1 1 2 3 3
4 8 
 
 9
11 13 1 Wigner Wigner [45] GUE GUE N × N-Hermite H N P N (dX) α exp(− Tr(X 2 ))dX
Hermite X Re X ij , Im X ij  (1 ≤ i < j ≤ N ), R N 2 dX X ii (1 ≤ i ≤ N) Lebesgue H N N²
Tr( X ² )
Re X ij , Im X ij (1 ≤ i < j ≤ N ), X ii (1 ≤ i ≤ N) N H N
R N
1
[29] GUE Hermite Lebesgue n n L 2
Laguarre 1.1. Fermion α- Tracy-Widom Boson Airy 2.1 Fermion μ(11) = det K (11) = ad − bc,  μ(10) = det K (10) = a(1 − d) + bc, μ(01) = det K (01) = (1 − a)d + bc,  μ(00) = det K (00) = (1 − a)(1 − d) − bc. μ(11) + μ(10) + μ(01) + μ(00) = 1 x1x2 E {0,1}² μ(x1x2) ≥ 0 K
Fermion
(Determinantal
2
R
Q
C
Λ = Λο □ Λι 2.2. {μ(0^o1^i); Λo,Λi ⊂ R,Λo ∩ Λi = 0} Λ 0, Λ 1  C  R x E R Λ := Λ 0 □ Λ 1 µ(0^o∪{x}1^1) + µ(0^o1^1∪{x}) = µ(0^o1^1).
K = K λ, k = K (x,x)

口
2.3. 2.2 (A = K, B = I – K) |Λ| × |Λ| A, B
2.2 Kolmogorov 2.4. K 2.1
μκ(0^o1^i) = det(P^o(I^ − K^) + P^i K^)
Fermion
μ κ
2.6.
J[A] = K Λ (IΛ − K Λ )− ¹ , J[A]− ¹ = (IΛ − K Λ )K− ¹ . . I Λ − K Λ det(P Λ o (I Λ − K Λ ) + P Λ 1 K Λ )    =   det(P Λ o + P Λ 1 J[Λ]) det(I Λ − K Λ ) =   det(Pn, + Pn, J[A]Pn,] det(In – Kn) =  det(J[A]A,) det(IA - KA). K Λ (PΛ 1 − PΛ 0 )² = IΛ det( P Λ 0 (I Λ − K Λ ) + P Λ 1 K Λ ) = det( P Λ 0 − P Λ 1 ) 2 (P Λ 0 (I Λ − K Λ ) + P Λ 1 K Λ ) =  det(P Λ 1 − P Λ 0 ) det(− P Λ 0 (I Λ − K Λ ) + P Λ 1 K Λ ) =   (−1)|^o| det(K Λ − P Λ o ).口
K (2) R = Z 1 x1 < x2 < ··· < x n ||K|| ≤ 1 totally positive y1 < y2 < ··· < yn
عLaplace 2.9. K 2.1
口2.10. 〈ξ, f〉
口2.11. (1) rank K Λ = n μ(ξ(Λ) ≤ n) = 1. μ(ξ(Λ) ≥ rank(I Λ − K Λ ) ≤ m Λ N − m) = 1. 1 ξ(Λ) = ⟨ξ, 1Λ⟩ (2) rank K = n μ(ξ(R) = n) = 1 K .  rank K Λ = n
μ(ξ(Λ) ≥ n + 1) = 0 2.12.
λι, . . . , λπ
2 μ(ξ(Λ) = 0) = det(ΙΛ − ΚΛ). Fermion Fermion
Fermion 2.13. A
K(x, ·) = (K(x,xi))n=1, K( ,y) = (K(xi,y))n=1 7 2.14. {0,1} R μ
μ 2.2
口
2.3
Fermion
K Hermite 2.15. (1) A, B
(2)
口2.16. K Toeplitz
[24, 25, 36] Q R {0, 1}- ξ ∈ A η ∈ A $ ≤ η increasing A Q A increasing μ(Α) ≤ ν(Α) A ∈ B(Q) Q μι, V μ ≤ ν μ V supp P ⊂ {(ξ,η) ∈ Q×Q; ξ ≤ η} 2.17 ([38]). μ,ν μ ≤ ν P
Fermion ll k 2.18. 0 ≤ k 1 ≤ k 2 ≤ 1 6 k1 k 1 = pk 2 (p, 1 − p)- || pk 2.19. 0 ≤ k 1 ≤ k 2 ≤ 1 μκ, ≤ μκ, Pk 2 l l k 1 P k
0 ≤ p ≤ 1 Q min μ ξ 1 || ||
3
Fermion
Boson
3.1 R R Radon B(Q)
μι
λ(dx) Radon Q = Q(R) ) , δ x, Q
Fermion
ξ ∈ Q (Q,B(Q),μ)
$1 S 3.1. fn E Cc(Rn)
R n n
(1.2) 2.7 Toeplitz [24]
Laplace 2.9 K
Fredholm n
2
3.3. Fermion
1 K 3.4. f R 10 Det(I − K ϕ) = Det(I − K ∧ ) Det(I + (J[Λ]) ε− s )
口
N 3.5. 1 R Fermion K Airy (x,y) K Bessel ( x, y ) 1
-∞, 0)
Poisson K sinc 3.2
Boson
α-Boson
Laplace Fermion Boson 3.8 ([35]). L 2 (R,dX) K K ≥ O
Laplace 3.9. L 2 (R, dλ) K
K ≥ O Laplace
Q = Q(R)
d(σ) σ
α = ±1 Fermion Poisson Ha,K Boson
R
Laplace α > 0
Fredholm (1 − x)− α
µ∧,α,κ(dξ) exp(− 〈ξ, ƒ〉) = Det( I + αΚφ)− 1/α JQ(A) Fermion
Pa,K
2
Pa,K K ≥ O
α = ±1 (per A ≥ det A ≥ 0, ∀A ≥ 0) α ∈ {±1/m, ; m ∈ N} //±1,K //±1,K Laplace //±1/m,K 3.12.
α ∈ {−1/m ; m ∈ N} ∪ {0} ∪ {2/m ; m ∈ N} α ∈ {−1/m ; m ∈ N} 3.3
Gauss
Boson
α = 2 Gauss 3.13. {X(x)}xer 0, K Gauss E[Π χ 2 (dξ)] = μ 2 ,κ (dξ) E
Poisson Π χ 2 intensity X(x)² · λ(dx) Q . (3.1) Laplace
4
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