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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 | 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 // s∈matrixReplicaSet n d D} where
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
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