The real Ginibre evolution
Roger Tribe and Oleg Zaboronski
Department of Mathematics, University of Warwick
Venice, 06-10.05.2013 – p. 1/19
Plan
Review of the model
Counting real eigenvalues: spin variables
Spin correlation functions for the real Ginibre evolution
Open problems
References
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Ginibre evolutions
The Ginibre(N) evolution
is a gLN (R)-valued
Brownian motion
2.5
2
1.5
1
[Mt ]ij = Bij (t),
independent BM’s.
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
The law of M1 is called
the real Ginibre random
matrix ensemble
Venice, 06-10.05.2013 – p. 3/19
Ginibre evolutions
2.5
The Ginibre(N) evolution
is a gLN (R)-valued
Brownian motion
2
1.5
1
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Note:
Stochastic motion of a
real eigenvalue
[Mt ]ij = Bij (t),
independent BM’s.
The law of M1 is called
the Ginibre random
matrix ensemble
Venice, 06-10.05.2013 – p. 4/19
Ginibre evolutions
2.5
2
1.5
1
The Ginibre(N) evolution
is a gLN (R)-valued
Brownian motion
0.5
0
−0.5
−1
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Note:
Stochastic motion of a
real eigenvalue
’Annihilation’ of two real
eigenvalues
[Mt ]ij = Bij (t),
independent BM’s.
The law of M1 is called
the Ginibre random
matrix ensemble
Venice, 06-10.05.2013 – p. 5/19
Ginibre evolutions
2.5
2
1.5
1
0.5
0
The Ginibre(N) evolution
is a gLN (R)-valued
Brownian motion
−0.5
−1
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Note:
[Mt ]ij = Bij (t),
independent BM’s.
Stochastic motion of a
real eigenvalue
’Annihilation’ of two real
eigenvalues
Stochastic motion
complex e.v.’s
The law of M1 is called
the Ginibre random
matrix ensemble
of
Venice, 06-10.05.2013 – p. 6/19
Ginibre evolutions
2.5
The Ginibre(N) evolution
is a gLN (R)-valued
Brownian motion
2
1.5
1
0.5
0
−0.5
−1
[Mt ]ij = Bij (t),
independent BM’s.
−1.5
−2
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
Note:
The law of M1 is called
the Ginibre random
matrix ensemble
Stochastic motion of a
real eigenvalue
’Annihilation’ of two real
eigenvalues
Stochastic motion
complex e.v.’s
of
Is there an interpretation
of the evolution of real
eigenvalues in terms of
an annihilation process?
Venice, 06-10.05.2013 – p. 7/19
ABM’s and random matrices
Theorem 1 (Borodin-Sinclair, 2009) For any N = 1, 2, . . .,
the law of real eigenvalues for the real Ginibre(N) random
matrix ensemble is a Pfaffian point process.
Corollary 2
KtABM (x, y)
1
Ginibre
= √ Krr
2t
y
x
√ ,√
2t 2t
,
Ginibre is the N → ∞ limit of the kernel of the
where Krr
Pfaffian point process characterising the law of real
eigenvalues in the real Ginibre(N ) ensemble.
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Spin variables for Ginibre evolutions.
Ginibre ensemble is mathematically the hardest of all
classical random matrix ensembles.
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Spin variables for Ginibre evolutions.
Ginibre ensemble is mathematically the hardest of all
classical random matrix ensembles.
Spin variables representation makes it tractable
Venice, 06-10.05.2013 – p. 10/19
Spin variables for Ginibre evolutions.
Ginibre ensemble is mathematically the hardest of all
classical random matrix ensembles.
Spin variables representation makes it tractable
Nt (x) = #{Real eigenvalues less or equal than x}
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Spin variables for Ginibre evolutions.
Ginibre ensemble is mathematically the hardest of all
classical random matrix ensembles.
Spin variables representation makes it tractable
Nt (x) = #{Real eigenvalues less or equal than x}
St (x) = (−1)Nt (x) - spin variable
Venice, 06-10.05.2013 – p. 12/19
Spin variables for Ginibre evolutions.
Ginibre ensemble is mathematically the hardest of all
classical random matrix ensembles.
Spin variables representation makes it tractable
Nt (x) = #{Real eigenvalues less or equal than x}
St (x) = (−1)Nt (x) - spin variable
St (x) = (−1)Ñt (x) , Ñt (x) = #{Eigenvalues with real part
less or equal than x}
Venice, 06-10.05.2013 – p. 13/19
Spin variables for Ginibre evolutions.
Ginibre ensemble is mathematically the hardest of all
classical random matrix ensembles.
Spin variables representation makes it tractable
Nt (x) = #{Real eigenvalues less or equal than x}
St (x) = (−1)Nt (x) - spin variable
St (x) = (−1)Ñt (x) , Ñt (x) = #{Eigenvalues with real part
less or equal than x}
St (x) = sign (det (Mt − x))
Venice, 06-10.05.2013 – p. 14/19
Tools.
1. Berezin integral: Let A ∈ RN ×N
Z
αT Aβ
det (A) =
dαdβe
R0|2N
2. Integral representation for spin variables
Z
dxdαdβ −xT (AT A+ǫ2 I)x+αT Aβ
sign (det (A)) = lim
e
N/2
ǫ↓0 RN |2N π
3. Fyodorov formula:
Z
m
Y
RN m k=1
dVk F ({Vi · Vj }1≤i,j≤m ) =
4. Laplace method for N → ∞ limit
Z
(+)
Symm
dQdet(Q)
N −m−1
2
F (Q)
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Matrix integral representation for the product expectation of spins
Theorem 3
lim ρ̃1 (x1 , x2 , . . . , xK ) = CK |∆(x)|
N →∞
Z
µH (dU )e
− 21 T r(H−H R )
2
,
U (K)
where CK is a positive constant, H = U XU † is a Hermitian
matrix with eigenvalues x1 , x2 , . . . , xK , µH is Haar measure
on the unitary group U (K), H R = JH T J is a symplectic
involution
Q of matrix H , J is a canonical symplectic matrix,
∆(x) = i>j (xi − xj ) is the Van-der-Monde determinant.
To restore Borodin-Sinclair result:
use HCIZ theorem, then de
Brujn formula.
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The two-time correlation functions of spins for Ginibre evolutions.
Theorem 4
lim EN (St+τ (x)St (0)) = 0
N →∞
lim EN St+ T (x)St (0) = erf c
N →∞
N
r
x2
T
+
t
2t
!
ABM’s 6= Ginibre(∞) evolutions as processes.
An interesting instance of a solvable long-range system?
Observation.
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Open problems
Multi-time correlation functions for Ginibre evolutions
Dynamic characterization of the real Ginibre evolutions
in terms of SDE’s and/or an interacting particle system
(similar to Dyson BM’s)
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References
1. Multi-Scaling of the n-Point Density Function for
Coalescing Brownian Motions , CMP Vol. 268, No. 3,
December 2006;
2. Pfaffian formulae for one dimensional coalescing and
annihilating systems, arXiv Math.PR: 1009.4565; EJP,
vol. 16, Article 76 (2011)
3. One dimensional annihilating and coalescing particle
systems as extended Pfaffian point processes, ECP,
vol. 17, Article 40 (2012)
4. The Ginibre evolution in the large-N limit,
arXiv:1212.6949 [math.PR] (2012), rejected by EJP
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