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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 | import OAI.Probability.InvariantIsing.Magnetic.MagneticDiagonalSelection
import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockTrialLower
namespace OAI
/-! Weak tests on growing physical blocks yield the magnetic variational trial lower bound. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron Filter
open scoped Topology BigOperators BoundedContinuousFunction
namespace InvariantIsing
theorem magnetic_block_diagonal_trial_lower {m : ℕ}
(ρ lam : Fin m → ℝ) (hρ : ∀ a, 0 < ρ a) (hsum : ∑ a, ρ a=1)
{K : ℝ} (hK : 0 ≤ K) (hlam : ∀ a, |lam a| ≤ K)
{A : Type*} [Fintype A] [DecidableEq A]
(N : ℕ → ℕ) (hN : ∀ r, 0 < N r) (hNlim : Tendsto N atTop atTop)
(group : ∀ r, Fin (N r) → A) (k : ℕ → A → ℕ)
(hk : ∀ r a, k r a ≤ spinGroupSize (group r) a)
(γ mag : A → ℝ) (hγ : ∀ a, 0 ≤ γ a) (hγsum : ∑ a, γ a=1)
(hcount : ∀ r a, (spinGroupSize (group r) a : ℝ)=N r*γ a)
{s : ℝ} (hs : s < 1) (hmag : ∀ a, |mag a| ≤ s)
(hc : ∀ r a, (k r a : ℝ)=spinGroupSize (group r) a*((1+mag a)/2))
(base : ℕ → OverlapPath)
(htest : ∀ r (Φ : ℝ →ᵇ ℝ), Tendsto (fun j => ∫ t, Φ (cavityStrictUniformPath (base r) j t)*
(restrictedBlockOverlapPath (hN r) (spinGroupSlice (group r) (k r))
(spinGroupSlice_nonempty (group r) (k r) (hk r))
(cavityStrictUniformField ρ lam hρ hsum (base r) j) t-
cavityStrictUniformPath (base r) j t) ∂pathMeasure) atTop (𝓝 0)) :
∃ d φ : ℕ → ℕ, StrictMono d ∧ StrictMono φ ∧
∀ ε > 0, ∀ᶠ r in atTop,
(magneticVariationalFunctional (finiteR ρ lam hρ hsum) γ mag).toReal-ε ≤ constrainedBlockValue (spinGroupSlice (group (φ r)) (k (φ r)))
(cavityStrictUniformField ρ lam hρ hsum (base (φ r)) (d (φ r)))+
fieldPairing (cavityStrictUniformPath (base (φ r)) (d (φ r)))
(cavityStrictUniformField ρ lam hρ hsum (base (φ r)) (d (φ r)))/2+
spectralFunctional (finiteR ρ lam hρ hsum)
(cavityStrictUniformPath (base (φ r)) (d (φ r))) := by
obtain ⟨d,φ,p,hd,hφ,hq,hself⟩ := exists_magnetic_consistent_diagonal
ρ lam hρ hsum hK hlam N hN hNlim group k hk (fun _ => mag) hs
(fun _ => hmag) hc base htest
refine ⟨d,φ,hd,hφ,?_⟩
exact magnetic_block_trial_eventually_lower ρ lam hρ hsum
(fun r => N (φ r)) (fun r => hN (φ r)) (hNlim.comp hφ.tendsto_atTop)
(fun r => group (φ r)) (fun r => k (φ r)) (fun r => hk (φ r)) γ mag hγ hγsum
(fun r => hcount (φ r)) hs hmag (fun r => hc (φ r))
(fun r => cavityStrictUniformField ρ lam hρ hsum (base (φ r)) (d (φ r)))
(fun r => cavityStrictUniformField_height_le ρ lam hρ hsum hK hlam
(base (φ r)) (d (φ r)) (Fin.last (d (φ r)))) p
(fun r => cavityStrictUniformPath (base (φ r)) (d (φ r))) hself hq
end InvariantIsing
end
end OAI
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