Sho Matsumoto Graduate School of Mathematics, Nagoya University Tomoyuki Shirai Institute of Mathematics for Industry, Kyushu University. Kac f n (t) = n k=0 a kt k ({a k } n k=0 i.i.d. ) N n E[N n ] = π R (t ) (n + ) t n (t n+ ) dt Littlewood-Offord n E[N n ] π log n [0] [4] Logan- Shepp {a k } α [] Shepp-Vanderbei f n (t) [8] Kac n = f(z) = a k z k ( z < ) [4] [4] k=0 {ζ k } k=0 Peres-Virág [5] f C (z) = k=0 ζ kz k Bergman Krishnapur (Ginibre ) [](7 ) ( ) Peres-Virág [9]. * RIMS, Dec. 8-, 0 * (Hafnian) E. R. Caianiello Hafnia [3, ]
n n B = (b ij ) i,j n Pf B n n A = (a ij ) i,j n Hf A Pf B = η F n ϵ(η)b η()η() b η(3)η(4) b η(n )η(n) Hf A = η F n a η()η() a η(3)η(4) a η(n )η(n) ϵ(η) η F n := {η S n η(i ) < η(i)(i =,,..., n), η() < η(3) < < η(n )}.. n =, F, F F = Pf Hf {}, F = 0 b = b b 0, Pf a a = a a a, Hf { 3 4, 3 4 3 4, 3 4 ( 3 )} 4 4 3 0 b b 3 b 4 b 0 b 3 b 4 b 3 b 3 0 b 34 = b b 34 b 3 b 4 + b 4 b 3 b 4 b 4 b 34 0 a a a 3 a 4 a a a 3 a 4 a 3 a 3 a 33 a 34 = a a 34 + a 3 a 4 + a 4 a 3 a 4 a 4 a 34 a 44 a ii, i =,,..., n (permanent) det A = n sgn(η) a iη(i), η S n i= per A = n η S n i= a iη(i) Borchardt 3. A n B n C n : det B = (Pf B). det A = ( ) n(n )/ O A Pf A T. O
Pf(C T BC) = det C Pf B. (Wick ). (X, X,..., X n ) 0 () E[X X X n ] = Hf(E[X i X j ]) n i,j=. (Z,..., Z n, W,..., W n ) 0 E[Z Z n W... W n ] = per ( E[Z i W j ] ) n i,j=.. n = Wick E[X X X 3 X 4 ] = E[X X ]E[X 3 X 4 ] + E[X X 3 ]E[X X 4 ] + E[X X 4 ]E[X X 3 ]. X = X = X 3 = X 4 = X E[X 4 ] = 3(E[X ]) Wick () X i = X (i =,,..., n) E[X n ] = F n (E[X ]) n F n = E[X n ]/(E[X ]) n = (n )!! Borchardt(855) [] ( det s i t j ) ( per s i t j ) = det ( s i t j ) 3 ([9]). ( si t j Pf s i t j ) ( Hf ) = Pf s i t j si t j ( s i t j ). f(z) 3. f(z) {f(t), t (, )} σ(s, t) := E[f(s)f(t)] = st f. f n ρ n (t,..., t n ) t, t,..., t n (, ) ρ n (t,..., t n ) = π n Pf(K(t i, t j )) i,j n.
K(s, t) (s, t (, )) Pf(K(t i, t j )) i,j n n n (K(t i, t j )) i,j n K(s, t) K(s, t) = ( s t K s K ) σ(s, t) t K, K (s, t) = sgn(t s) arcsin. K (s, t) σ(s, s)σ(t, t) sgn t t > 0 sgn t = + t < 0 sgn t = t = 0 sgn 0 = 0 K(s, t) s t t K (s, t) = ( s )( t )( st), K (s, t) = K (s, t) = s st, s t st, K ( s )( t = sgn(t s) arcsin ) st. 3. : s (, ) : s, t (, ) ρ (s) = π K (s, s) = π( s ). ρ (s, t) = π {K (s, s)k (t, t) K (s, t)k (s, t) + K (s, t)k (s, t)} = π( s ) 3 t s + O( t s ). s, t (, ) 4. ρ (s, t) ρ (s)ρ (t). t, t,..., t n (, ) E[ f(t )f(t ) f(t n ) ] = Σ = (σ(t i, t j )) i,j n n/ (det Σ) Pf(K(ti, t j )) i,j n π E[ f(t )f(t ) f(t n ) ] f(t) E[sgn f(t ) sgn f(t n )] n E[sgn f(t ) sgn f(t n )]
3. t, t,..., t n (, ) E[sgn f(t ) sgn f(t ) sgn f(t n )] =. 3 n sgn(t j t i ) Pf(K (t i, t j )) i,j n π i<j n E[sgn f(t ) sgn f(t ) sgn f(t n )] = Pf(E[f(t i )f(t j )]) i,j n Wick 5. f(z) 4. D + = {z C; z <, Iz > 0} z,..., z n D + f(z) n ρ c n(z,..., z n ) = (π ) n n j= z j Pf(Kc (z i, z j )) i,j n K c (z, w) K c (z, w) = ( z w ( zw) z w ( zw) z w ( z w) z w ( z w) ) 4. ρ c (z) = z z π z ( z ), ( ρ c (z, w) = ρ c (z)ρ c z w (w) + π z w zw z w z w z, w D + ρ c (z, w) < ρ c (z)ρ c (w). 4 Forrester [5] Hammersley [7, 8] 3 )
6. i.i.d. N R (0, ) N N N [, 6]. K Gin (s, t) = ( s t K s K ) t K. K (s, t) K (s, t) = sgn(t s) e x / dx. s t π λ > 0 annihilating B.M. {B λ (t), t > 0} Maximal entrance law annihilating B.M. {B λ (t), t > 0} λ t > 0 ( ) [0] K abm t (x, y) = x y K Gin t, t t K(s, t) = ( s t K s K ) t K K (s, t) K 7. (X,..., X n ) n (n )- S n ( n) k (X,..., X k ) d N(0, I k ) Poincaré [6] McKean [3] S ( ) Zyczkowski-Sommers [3] Krishnapur Peres-Virág [] Haar (k + N) (k + N) ( ) Ak k B U = k N C N k V N N
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