| import OAI.Probability.InvariantIsing.Pressure.LimitingPressure |
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| namespace OAI |
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| /-! Positive temperature scaling preserves the precise spectral hypotheses. -/ |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology |
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| namespace InvariantIsing |
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| def scaledSpectralLaw (ν : ProbabilityMeasure ℝ) (β : ℝ) : ProbabilityMeasure ℝ := |
| ν.map (fun x => β*x) |
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| 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⟩ |
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| 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) |
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| 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) |
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| 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⟩ |
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| 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) |
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| end InvariantIsing |
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| end |
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| end OAI |
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