We analyse statistics of the real eigenvalues of gl(N, R)-valued Brownian motion (the Ginibre evolution) in the limit of large N. In particular, we calculate the limiting two-time correlation function of spin variables associated with real eigenvalues of the Ginibre evolution. We also show how the formalism of spin variables can be used to compute the fixed time correlation functions of real eigenvalues discovered originally by Forrester and Nagao [“Eigenvalue statistics of the real Ginibre ensemble,” Phys. Rev. Lett. 99(5), 050603 (2007)] and Borodin and Sinclair [“The Ginibre ensemble of real random matrices and its scaling limits,” Commun. Math. Phys. 291(1), 177–224 (2009)].
REFERENCES
1.
G.
Akemann
, M. J.
Phillips
, and H. J.
Sommers
, “Characteristic polynomials in real Ginibre ensembles
,” J. Phys. A: Math. Theor.
42
(1
), 012001
(2009
).2.
G.
Akemann
, J.
Baik
, and P.
Di Francesco
, The Oxford Handbook of Random Matrix Theory
(Oxford University Press
, 2011
).3.
G. W.
Anderson
, A.
Guionnet
, and O.
Zeitouni
, An Introduction to Random Matrices
(Cambridge University Press
, 2010
), Vol. 118
.4.
N.
Berline
, E.
Getzler
, and M.
Vergne
, Heat Kernels and Dirac Operators
(Springer-Verlag
, 2004
).5.
A.
Borodin
and C. D.
Sinclair
, “The Ginibre ensemble of real random matrices and its scaling limits
,” Commun. Math. Phys.
291
(1
), 177
–224
(2009
).6.
F.
Bouchet
, “Stochastic process of equilibrium fluctuations of a system with long-range interactions
,” Phys. Rev. E
70
(3
), 036113
(2004
).7.
N. G.
De Bruijn
, “On some multiple integrals involving determinants
,” J. Indian Math. Soc
19
, 133
–151
(1955
).8.
J. E.
Bunder
, K. B.
Efetov
, V. E.
Kravtsov
, O. M.
Yevtushenko
, and M. R.
Zirnbauer
, “Superbosonization formula and its application to random matrix theory
,” J. Stat. Phys.
129
(5-6
), 809
–832
(2007
).9.
B. A.
Dubrovin
, A. T.
Fomenko
, S. P.
Novikov
, and R. G.
Burns
, Modern Geometry - Methods and Applications: Part II: The Geometry and Topology of Manifolds
, Graduate Texts in Mathematics
(Springer
, 1991
).10.
A.
Edelman
, E.
Kostlan
, M.
Shub
et al., “How many eigenvalues of a random matrix are real?
,” J. Am. Math. Soc.
7
(1
), 247
(1994
).11.
12.
P. J.
Forrester
and T.
Nagao
, “Eigenvalue statistics of the real Ginibre ensemble
,” Phys. Rev. Lett.
99
(5
), 050603
(2007
).13.
R. J.
Glauber
, “Time-dependent statistics of the Ising model
,” J. Math. Phys.
4
, 294
(1963
).14.
H.
Chandra
, “Differential operators on a semisimple Lie algebra
,” Am. J. Math.
79
, 87
–120
(1957
).15.
S.
Helgason
, Differential Geometry and Symmetric Spaces
(American Mathematical Society
, 1962
).16.
A. S.
Householder
, “Unitary triangularization of a nonsymmetric matrix
,” J. Assoc. Comput. Mach.
5
(4
), 339
–342
(1958
).17.
C.
Itzykson
and J. M.
Drouffe
, Statistical Field Theory: Volume 1, From Brownian Motion to Renormalization and Lattice Gauge Theory
, Cambridge Monographs on Mathematical Physics
(Cambridge University Press
, 1991
).18.
C.
Itzykson
and J. B.
Zuber
, “The planar approximation. II
,” J. Math. Phys.
21
, 411
(1980
).19.
M.
Kieburg
and T.
Guhr
, “A new approach to derive Pfaffian structures for random matrix ensembles
,” J. Phys. A: Math. Theor.
43
, 135204
(2010
).20.
T. O.
Masser
and D.
Ben-Avraham
, “Method of intervals for the study of diffusion-limited annihilation, A + A → 0
,” Phys. Rev. E
63
(6
), 066108
(2001
).21.
M. L.
Mehta
, Matrix Theory: Selected Topics and Useful Results
, Editions de Physique
, Orsay, France
(1989
).22.
M. L.
Mehta
, Random Matrices
, Pure and Applied Mathematics
(Academic Press
, San Diego
, 2004
), Vol. 142
.23.
M. L.
Mehta
and A.
Pandey
, “On some Gaussian ensembles of Hermitian matrices
,” J. Phys. A: Math. Gen.
16
(12
), 2655
(1982
).24.
A.
Pandey
and M. L.
Mehta
, “Gaussian ensembles of random Hermitian matrices intermediate between orthogonal and unitary ones
,” Commun. Math. Phys.
87
, 449
–468
(1983
).25.
H. J.
Sommers
, “Symplectic structure of the real Ginibre ensemble
,” J. Phys. A: Math. Theor.
40
(29
), F671
(2007
).26.
H. J.
Sommers
and B. A.
Khoruzhenko
, “Schur function averages for the real Ginibre ensemble
,” J. Phys. A: Math. Theor.
42
(22
), 222002
(2009
).27.
H. J.
Sommers
and W.
Wieczorek
, “General eigenvalue correlations for the real Ginibre ensemble
,” J. Phys. A: Math. Theor.
41
(40
), 405003
(2008
).28.
29.
M.
Stone
, “Supersymmetry and the quantum mechanics of spin
,” Nucl. Phys. B
314
, 557
(1989
).30.
R.
Tribe
, J.
Yip
, and O.
Zaboronski
, “One dimensional annihilating and coalescing particle systems as extended Pfaffian point processes
,” Electron. Commun. Probab.
17
(40
), 1
–7
(2012
).31.
R.
Tribe
and O.
Zaboronski
, “Pfaffian formulae for one dimensional coalescing and annihilating systems
,” Electron. J. Probab.
16
(76
), 2080
–2103
(2011
).32.
R.
Tribe
and O.
Zaboronski
, “Multi-time correlation functions for the real Ginibre evolution
” (unpublished).33.
M.
Zirnbauer
, “Riemannian symmetric superspaces and their origin in random matrix theory
,” J. Math. Phys.
37
, 4986
(1996
).34.
As conjectured by Yan Fyodorov during an after-seminar discussion.
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