| import OAI.Probability.InvariantIsing.Pressure.PositiveTemperaturePressure |
| import OAI.Probability.InvariantIsing.Pressure.GroundStateMean |
| import OAI.Probability.InvariantIsing.Pressure.ThermalLimitSqueeze |
|
|
| namespace OAI |
|
|
| /-! The deterministic ground-state limit, almost surely and in expectation, |
| and the existence of the full real-temperature variational limit. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
|
|
| namespace InvariantIsing |
|
|
| def thermalVariationalValue (ν : ProbabilityMeasure ℝ) (b β : ℝ) : ℝ := |
| (variationalFunctional (measureR (scaledSpectralLaw ν β : Measure ℝ) (β*b))).toReal/β |
|
|
| theorem ground_state_limit |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) |
| (hpub : PanchenkoTalagrandFieldPairInput) |
| {Ω : Type*} [MeasurableSpace Ω] (P : Measure Ω) [IsProbabilityMeasure P] |
| (U : (N : ℕ) → Ω → Orthogonal N) (hU : ∀ N, Measurable (U N)) |
| (hHaar : ∀ N, (P.map (U N)).IsMulRightInvariant) |
| (eig : (N : ℕ) → Fin N → ℝ) (ν : ProbabilityMeasure ℝ) (a b : ℝ) |
| (hbound : (ν : Measure ℝ).support ⊆ Icc a b) |
| (ha : a∈(ν : Measure ℝ).support) (hb : b∈(ν : Measure ℝ).support) |
| (hno : ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i, a-ε ≤ eig N i ∧ eig N i ≤ b+ε) |
| (hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1))) |
| atTop (𝓝 ν)) : |
| ∃ M : ℝ, |
| Tendsto (fun N => ∫ ω, groundStateEnergy (eig N) (matrixRotation (U N ω)⁻¹) ∂P) |
| atTop (𝓝 M) ∧ |
| (∀ᵐ ω ∂P, Tendsto (fun N => groundStateEnergy (eig N) (matrixRotation (U N ω)⁻¹)) |
| atTop (𝓝 M)) ∧ |
| Tendsto (thermalVariationalValue ν b) atTop (𝓝 M) := by |
| let p := fun (k N : ℕ) => (∫ ω, rotatedPressure (fun i => ((k : ℝ)+1)*eig N i) |
| (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P)/((k : ℝ)+1) |
| let v := fun k : ℕ => thermalVariationalValue ν b ((k : ℝ)+1) |
| let g := fun N => ∫ ω, groundStateEnergy (eig N) (matrixRotation (U N ω)⁻¹) ∂P |
| have hc : 0 ≤ Real.log 2 := (Real.log_pos (by norm_num : (1 : ℝ) < 2)).le |
| have hp (k : ℕ) : Tendsto (p k) atTop (𝓝 (v k)) := |
| ((positive_temperature_pressure hhaar hgauss hpub P U hU hHaar eig ν a b |
| hbound ha hb hno ((k : ℝ)+1) (by positivity) hweak).1).div_const _ |
| have hmean (k : ℕ) : ∀ᶠ N in atTop, p k N ≤ g N ∧ g N ≤ p k N+Real.log 2/(k+1) := by |
| filter_upwards [eventually_ge_atTop 1] with N hN |
| exact mean_groundState_squeeze (by omega) P (U N) (hU N) (eig N) _ (by positivity) |
| obtain ⟨M,hv,hg⟩ := exists_thermal_ground_limit p g v (Real.log 2) hc hp hmean |
| refine ⟨M,hg,?_,?_⟩ |
| · have hall : ∀ᵐ ω ∂P, ∀ k : ℕ, Tendsto |
| (fun N => rotatedPressure (fun i => ((k : ℝ)+1)*eig N i) |
| (matrixRotation (U N ω)⁻¹) (fun _ => 0)/((k : ℝ)+1)) atTop (𝓝 (v k)) := by |
| apply ae_all_iff.mpr |
| intro k |
| filter_upwards [(positive_temperature_pressure hhaar hgauss hpub P U hU hHaar eig ν a b |
| hbound ha hb hno ((k : ℝ)+1) (by positivity) hweak).2] with ω hω |
| exact hω.div_const _ |
| filter_upwards [hall] with ω hω |
| apply thermal_squeeze_tendsto _ _ v (Real.log 2) M hc hv hω |
| intro k |
| filter_upwards [eventually_ge_atTop 1] with N hN |
| exact groundStateEnergy_squeeze (by omega) (eig N) (matrixRotation (U N ω)⁻¹) _ (by positivity) |
| · have hbounds (β : ℝ) (hβ : 0 < β) : |
| thermalVariationalValue ν b β ≤ M ∧ M ≤ thermalVariationalValue ν b β+Real.log 2/β := by |
| have ht := ((positive_temperature_pressure hhaar hgauss hpub P U hU hHaar eig ν a b |
| hbound ha hb hno β hβ hweak).1).div_const β |
| have he : ∀ᶠ N in atTop, |
| (∫ ω, rotatedPressure (fun i => β*eig N i) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P)/β ≤ g N ∧ |
| g N ≤ (∫ ω, rotatedPressure (fun i => β*eig N i) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P)/β+ |
| Real.log 2/β := by |
| filter_upwards [eventually_ge_atTop 1] with N hN |
| exact mean_groundState_squeeze (by omega) P (U N) (hU N) (eig N) β hβ |
| exact ⟨le_of_tendsto_of_tendsto ht hg (he.mono fun _ h => h.1), |
| le_of_tendsto_of_tendsto hg (ht.add_const _) (he.mono fun _ h => h.2)⟩ |
| apply tendsto_sub_nhds_zero_iff.mp |
| apply squeeze_zero_norm' _ (tendsto_id.const_div_atTop (Real.log 2)) |
| filter_upwards [eventually_gt_atTop (0 : ℝ)] with β hβ |
| rw [Real.norm_eq_abs,abs_le] |
| simp only [id_eq] |
| have hh := hbounds β hβ |
| constructor <;> linarith [hh.1,hh.2,div_nonneg hc hβ.le] |
| |
| end InvariantIsing |
| |
| end |
| |
| end OAI |
| |
