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c4cdbbd | 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 77 78 | import OAI.Probability.InvariantIsing.Arrays.TensorContactAsymptotic
import OAI.Probability.InvariantIsing.Arrays.TensorZeroFieldPressure
namespace OAI
/-! The finite-spectrum pressure upper bound against a fixed finite-step
trial. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter
open scoped BigOperators Topology
namespace InvariantIsing
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
end InvariantIsing
end
end OAI
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