| import OAI.Probability.ContinuumTransition.RootVolume |
|
|
| namespace OAI |
|
|
| section |
| open scoped BigOperators ENNReal |
| open MeasureTheory Set |
| namespace ContinuumTemperature |
| open Classical |
|
|
| lemma measurableSet_freeConfig {ι : Type} [Fintype ι] (t : ℝ) : |
| MeasurableSet {x : ι → Space | FreeConfig t x} := by |
| apply measurableSet_forall_labels |
| intro i |
| apply measurableSet_forall_labels |
| intro j |
| by_cases hij : i = j |
| · simp only [hij, ne_eq, not_true_eq_false, false_implies, ofPred_true, MeasurableSet.univ] |
| have hm : Measurable (fun x : ι → Space => ‖x i-x j‖) := |
| ((measurable_pi_apply i).sub (measurable_pi_apply j)).norm |
| have he : {x : ι → Space | i ≠ j → ‖x i-x j‖ < 1 → |
| Real.exp (-t) ≤ ‖x i-x j‖ ∧ ‖x i-x j‖ ≤ (1+shellDelta)*Real.exp (-t)} = |
| {x | 1 ≤ ‖x i-x j‖} ∪ |
| ({x | Real.exp (-t) ≤ ‖x i-x j‖} ∩ {x | ‖x i-x j‖ ≤ (1+shellDelta)*Real.exp (-t)}) := by |
| ext x |
| simp only [mem_ofPred_eq, mem_union, mem_inter_iff, ne_eq, hij, not_false_eq_true, imp_iff_not_or, not_lt, not_true_eq_false, false_or] |
| rw [he] |
| exact (measurableSet_le measurable_const hm).union |
| ((measurableSet_le measurable_const hm).inter (measurableSet_le hm measurable_const)) |
|
|
| noncomputable def freeFamily (t d : ℝ) : ConfigFamily := |
| fun ι _ x => if FreeConfig t x then Real.exp (d*Fintype.card ι) else 0 |
|
|
| lemma freeFamily_relabel (t d : ℝ) : RelabelInvariant (freeFamily t d) := by |
| intro ι κ _ _ e x |
| have he : FreeConfig t (x ∘ e) ↔ FreeConfig t x := by |
| constructor |
| · intro hh i j hij hdist |
| have hn : e.symm i ≠ e.symm j := fun he => hij (e.symm.injective he) |
| simpa only [Function.comp_apply, Equiv.apply_symm_apply] using |
| hh (e.symm i) (e.symm j) hn (by simpa only [Function.comp_apply, Equiv.apply_symm_apply] using hdist) |
| · intro hh i j hij hdist |
| exact hh (e i) (e j) (fun he => hij (e.injective he)) hdist |
| simp only [freeFamily, he, Fintype.card_congr e] |
|
|
| lemma freeFamily_measurable (t d : ℝ) (ι : Type) [Fintype ι] : Measurable (freeFamily t d ι) := |
| Measurable.ite (measurableSet_freeConfig t) measurable_const measurable_const |
|
|
| lemma freeFamily_nonneg (t d : ℝ) (ι : Type) [Fintype ι] (x : ι → Space) : |
| 0 ≤ freeFamily t d ι x := by |
| unfold freeFamily |
| split_ifs <;> positivity |
|
|
| lemma freeFamily_le_exp (t d : ℝ) (ι : Type) [Fintype ι] (x : ι → Space) : |
| freeFamily t d ι x ≤ Real.exp (d*Fintype.card ι) := by |
| unfold freeFamily |
| split_ifs |
| · exact le_rfl |
| · exact (Real.exp_pos _).le |
|
|
| lemma freeFamily_summable (t d L : ℝ) : Summable (familyTerm (freeFamily t d) (cube L)) := |
| summable_familyTerm_of_bound _ (isCompact_cube L).measure_lt_top.ne (freeFamily_measurable t d) |
| (freeFamily_nonneg t d) (freeFamily_le_exp t d) |
|
|
| lemma rootCover_volume_real {ι : Type} [Fintype ι] (s : Finset ι) (r L : ℝ) : |
| volume.real ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ rootCover s r) ≤ |
| (volume.real (cube L))^s.card * |
| ((s.card : ℝ)*volume.real (Metric.closedBall (0 : Space) r))^(Fintype.card ι-s.card) := by |
| have hh := ENNReal.toReal_mono (show |
| (volume (cube L))^s.card * |
| ((s.card : ℝ≥0∞)*volume (Metric.closedBall (0 : Space) r))^(Fintype.card ι-s.card) ≠ ⊤ by |
| exact ENNReal.mul_ne_top (ENNReal.pow_ne_top (isCompact_cube L).measure_lt_top.ne) |
| (ENNReal.pow_ne_top (ENNReal.mul_ne_top (by simp) (measure_closedBall_lt_top.ne)))) (rootCover_volume s r L) |
| simpa only [Measure.real, ENNReal.toReal_mul, ENNReal.toReal_pow, ENNReal.toReal_natCast] using hh |
|
|
| lemma freeConfig_volume_le {ι : Type} [Fintype ι] {t : ℝ} |
| (ht : 2*(1+shellDelta)*Real.exp (-t) < 1) (L : ℝ) : |
| volume.real ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ {x | FreeConfig t x}) ≤ |
| ∑ k ∈ Finset.range (Fintype.card ι+1), |
| if k ≤ packingNumber L ∧ Fintype.card ι ≤ 4*k then |
| ((Fintype.card ι).choose k : ℝ)*(volume.real (cube L))^k * |
| ((k : ℝ)*volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t))))^(Fintype.card ι-k) |
| else 0 := by |
| let T : Finset (Finset ι) := Finset.univ.filter (fun s => s.card ≤ packingNumber L ∧ Fintype.card ι ≤ 4*s.card) |
| let A : Finset ι → Set (ι → Space) := fun s => |
| {x | ∀ i, x i ∈ cube L} ∩ rootCover s ((1+shellDelta)*Real.exp (-t)) |
| have hsub : ({x : ι → Space | ∀ i, x i ∈ cube L} ∩ {x | FreeConfig t x}) ⊆ ⋃ s ∈ T, A s := by |
| intro x hx |
| obtain ⟨s,hs,hn,hc⟩ := freeConfig_roots ht hx.2 hx.1 |
| apply mem_iUnion₂.mpr |
| refine ⟨s,by simp only [T, Finset.mem_filter, Finset.mem_univ, true_and]; exact ⟨hs,hn⟩,hx.1,?_⟩ |
| intro j |
| obtain ⟨i,hi,hij⟩ := hc j |
| exact ⟨⟨i,hi⟩,hij⟩ |
| have hfin : volume (⋃ s ∈ T, A s) ≠ ⊤ := by |
| apply ne_top_of_le_ne_top (show volume {x : ι → Space | ∀ i, x i ∈ cube L} ≠ ⊤ by |
| rw [volume_generalConfigurations]; exact ENNReal.pow_ne_top (isCompact_cube L).measure_lt_top.ne) |
| apply measure_mono |
| intro x hx |
| obtain ⟨s,_,hs⟩ := mem_iUnion₂.mp hx |
| exact hs.1 |
| calc |
| _ ≤ volume.real (⋃ s ∈ T, A s) := measureReal_mono hsub hfin |
| _ ≤ ∑ s ∈ T, volume.real (A s) := measureReal_biUnion_finset_le _ _ |
| _ ≤ ∑ s ∈ T, (volume.real (cube L))^s.card * |
| ((s.card : ℝ)*volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t))))^(Fintype.card ι-s.card) := by |
| exact Finset.sum_le_sum (fun s _ => rootCover_volume_real s _ L) |
| _ = _ := by |
| rw [Finset.sum_filter] |
| have hh := Finset.sum_powerset_apply_card |
| (fun k => if k ≤ packingNumber L ∧ Fintype.card ι ≤ 4*k then |
| (volume.real (cube L))^k * |
| ((k : ℝ)*volume.real (Metric.closedBall (0 : Space) ((1+shellDelta)*Real.exp (-t))))^(Fintype.card ι-k) else 0) |
| (x := (Finset.univ : Finset ι)) |
| simp only [Finset.powerset_univ, Finset.card_univ, nsmul_eq_mul] at hh |
| rw [hh] |
| apply Finset.sum_congr rfl |
| intro k hk |
| split_ifs <;> simp [mul_assoc] |
|
|
| end ContinuumTemperature |
| end |
|
|
| end OAI |
|
|
