File size: 4,728 Bytes
135052a | 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 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 | import Mathlib
import OAI.Probability.SKBarriers.Replicas.MatrixSKGuerra
import OAI.Probability.SKBarriers.Scalar.VectorBlocks
import OAI.Probability.SKBarriers.SpinGlass.FinitePrefix
namespace OAI
section
noncomputable section
open scoped BigOperators
open MeasureTheory ProbabilityTheory Set
namespace SK.Analytic
def replicaSpinMap (d : ℕ) : (Fin d → ℝ) →L[ℝ] (Config d → ℝ) :=
ContinuousLinearMap.pi (fun s => ∑ i, spin (s i) • ContinuousLinearMap.proj i)
@[simp] theorem replicaSpinMap_apply (d : ℕ) (x : Fin d → ℝ) (s : Config d) :
replicaSpinMap d x s=∑ i, spin (s i)*x i := by
simp [replicaSpinMap]
theorem replicaSpin_terminal (d : ℕ) :
siteValueTerminal (fun _ : Config d => 0) ∘ replicaSpinMap d=
fun x => ∑ i, scalarSpinTerminal (x i) := by
funext x
have H := congrFun (affineLogPartition_product d
(fun (_ : Fin d) (_ : Bool) => (0:ℝ))
(fun i b => spin b • (ContinuousLinearMap.proj i : (Fin d → ℝ) →L[ℝ] ℝ))) x
simp only [affineLogPartition_bool,ContinuousLinearMap.proj_apply] at H
simpa only [Function.comp_def,siteValueTerminal,affineLogPartition,
Finset.sum_const_zero,zero_add,replicaSpinMap_apply,sum_apply,
smul_apply,smul_eq_mul,ContinuousLinearMap.proj_apply,scalarSpinTerminal] using H
section Vector
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E]
theorem vectorBlockHierarchy_one (n : ℕ) (m : Fin n → ℝ) (v : Fin n → E) (f : E → ℝ) :
vectorBlockHierarchy 1 n m (fun i _ => v i) f=vectorHierarchy n m v f := by
induction n generalizing f with
| zero => rfl
| succ n ih =>
change vectorBlockHierarchy 1 n _ _ (vectorStep (m (Fin.last n)) (v (Fin.last n)) f)=_
exact ih _ _ _
end Vector
def rankOnePath {n : ℕ} (d : ℕ) (b : Fin n → Fin d → ℝ) (j : Fin n) : Fin d → Fin d → ℝ :=
fun u v => ∑ i, if i ≤ j then b i u*b i v else 0
theorem factorPath_rankOne {k : ℕ} (d : ℕ) (b : Fin (k+1) → Fin d → ℝ) (j : Fin (k+1)) :
factorPath d 1 k (fun i _ => b i) j=rankOnePath d b j := by
funext u v
simp [factorPath,factorCovariance,rankOnePath]
theorem rankOne_siteValue {k : ℕ} (d : ℕ) (β : ℝ) (m : Fin (k+1) → ℝ) (hm : m 0=0)
(b : Fin (k+1) → Fin d → ℝ) :
vectorHierarchy ((k+1)*1) (siteBlockMass 1 k (zeroInitialAtoms k m))
(matrixSiteVector β (fun i _ => b i) (fun (s : Config d) u => spin (s u)))
(siteValueTerminal (fun _ : Config d => 0)) 0=
vectorHierarchy (k+1) m (fun i => β • b i) (fun x => ∑ u, scalarSpinTerminal (x u)) 0 := by
rw [funext (siteBlockMass_zeroInitial m hm)]
have hev : matrixSiteVector β (fun i (_ : Fin 1) => b i) (fun (s : Config d) u => spin (s u))=
replicaSpinMap d ∘ (fun i => β • b (finProdFinEquiv.symm i).1) := by
funext i s
simp only [matrixSiteVector,Function.comp_apply,replicaSpinMap_apply,Pi.smul_apply,smul_eq_mul,Finset.mul_sum]
apply Finset.sum_congr rfl
intro u _
ring
rw [hev]
have H := congrFun (vectorHierarchy_pullback (replicaSpinMap d) ((k+1)*1)
(fun i => m (finProdFinEquiv.symm i).1)
(fun i => β • b (finProdFinEquiv.symm i).1)
(siteValueTerminal (fun _ : Config d => 0))) (0 : Fin d → ℝ)
simp only [Function.comp_apply,map_zero] at H
rw [← H,replicaSpin_terminal]
rw [vectorHierarchy_blocks (k+1) 1 m (fun i _ => β • b i),vectorBlockHierarchy_one]
theorem matrixConstrainedPressure_rankOne {N d k : ℕ} (hN : 0<N) (β : ℝ)
(D : Fin d → Fin d → ℝ) (hD : Nonempty (MatrixStates N d D))
(m : Fin (k+1) → ℝ) (hm0 : m 0=0) (hm : ∀ i,m i∈Icc (0:ℝ) 1) (hmono : Monotone m)
(b : Fin (k+1) → Fin d → ℝ) :
matrixConstrainedPressure N d β D ≤
vectorHierarchy (k+1) m (fun i => β • b i) (fun x => ∑ u,scalarSpinTerminal (x u)) 0+
β^2/4*matrixSquare d (fun i j => D i j-rankOnePath d b (Fin.last k) i j)-
β^2/4*∑ i : Fin k,m i.succ*(matrixSquare d (rankOnePath d b i.succ)-
matrixSquare d (rankOnePath d b i.castSucc)) := by
have H := matrixConstrainedPressure_le hN (by norm_num : 0<1) β D (fun _ _ => 0) hD
(fun i (_ : Fin 1) => b i) (zeroInitialAtoms k m)
(zeroInitialAtoms_nonneg m hm hmono) (zeroInitialAtoms_sum m)
have hz : matrixSiteWeight (fun (_ _ : Fin d) => (0:ℝ)) (fun (s : Config d) u => spin (s u))=fun _ => 0 := by
funext s; simp [matrixSiteWeight]
rw [hz,rankOne_siteValue d β m hm0 b] at H
simp only [matrixInner,zero_mul,Finset.sum_const_zero,sub_zero] at H
simp_rw [factorPath_rankOne] at H
rw [zeroInitialAtoms_sum_mul_parts] at H
rw [matrixSquare_sub]
unfold matrixInner
convert H using 1
ring
end SK.Analytic
end
end
end OAI
|