File size: 3,259 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 | import OAI.Probability.InvariantIsing.Pressure.LimitingPressure
namespace OAI
/-! Positive temperature scaling preserves the precise spectral hypotheses. -/
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology
namespace InvariantIsing
def scaledSpectralLaw (ν : ProbabilityMeasure ℝ) (β : ℝ) : ProbabilityMeasure ℝ :=
ν.map (fun x => β*x)
lemma scaled_spectral_support_bound (ν : ProbabilityMeasure ℝ) (a b β : ℝ)
(hβ : 0 ≤ β) (hbound : (ν : Measure ℝ).support ⊆ Icc a b) :
(scaledSpectralLaw ν β : Measure ℝ).support ⊆ Icc (β*a) (β*b) := by
apply Measure.support_subset_of_isClosed isClosed_Icc
change ∀ᵐ y ∂((ν : Measure ℝ).map (fun x => β*x)), y∈Icc (β*a) (β*b)
apply (ae_map_iff (measurable_const.mul measurable_id).aemeasurable measurableSet_Icc).mpr
filter_upwards [(ν : Measure ℝ).support_mem_ae] with x hx
exact ⟨mul_le_mul_of_nonneg_left (hbound hx).1 hβ,
mul_le_mul_of_nonneg_left (hbound hx).2 hβ⟩
lemma scaled_spectral_support_mem (ν : ProbabilityMeasure ℝ) (β x : ℝ)
(hx : x∈(ν : Measure ℝ).support) :
β*x∈(scaledSpectralLaw ν β : Measure ℝ).support := by
rw [Measure.support_eq_forall_isOpen]
intro S hS hopen
change ((ν : Measure ℝ).map (fun x => β*x)) S > 0
have hm : Measurable (fun y : ℝ => β*y) := measurable_const.mul measurable_id
rw [Measure.map_apply hm hopen.measurableSet]
exact (Measure.mem_support_iff_forall x).mp hx _
((hopen.preimage (continuous_const.mul continuous_id)).mem_nhds hS)
lemma scaled_spectral_compact (ν : ProbabilityMeasure ℝ) (a b β : ℝ)
(hβ : 0 ≤ β) (hbound : (ν : Measure ℝ).support ⊆ Icc a b) :
IsCompact (scaledSpectralLaw ν β : Measure ℝ).support :=
isCompact_Icc.of_isClosed_subset Measure.isClosed_support
(scaled_spectral_support_bound ν a b β hβ hbound)
lemma scaled_spectral_no_outliers (eig : (N : ℕ) → Fin N → ℝ) (a b β : ℝ)
(hβ : 0 < β)
(hno : ∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i, a-ε ≤ eig N i ∧ eig N i ≤ b+ε) :
∀ ε : ℝ, 0 < ε → ∀ᶠ N in atTop, ∀ i,
β*a-ε ≤ β*eig N i ∧ β*eig N i ≤ β*b+ε := by
intro ε hε
filter_upwards [hno (ε/β) (div_pos hε hβ)] with N hN i
have hl := mul_le_mul_of_nonneg_left (hN i).1 hβ.le
have hu := mul_le_mul_of_nonneg_left (hN i).2 hβ.le
have he : β*(ε/β)=ε := mul_div_cancel₀ ε hβ.ne'
rw [mul_sub,he] at hl
rw [mul_add,he] at hu
exact ⟨hl,hu⟩
lemma scaled_spectral_weak (eig : (N : ℕ) → Fin N → ℝ)
(ν : ProbabilityMeasure ℝ) (β : ℝ)
(hweak : Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (eig (k+1)))
atTop (𝓝 ν)) :
Tendsto (fun k => empiricalSpectralLaw (Nat.succ_pos k) (fun i => β*eig (k+1) i))
atTop (𝓝 (scaledSpectralLaw ν β)) := by
have hh := ProbabilityMeasure.tendsto_map_of_tendsto_of_continuous _ ν hweak
(show Continuous (fun x : ℝ => β*x) from continuous_const.mul continuous_id)
apply hh.congr
intro k
apply ProbabilityMeasure.toMeasure_injective
exact empiricalSpectralLaw_map _ _ _ (measurable_const.mul measurable_id)
end InvariantIsing
end
end OAI
|