| import OAI.Probability.InvariantIsing.Arrays.TensorContactAsymptotic |
| import OAI.Probability.InvariantIsing.Arrays.TensorZeroFieldPressure |
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| namespace OAI |
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| /-! The finite-spectrum pressure upper bound against a fixed finite-step |
| trial. -/ |
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| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter |
| open scoped BigOperators Topology |
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| namespace InvariantIsing |
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| theorem finiteSpectrum_meanPressure_step_upper |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) |
| (N : ℕ → ℕ) (hN : ∀ k, 3 ≤ N k) (hNlim : Tendsto N atTop atTop) (m n : ℕ) |
| (μ : (k : ℕ) → Measure (SpecialOrthogonal (N k))) [∀ k, IsProbabilityMeasure (μ k)] |
| (hμinv : ∀ k, (μ k).IsMulLeftInvariant) |
| (eig : (k : ℕ) → Fin (N k) → ℝ) |
| (K : ℝ) (hK : 0 < K) (heig : ∀ k i, |eig k i| ≤ K) |
| (I : (k : ℕ) → Fin m → Finset (Fin (N k))) |
| (hdis : ∀ k, Set.PairwiseDisjoint (Set.univ : Set (Fin m)) (I k)) |
| (hcover : ∀ k, Finset.univ.biUnion (I k) = Finset.univ) |
| (lam : Fin m → ℝ) (hlam : ∀ k a i, i ∈ I k a → eig k i = lam a) |
| (ρ : Fin m → ℝ) (hρpos : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) |
| (hρ : Tendsto (fun k a => ((I k a).card : ℝ) / N k) atTop (𝓝 ρ)) |
| (cut : Fin (n + 2) → ℝ) (hc : StrictMono cut) |
| (hfirst : cut 0 = 0) (hlast : cut (Fin.last (n + 1)) = 1) |
| (trial : OverlapPath) (values : Fin (n + 1) → ℝ) |
| (hvalues : ∀ i s, s ∈ Ioo (cut i.castSucc) (cut i.succ) → trial s = values i) |
| (d : ℝ) (hd : 0 < d) (htrial : ∀ᵐ s ∂pathMeasure, trial s ≤ 1 - d) |
| (S : ℝ) (hS : entropyFunctional trial ≤ (S : EReal)) : |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, |
| (∫ U : SpecialOrthogonal (N k), |
| rotatedPressure (eig k) (specialRotation U) (fun _ => 0) ∂μ k) ≤ |
| S + spectralFunctional (finiteR ρ lam hρpos hρsum) trial + ε := by |
| intro ε hε |
| have hthird : 0 < ε / 3 := div_pos hε (by norm_num) |
| obtain ⟨H, hH, hevent⟩ := tensorContact_eventually_nonnegative hhaar hgauss |
| N hN hNlim m n μ hμinv eig K hK heig I hdis hcover lam hlam ρ hρpos hρsum hρ |
| cut hc hfirst hlast trial values hvalues d hd htrial (ε / 3) S hthird hS |
| have hcost : ∀ᶠ k in atTop, |
| 2 * m * perturbationScale (N k) + 8 * perturbationScale (N k) ^ 2 < ε / 3 := |
| ((fullPerturbationCost_tendsto m).comp hNlim).eventually (gt_mem_nhds hthird) |
| filter_upwards [hevent (ε / 3) hthird, hcost] with k hk hck |
| let p : TensorContactParameter (N k) m n := |
| (1, (fun _ => 0), (fun _ => 3 / 2), (fun _ => 3 / 2)) |
| have hp : p ∈ tensorContactRegion (N k) m n H := by |
| apply tensorContactRegion_mem (by norm_num) (fun _ => le_rfl) |
| · simpa only [p, Finset.sum_const_zero] using hH.le |
| · intro j; norm_num |
| · intro a; norm_num |
| have hj := hk p hp |
| have hz := tensorContactPressure_one_zero_field_cost hhaar hgauss (hN k) (μ k) (hμinv k) |
| (eig k) (fun _ => 0) (I k) (fun _ => 3 / 2) (by intro j; norm_num) |
| (fun _ => 3 / 2) (by intro j; norm_num) n (chainExponent cut) |
| (chainExponent_admissible hc hfirst hlast) |
| have hg : finiteTemperatureFunctional ρ lam hρpos hρsum trial 1 = |
| spectralFunctional (finiteR ρ lam hρpos hρsum) trial := by |
| simpa only [one_mul] using finiteTemperatureFunctional_eq ρ lam hρpos hρsum trial zero_le_one |
| have hzero : finiteFieldPath (fun _ : Fin (n + 1) => (0 : ℝ)) = fun _ => 0 := by |
| funext i |
| simp only [finiteFieldPath, ite_self, Finset.sum_const_zero] |
| simp only [tensorContactObjective, p, hzero, mul_zero, Finset.sum_const_zero, zero_div, |
| sub_zero, one_mul, sub_self, zero_pow (by norm_num : 2 ≠ 0), add_zero, hg] at hj |
| have hdiff := neg_le_abs (tensorContactPressure (μ k) (eig k) (fun _ => 0) (I k) |
| (chainExponent cut) p - ∫ U : SpecialOrthogonal (N k), |
| rotatedPressure (eig k) (specialRotation U) (fun _ => 0) ∂μ k) |
| change |tensorContactPressure (μ k) (eig k) (fun _ => 0) (I k) (chainExponent cut) p - _| ≤ _ at hz |
| linarith |
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| end InvariantIsing |
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| end |
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| end OAI |
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