Sho Matsumoto Graduate School of Mathematics, Nagoya University. Tomoyuki Shirai Institute of Mathematics for Industry, Kyushu University

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1 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, ]

2 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 )} 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

3 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.

4 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 )]

5 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 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

7 V = V N N λ,..., λ N f N (z) := ( ) N det U det(zi V ) det(i zv ) = ( )N det U N k= z λ k zλ k N k/ f N (z) }{{} d det( G j z j ) j=0 }{{} G j k k i.i.d. Ginibre U k = f(z) {X k (t)} k=0 Ornstein-Uhlenbeck i.i.d. f t (z) = X k (t)z k k=0 OU f t ξ t t ξ t Krishnapur ξ t [] C. W. Borchardt, Bestimmung der symmetrischen Verbindungen vermittelst ihrer erzeugenden Funktion, Crelle s Journal 53 (855), [] A. Borodin and C. D. Sinclair, The Ginibre ensemble of real random matrices and its scaling limits, Comm. Math. Phys. 9 (009), no., [3] E. R. Caianiello, On quantum field theoy I, Nuovo Cimento (9) 0 (953), [4] A. Edelman and E. Kostlan, How many zeros of a random polynomial are real?, Bull. Amer. Math. Soc. 3 (995), 37. [5] P. J. Forrester, The limiting Kac random polynomial and truncated random orthogonal matrices, J. Stat. Mech. (00), P08. available at arxiv: v [6] P. J. Forrester and T. Nagao, Eigenvalue statistics of the real Ginibre ensemble, Phys. Rev. Lett. 99, (007), , 4 pp. [7] J. M. Hammersley, The zeros of a random polynomial, Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, , vol. II, pp. 89. University of California Press, Berkeley and Los Angeles, 956.

8 [8] J. B. Hough, M. Krishnapur, Y. Peres and B. Virág, Zeros of Gaussian Analytic Functions and Determinantal Point Processes, University Lecture Series, 5. American Mathematical Society, Providence, RI, 009. [9] M. Ishikawa, H. Kawamuko, and S. Okada, A Pfaffian-Hafnian analogue of Borchardt s identity, Electron. J. Combin. (005), Note 9, 8 pp. (electronic). [0] M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49 (943), [] M. Krishnapur, From random matrices to random analytic functions, Ann. Probab. 37 (009), [] B. F. Logan and L. A. Shepp, Real zeros of random polynomials. II, Proc. London Math. Soc. 8 (968), [3] P. McKean, Geometry of differential space, Ann. Probab. (973), [4] S. Matsumoto and T. Shirai, Correlation functions for zeros of a Gaussian power series and Pfaffians, available at [5] Y. Peres and B. Virág, Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process, Acta Math. 94 (005), 35. [6] H. Poincaré, Calcul des Probabilités, Gauthier-Villars, Paris, 9. [7] S. O. Rice, Mathematical theory of random noise, Bell. System Tech. J. 5 (945), [8] L. A. Shepp and R. J. Vanderbei, The complex zeros of random polynomials, Trans. Amer. Math. Soc. 347 (995), [9] T. Shirai, Limit theorem for random analytic functions and their zeros, RIMS Kôkyûroku Bessatsu 34 (0), [0] R. Tribe and O. Zaboronski, Pfaffian formulae for one dimensional coalescing and annihilating systems, Electronic Journal of Probability, 6, Article 76 (0). [] T. Umeda, CAPELLI 009, available at [] A. Zvonkin, Matrix integrals and map enumeration: an accessible introduction, Combinatorics and physics (Marseilles, 995), Math. Comput. Modelling 6 (997), no. 8 0, [3] K. Zyczkowski and H. J. Sommers, Truncations of random unitary matrices, J. Phys. A, 33 (000),

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