| import Mathlib |
| import OAI.Probability.SKBarriers.Coverage.PressureGapFiniteInf |
| import OAI.Probability.SKBarriers.Replicas.MatrixSKGuerra |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
|
|
| def replicaGram {n d : ℕ} (s : ReplicaConfig n d) (a b : Fin d) : ℝ := overlap (s a) (s b) |
|
|
| def matrixReplicaSet (n d : ℕ) (D : Fin d → Fin d → ℝ) : Finset (ReplicaConfig n d) := by |
| classical |
| exact Finset.univ.filter (fun s => replicaGram s=D) |
|
|
| @[simp] theorem mem_matrixReplicaSet {n d : ℕ} (D : Fin d → Fin d → ℝ) (s : ReplicaConfig n d) : |
| s∈matrixReplicaSet n d D ↔ replicaGram s=D := by |
| classical |
| simp [matrixReplicaSet] |
|
|
| def matrixReplicaEquiv (n d : ℕ) (D : Fin d → Fin d → ℝ) : |
| MatrixStates n d D ≃ {s : ReplicaConfig n d |
| toFun s := ⟨fun a i => s.val i a,(mem_matrixReplicaSet D _).mpr s.property⟩ |
| invFun s := ⟨fun i a => s.val a i,(mem_matrixReplicaSet D _).mp s.property⟩ |
| left_inv _ := rfl |
| right_inv _ := rfl |
|
|
| theorem matrixReplicaSet_nonempty_iff {n d : ℕ} (D : Fin d → Fin d → ℝ) : |
| (matrixReplicaSet n d D).Nonempty ↔ Nonempty (MatrixStates n d D) := by |
| rw [← Finset.nonempty_coe_sort] |
| exact (matrixReplicaEquiv n d D).nonempty_congr.symm |
|
|
| theorem restrictedPressure_matrixReplicaSet (n d : ℕ) (β : ℝ) (D : Fin d → Fin d → ℝ) : |
| restrictedPressure β (matrixReplicaSet n d D)=matrixConstrainedPressure n d β D := by |
| classical |
| unfold restrictedPressure matrixConstrainedPressure |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards [] with J |
| unfold restrictedLogPartition restrictedPartition replicaEnergy |
| congr 1 |
| rw [← Finset.sum_coe_sort] |
| exact ((matrixReplicaEquiv n d D).sum_comp (fun s => Real.exp (β*∑ a,hamiltonian J (s.val a)))).symm |
|
|
| end SK.Analytic |
|
|
| end |
| end |
|
|
| end OAI |
|
|
