| import OAI.Probability.InvariantIsing.Pressure.GroundStateFinite |
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| namespace OAI |
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| /-! Integrable physical energies and the expected soft-maximum inequality. -/ |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped Classical BigOperators |
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| namespace InvariantIsing |
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| lemma integrable_physicalPressure {Ω : Type*} [MeasurableSpace Ω] |
| {N : ℕ} (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P] |
| (U : Ω → Orthogonal N) (hU : Measurable U) (eig c : Fin N → ℝ) : |
| Integrable (fun ω => rotatedPressure eig (matrixRotation (U ω)⁻¹) c) P := by |
| let K := ∑ i, |eig i| |
| have he i : |eig i| ≤ K := Finset.single_le_sum (f := fun i => |eig i|) (fun _ _ => abs_nonneg _) (Finset.mem_univ i) |
| apply (integrable_const (K/2+|noninteractingPressure c|)).mono' |
| ((measurable_physicalPressure hN eig c).comp hU).aestronglyMeasurable |
| filter_upwards [] with ω |
| rw [Real.norm_eq_abs] |
| have hb := abs_pressure_sub_noninteracting_le hN eig c (matrixRotation (U ω)⁻¹) K he |
| have ha := abs_add_le (rotatedPressure eig (matrixRotation (U ω)⁻¹) c-noninteractingPressure c) |
| (noninteractingPressure c) |
| rw [sub_add_cancel] at ha |
| exact ha.trans (add_le_add hb le_rfl) |
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| lemma integrable_physicalGroundState {Ω : Type*} [MeasurableSpace Ω] |
| {N : ℕ} (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P] |
| (U : Ω → Orthogonal N) (hU : Measurable U) (eig : Fin N → ℝ) : |
| Integrable (fun ω => groundStateEnergy eig (matrixRotation (U ω)⁻¹)) P := by |
| let K := ∑ i, |eig i| |
| have he i : |eig i| ≤ K := Finset.single_le_sum (f := fun i => |eig i|) (fun _ _ => abs_nonneg _) (Finset.mem_univ i) |
| exact (integrable_const (K/2)).mono' |
| ((measurable_physical_groundStateEnergy hN eig).comp hU).aestronglyMeasurable |
| (ae_of_all _ fun ω => by simpa only [Real.norm_eq_abs] using |
| abs_groundStateEnergy_le hN eig (matrixRotation (U ω)⁻¹) K he) |
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| lemma mean_groundState_squeeze {Ω : Type*} [MeasurableSpace Ω] |
| {N : ℕ} (hN : 0 < N) (P : Measure Ω) [IsProbabilityMeasure P] |
| (U : Ω → Orthogonal N) (hU : Measurable U) (eig : Fin N → ℝ) |
| (β : ℝ) (hβ : 0 < β) : |
| (∫ ω, rotatedPressure (fun i => β*eig i) (matrixRotation (U ω)⁻¹) (fun _ => 0) ∂P)/β ≤ |
| (∫ ω, groundStateEnergy eig (matrixRotation (U ω)⁻¹) ∂P) ∧ |
| (∫ ω, groundStateEnergy eig (matrixRotation (U ω)⁻¹) ∂P) ≤ |
| (∫ ω, rotatedPressure (fun i => β*eig i) (matrixRotation (U ω)⁻¹) (fun _ => 0) ∂P)/β+ |
| Real.log 2/β := by |
| have hp := integrable_physicalPressure hN P U hU (fun i => β*eig i) (fun _ => 0) |
| have hg := integrable_physicalGroundState hN P U hU eig |
| have hl := integral_mono (hp.div_const β) hg (fun ω => |
| (groundStateEnergy_squeeze hN eig (matrixRotation (U ω)⁻¹) β hβ).1) |
| have hu := integral_mono hg ((hp.div_const β).add (integrable_const (Real.log 2/β))) (fun ω => |
| (groundStateEnergy_squeeze hN eig (matrixRotation (U ω)⁻¹) β hβ).2) |
| rw [integral_div] at hl |
| simp only [Pi.add_apply] at hu |
| rw [integral_add (hp.div_const β) (integrable_const _),integral_div,integral_const, |
| probReal_univ,one_smul] at hu |
| exact ⟨hl,hu⟩ |
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| end InvariantIsing |
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| end |
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| end OAI |
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