File size: 2,695 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 | import OAI.Probability.InvariantIsing.Pressure.PressureLipschitz
namespace OAI
/-!
# Haar concentration for Ising pressure
Frobenius Lipschitz continuity and measurability give concentration of the
Ising pressure about a median.
-/
noncomputable section
open MeasureTheory
open scoped BigOperators
namespace InvariantIsing
lemma measurable_specialRotation_eval {N : ℕ} (v : EuclideanSpace ℝ (Fin N))
(i : Fin N) : Measurable (fun U : SpecialOrthogonal N => specialRotation U v i) := by
simp only [specialRotation_apply]
apply Finset.measurable_sum
intro j _
have he : Measurable (fun U : SpecialOrthogonal N =>
(U : Matrix (Fin N) (Fin N) ℝ) i j) :=
(measurable_pi_apply j).comp ((measurable_pi_apply i).comp measurable_subtype_coe)
exact he.mul_const (v j)
lemma measurable_rotatedPressure {N : ℕ} (eig c : Fin N → ℝ) :
Measurable (fun U : SpecialOrthogonal N => rotatedPressure eig (specialRotation U) c) := by
have hE (σ : Spin N) : Measurable (fun U : SpecialOrthogonal N =>
rotatedEnergy eig (specialRotation U) σ + fieldEnergy c σ) := by
exact ((Finset.measurable_sum _ fun i _ =>
((measurable_specialRotation_eval (spinVector σ) i).pow_const 2).const_mul
(eig i)).const_mul (1 / 2 : ℝ)).add_const _
unfold rotatedPressure logPartition
exact ((Finset.measurable_sum _ fun σ _ => (hE σ).exp).const_mul
(Fintype.card (Spin N) : ℝ)⁻¹).log.const_mul (N : ℝ)⁻¹
/-- Uniform concentration about a median for the actual spectral-orbit pressure,
conditional only on the cited concentration theorem. No pressure limit is an
assumption or a conclusion of this finite-volume estimate. -/
theorem haar_pressure_median_tail (hpub : HaarConcentrationInput) :
∃ C c₀ : ℝ, 0 < C ∧ 0 < c₀ ∧
∀ N : ℕ, 3 ≤ N →
∀ μ : Measure (SpecialOrthogonal N),
IsProbabilityMeasure μ → μ.IsMulLeftInvariant →
∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) →
∃ m : ℝ,
(1 / 2 : ℝ) ≤ μ.real {U | rotatedPressure eig (specialRotation U) c ≤ m} ∧
(1 / 2 : ℝ) ≤ μ.real {U | m ≤ rotatedPressure eig (specialRotation U) c} ∧
∀ r : ℝ, 0 < r →
μ.real {U | r ≤ |rotatedPressure eig (specialRotation U) c - m|} ≤
C * Real.exp (-c₀ * (N : ℝ) * r ^ 2 / K ^ 2) := by
obtain ⟨C, c₀, hC, hc, hp⟩ := hpub
refine ⟨C, c₀, hC, hc, ?_⟩
intro N hN μ hμ hμinv eig c K hK heig
exact hp N hN μ hμ hμinv _ (measurable_rotatedPressure eig c) K hK
(fun U V => abs_rotatedPressure_sub_rotation_le (by omega) eig U V c K hK.le heig)
end InvariantIsing
end
end OAI
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