Tiểu Luận Multi-critical unitary random matrix ensembles and the general Painlev e II equation

Multi-critical unitary random
matrix ensembles and the
general Painleve II equation
By T. Claeys, A.B.J. Kuijlaars, and M. Vanlessen
Abstract
We study unitary random matrix ensembles of the form
Z
1
n;N
j det M j
2
e
N Tr V (M)
dM;
where  > 1=2 and V is such that the limiting mean eigenvalue density for
n;N ! 1 and n=N ! 1 vanishes quadratically at the origin. In order to
compute the double scaling limits of the eigenvalue correlation kernel near
the origin, we use the Deift/Zhou steepest descent method applied to the
Riemann-Hilbert problem for orthogonal polynomials on the real line with
respect to the weight jxj
2
e
NV (x)
. Here the main focus is on the construction
of a local parametrix near the origin with -functions associated with a special
solution q of the Painleve II equation q
00
= sq + 2q
3
. We show that q
has no real poles for  > 1=2, by proving the solvability of the corresponding
Riemann-Hilbert problem. We also show that the asymptotics of the recurrence
coecients of the orthogonal polynomials can be expressed in terms of q in
the double scaling limit.
1. Introduction and statement of results
1.1. Unitary random matrix ensembles. For n 2 N, N > 0, and  > 1=2,
we consider the unitary random matrix ensemble
(1.1) Z
1
n;N
j det M j
2
e
N Tr V (M)
dM;
on the space of n n Hermitian matrices M , where V : R ! R is a real analytic
function satisfying
(1.2) lim
x!1
V (x)
log(x
2
+ 1)
= +1:
Because of (1.2) and  > 1=2, the integral
(1.3) Zn;N =
Z
j det M j
2
e
N Tr V (M)
dM
602 T. CLAEYS, A.B.J. KUIJLAARS, AND M. VANLESSEN
converges and the matrix ensemble (1.1) is well- dened. It is well known, see
for example [11], [36], that the eigenvalues of M are distributed according to
a determinantal point process with a correlation kernel given by
(1.4) Kn;N
(x; y) = jxj e
N
2
V (x)
jyj e
N
2
V (y)
n 1 X
k=0
p
k;N
(x)p
k;N
(y);
where p
k;N = 
k;N
x
k
+


, 
k;N > 0, denotes the k-th degree orthonormal
polynomial with respect to the weight jxj
2
e
NV (x)
on R.
Scaling limits of the kernel (1.4) as n;N ! 1, n=N ! 1, show a remark-able universal behavior which is determined to a large extent by the limiting
mean density of eigenvalues
(1.5) V
(x) = lim
n!1
1
n
Kn;n
(x; x):
Indeed, for the case  = 0, Bleher and Its [5] (for quartic V ) and Deift et al.
[16] (for general real analytic V ) showed that the sine kernel is universal in the
bulk of the spectrum, i.e.,
lim
n!1
1
n
V
(x
0
)
Kn;n

x
0 +
u
n
V
(x
0
)
; x
0 +
v
n
V
(x
0
)

=
sin (u v)
(u v)
whenever V
(x
0
) > 0. In addition, the Airy kernel appears generically at
endpoints of the spectrum. If x
0
is a right endpoint and V
(x)  (x
0 x)
1=2
as x ! x
0 , then there exists a constant c > 0 such that
lim
n!1
1
cn
2=3
Kn;n

x
0 +
u
cn
2=3
; x
0 +
v
cn
2=3

=
Ai (u)Ai
0
(v) Ai
0
(u)Ai (v)
u v
;
where Ai denotes the Airy function; see also [13].
The extra factor j det M j
2
in (1.1) introduces singular behavior at 0 if
6 = 0. The pointwise limit (1.5) does not hold if V
(0) > 0, since Kn;n
(0; 0) =
0 if  > 0 and Kn;n
(0; 0) = +1 if  < 0, due to the factor jxj jyj in (1.4).
However (1.5) continues to hold for x 6 = 0 and also in the sense of weak

convergence of probability measures
1
n
Kn;n
(x; x)dx

! V
(x)dx; as n ! 1.
Therefore we can still call V
the limiting mean density of eigenvalues. Observe
that V
does not depend on .
However, at a microscopic level the introduction of the factor j det M j
2
changes the eigenvalue correlations near the origin. Indeed, for the case of a
noncritical V for which V
(0) > 0, it was shown in [35] that