| import Mathlib |
| import OAI.Probability.SKBarriers.Replicas.MatrixSKGuerra |
| import OAI.Probability.SKBarriers.Scalar.VectorMultiplier |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Set |
| namespace SK.Analytic |
|
|
| theorem matrixConstrainedPressure_quadratic_gap {N d r k : ℕ} (hN : 0<N) |
| (hr : 0<r) (β : ℝ) (D A : Fin d → Fin d → ℝ) |
| (hD : Nonempty (MatrixStates N d D)) |
| (B : Fin (k+1) → Fin r → Fin d → ℝ) |
| (w : Fin (k+1) → ℝ) (hw : ∀ b, 0 ≤ w b) (hs : ∑ b, w b=1) |
| (hO : ∀ s : Config d, |matrixSiteWeight A (fun a u => spin (a u)) s|≤1) : |
| let V := matrixSiteVector β B (fun (a : Config d) u => spin (a u)) |
| let O := matrixSiteWeight A (fun (a : Config d) u => spin (a u)) |
| let f := siteValueTerminal (fun _ : Config d => 0) |
| let g := vectorHierarchyAverage ((k+1)*r) (siteBlockMass r k w) V f |
| (affineMoment (fun _ : Config d => 0) (fun s => ContinuousLinearMap.proj s) O) 0 |
| matrixConstrainedPressure N d β D ≤ |
| vectorHierarchy ((k+1)*r) (siteBlockMass r k w) V f 0+ |
| (β^2/4)*(matrixSquare d D-2*matrixInner d (factorPath d r k B (Fin.last k)) D+ |
| ∑ j, w j*matrixSquare d (factorPath d r k B j))- |
| (g-matrixInner d A D)^2/2 := by |
| dsimp only |
| let V := matrixSiteVector β B (fun (a : Config d) u => spin (a u)) |
| let O := matrixSiteWeight A (fun (a : Config d) u => spin (a u)) |
| let f := siteValueTerminal (fun _ : Config d => 0) |
| let g := vectorHierarchyAverage ((k+1)*r) (siteBlockMass r k w) V f |
| (affineMoment (fun _ : Config d => 0) (fun s => ContinuousLinearMap.proj s) O) 0 |
| let L := matrixInner d A D-g |
| have hm (i : Fin ((k+1)*r)) : siteBlockMass r k w i∈Icc (0:ℝ) 1 := by |
| refine ⟨Finset.sum_nonneg (fun b _ => ?_),?_⟩ |
| · split_ifs; exact hw b; rfl |
| · calc |
| siteBlockMass r k w i ≤ ∑ b, w b := Finset.sum_le_sum (fun b _ => by |
| change (if _ then w b else 0) ≤ w b |
| split_ifs; rfl; exact hw b) |
| _ = 1 := hs |
| have HB := matrixConstrainedPressure_le hN hr β D (fun u t => L*A u t) hD B w hw hs |
| have he : matrixSiteWeight (fun u t => L*A u t) (fun (a : Config d) u => spin (a u))= |
| fun s => 0+L*O s := by |
| funext s |
| simp only [O,matrixSiteWeight,zero_add,Finset.mul_sum,mul_assoc] |
| have hi : matrixInner d (fun u t => L*A u t) D=L*matrixInner d A D := by |
| simp only [matrixInner,Finset.mul_sum,mul_assoc] |
| rw [he,hi] at HB |
| have HM := vectorHierarchy_bounded_multiplier ((k+1)*r) (siteBlockMass r k w) hm |
| V (fun _ : Config d => 0) O hO L |
| change _ ≤ vectorHierarchy ((k+1)*r) (siteBlockMass r k w) V f 0+L*g+L^2/2 at HM |
| change matrixConstrainedPressure N d β D ≤ _ |
| dsimp only [L] at HB HM |
| dsimp only [V,f,g] at HB HM ⊢ |
| nlinarith |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
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|
