File size: 3,679 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 | import OAI.Probability.InvariantIsing.Pressure.PhysicalPressureMean
import OAI.Probability.InvariantIsing.Pressure.PhysicalPressureConcentration
namespace OAI
/-! Manuscript Theorem 1.1, conditional only on the explicitly cited
published concentration and finite hierarchical field inputs. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology
namespace InvariantIsing
theorem limiting_pressure
(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)
(hcompact : IsCompact (ν : Measure ℝ).support)
(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)) :
Tendsto (fun N => ∫ ω, rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0) ∂P)
atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) ∧
∀ᵐ ω ∂P, Tendsto (fun N => rotatedPressure (eig N) (matrixRotation (U N ω)⁻¹) (fun _ => 0))
atTop (𝓝 (variationalFunctional (measureR (ν : Measure ℝ) b)).toReal) := by
refine ⟨physical_mean_pressure_tendsto hhaar hgauss hpub P U hU hHaar eig ν a b
hcompact hbound ha hb hno hweak,?_⟩
have hab : a ≤ b := (hbound ha).2
let clip := fun N (i : Fin N) => spectralClip a b (eig N i)
have hcmean := physical_mean_pressure_tendsto hhaar hgauss hpub P U hU hHaar clip ν a b
hcompact hbound ha hb (fun ε hε => Eventually.of_forall (fun N i => by
have hh := spectralClip_mem hab (eig N i)
constructor <;> linarith [hh.1,hh.2]))
(spectral_clip_empirical_tendsto eig ν a b hbound hweak)
let K := |a|+|b|+1
have hK : 0 < K := by dsimp [K]; positivity
have hclip N i : |clip N i| ≤ K := by
have hh := spectralClip_mem hab (eig N i)
rw [abs_le]
dsimp [K]
constructor <;> linarith [hh.1,hh.2,neg_abs_le a,le_abs_self b,abs_nonneg a,abs_nonneg b]
have hconc := ae_physical_pressure_sub_mean_tendsto_zero hhaar P
(fun n => U (n+3)) (fun n => hU (n+3)) (fun n => hHaar (n+3))
(fun n => clip (n+3)) (fun _ _ => 0) K hK (fun n => hclip (n+3))
have hη := (spectralExcess_tendsto eig a b hno).comp (tendsto_add_atTop_nat 3)
filter_upwards [hconc] with ω hω
have hclim := hω.add (hcmean.comp (tendsto_add_atTop_nat 3))
have hd : Tendsto (fun n => rotatedPressure (eig (n+3)) (matrixRotation (U (n+3) ω)⁻¹) (fun _ => 0)-
rotatedPressure (clip (n+3)) (matrixRotation (U (n+3) ω)⁻¹) (fun _ => 0))
atTop (𝓝 0) := by
apply squeeze_zero_norm _ (by simpa only [zero_div] using hη.div_const 2)
intro n
rw [Real.norm_eq_abs,abs_sub_comm]
exact abs_rotatedPressure_sub_le (by omega) (clip (n+3)) (eig (n+3))
(matrixRotation (U (n+3) ω)⁻¹) (fun _ => 0) (spectralExcess (eig (n+3)) a b)
(fun i => spectralClip_close hab (spectralExcess_nonneg _ _ _) (spectralExcess_bounds _ _ _ i))
apply (tendsto_add_atTop_iff_nat 3).mp
have hh := hclim.add hd
convert hh using 1
· funext n
dsimp only [Function.comp_apply]
ring
· simp only [zero_add,add_zero]
end InvariantIsing
end
end OAI
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