (t β : ℝ) (s : Multiset Position) : ℝ≥0∞ := |
| ENNReal.ofReal (Real.exp (-β*t^2*s.card/2)) |
|
|
| lemma measurable_removalWeight (t β : ℝ) : MeasurableConfiguration (removalWeight t β) := by |
| intro s ι _ |
| simp only [removalWeight, Multiset.card_add, configurationMultiset_card] |
| exact measurable_const |
|
|
| lemma particleBase_removalWeight {μ : Measure Position} (hμ : μ univ ≠ ⊤) (t β z : ℝ) : |
| (∫⁻ x, removalWeight t β (particleMultiset x) ∂particleBase μ z) = |
| ENNReal.ofReal (Real.exp ((μ univ).toReal*Real.exp (z-β*t^2/2))) := by |
| have hm := measurable_particleObservable _ (measurable_removalWeight t β) 0 |
| simp only [zero_add] at hm |
| have hh := lintegral_particleBase μ (z-β*t^2/2) (f := fun _ => (1:ℝ≥0∞)) measurable_const |
| simp only [lintegral_one] at hh |
| rw [lintegral_particleBase μ z hm, ← particleBase_univ hμ (z-β*t^2/2), hh] |
| apply tsum_congr |
| intro n |
| simp only [particleMultiset, removalWeight, configurationMultiset_card, Fintype.card_fin, |
| lintegral_const] |
| rw [← mul_assoc, ← ENNReal.ofReal_mul (by positivity)] |
| congr 2 |
| rw [div_mul_eq_mul_div, ← Real.exp_add] |
| congr 1 |
| ring_nf |
|
|
| end RadialTransition |
| end |
|
|
| end OAI |
|
|
