File size: 4,716 Bytes
a89d282 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 | 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
|