Table Of ContentUniversality of the double scaling limit in
random matrix models
5
0
0
2
Tom Claeys and Arno B.J. Kuijlaars
n
a
J February 7, 2008
1
3
1 Abstract
v
4 We study unitary random matrix ensembles in the critical case
7
where the limiting mean eigenvalue density vanishes quadratically at
0
1 aninteriorpointofthesupport. Weestablishuniversalityofthelimits
0 of the eigenvalue correlation kernel at such a critical point in a double
5
scaling limit. The limiting kernels are constructed out of functions
0
/ associated with the second Painlev´e equation. This extends a result
h
p of Bleher and Its for the special case of a critical quartic potential.
- ThetwomaintoolsweuseareequilibriummeasuresandRiemann-
h
t Hilbert problems. In our treatment of equilibrium measures we allow
a
m a negative density near the critical point, which enables usto treat all
casessimultaneously. TheasymptoticanalysisoftheRiemann-Hilbert
:
v
problemisdonewiththeDeift/Zhousteepestdescentanalysis. Forthe
i
X construction of a local parametrix at the critical point we introduce
r a modification of the approach of Baik, Deift, and Johansson so that
a
we are able to satisfy the required jump properties exactly.
1 Introduction
We consider the unitary random matrix model
Z−1 exp( N TrV(M))dM (1.1)
n,N −
defined on Hermitian n n matrices M in a critical regime where the limiting
×
mean density of eigenvalues vanishes at an interior point. It is a basic fact
1
of random matrix theory [17, 34] that the eigenvalues of the random matrix
ensemble (1.1) follow a determinantal point process with correlation kernel
n−1
Kn,N(x,y) = e−N2V(x)e−N2V(y) pk,N(x)pk,N(y), (1.2)
k=0
X
where p denotes the kth degree orthonormal polynomial with respect to
k,N
the weight e−NV(x) on R.
We assume in this paper that the confining potential V : R R in (1.1)
→
is real analytic and that it satisfies the growth condition
V(x)
+ as x + . (1.3)
log(x2 +1) → ∞ | | → ∞
These assumptions ensure that the mean eigenvalue density 1K (x,x) has
n n,N
a limit as n,N , n/N 1, see e.g. [17], which we denote by ψ (x).
V
→ ∞ →
It is known that ψ is the density of the measure µ which minimizes the
V V
weighted energy
1
I (µ) = log dµ(s)dµ(t)+ V(t)dµ(t) (1.4)
V
s t
ZZ | − | Z
among all probability measure on R. The measure µ is called the equilib-
V
rium measure in the external field V. The fact that V is real analytic ensures
that the support S = supp(µ ) consists of a finite union of intervals [18].
V V
It is a remarkable fact that local scaling limits of the kernel (1.2) depend
only on the nature of the density ψ . This has been proved rigorously in the
V
bulk of the spectrum for a quartic V in [9] and for general real analytic V in
∗
[20]. Indeed, if ψ (x ) > 0, then
V
1 u v
lim K x∗ + ,x∗ + = Kbulk(u,v) (1.5)
n→∞ nψV(x∗) n,n(cid:18) nψV(x∗) nψV(x∗)(cid:19)
exists, and
sinπ(u v)
Kbulk(u,v) = − . (1.6)
π(u v)
−
The scaling limits are different at special points of the spectrum. At edge
points of the spectrum the density ψ typically vanishes like a square root,
V
and then it is known that for some constant c > 0,
1 u v
lim K x∗ + ,x∗ + = Kedge(u,v) (1.7)
n→∞ (cn)2/3 n,n (cn)2/3 (cn)2/3
(cid:18) (cid:19)
2
where
′ ′
Ai(u)Ai(v) Ai(u)Ai(v)
Kedge(u,v) = − (1.8)
u v
−
and Ai is the Airy function. The Airy kernel is related to the Tracy-Widom
∗
distribution [39]. In (1.7) we have assumed that x is a right edge point. For
a left edge point we change u u, v v in the left-hand side of (1.7).
7→ − 7→ −
Other special points in the spectrum include
Edge points where the density vanishes to a higher order. The possible
• ∗
edge point behaviors (at a right end point x ) are
ψV(x) = c(x∗ x)2k+21(1+o(1)) as x x∗+ (1.9)
− →
where c > 0 and k is a non-negative integer.
Interior points where the density vanishes. Then
•
ψ (x) = c(x x∗)2k(1+o(1)) as x x∗ (1.10)
V
− →
where c > 0 and k is a positive integer.
∗
In these critical cases it is believed that the local scaling limit at x of the
∗
kernel only depends on the order of vanishing of the density at x [8].
The case where ψ vanishes quadratically at an interior point of S ,
V V
that is, the case k = 1 in (1.10), was considered by Bleher and Its [10] for
the case of a critical quartic potential V(x) = gx4 + tx2, with g > 0 and
4 2
t = t = 2√g. Then
c
−
1 4
ψ (x) = gx2 x2, for x [ 2g−1/4,2g−1/4],
V
2π s√g − ∈ −
so that ψ vanishes quadratically at the origin. Bleher and Its consider the
V
double scaling limit where t changes with n and tends to t as n in
c
→ ∞
such a way that n2/3(t t ) remains constant. For the quartic potential this
c
−
is equivalent to considering (1.1) where n,N , n/N 1, such that
→ ∞ →
n
lim n2/3 1 (1.11)
n,N→∞ N −
(cid:16) (cid:17)
3
exists. Bleher and Its gave a one-paremeter family Kcrit(u,v;s) of limiting
kernels, depending on s R, so that for some c > 0,
∈
1 u v
lim K , = Kcrit(u,v;s) (1.12)
n,N→∞ (cn)1/3 n,N (cn)1/3 (cn)1/3
(cid:18) (cid:19)
where s is proportional to the value of the limit (1.11).
The critical kernels are expressed in terms of so-called ψ-functions asso-
ciated with the Hastings-McLeod solution of the Painlev´e II equation [28].
Consider as in [10] the linear differential equations for a 2-vector (or 2 2
×
matrix) Ψ = Ψ(ζ;s),
d ∂
Ψ = AΨ, Ψ = BΨ (1.13)
dζ ∂s
where
4ζq 4ζ2 +s+2q2 +2r
A = A(ζ;s) = , (1.14)
4ζ2 s 2q2 +2r 4ζq
(cid:18)− − − − (cid:19)
and
q ζ
B = B(ζ;s) = . (1.15)
ζ q
(cid:18)− − (cid:19)
The compatibility condition for (1.13) is that q = q(s) satisfies the Painlev´e
II equation q′′ = sq +2q3 and that r = r(s) = q′(s). We assume that q(s) is
the Hastings-McLeod solution of Painlev´e II, which is characterized by the
asymptotic condition
q(s) = Ai(s)(1+o(1)) as s + .
→ ∞
The critical kernels op [10] are given by
Φ1(u;s)Φ2(v;s) Φ2(u;s)Φ1(v;s)
Kcrit(u,v;s) = − (1.16)
π(u v)
−
Φ1(ζ;s)
where is the special solution to (1.13) which is real for real ζ,
Φ2(ζ;s)
(cid:18) (cid:19)
satisfies
Φ1( ζ;s) = Φ1(ζ;s), Φ2( ζ;s) = Φ2(ζ;s)
− − −
4
and has asymptotics on the real line
4
Φ1(ζ;s) = cos ζ3 +sζ +O(ζ−1),
3
(cid:18) (cid:19) as ζ .
4 → ±∞
Φ2(ζ;s) = sin ζ3+sζ +O(ζ−1)
− 3
(cid:18) (cid:19)
If we put
Φ = Φ1 +iΦ2, Φ = Φ1 iΦ2 (1.17)
1 2
−
then
Φ (u;s)Φ (v;s)+Φ (u;s)Φ (v;s)
Kcrit(u,v;s) = − 1 2 2 1 (1.18)
2πi(u v)
−
Φ
and 1 is a special solution of the differential equations
Φ
2
(cid:18) (cid:19)
d 4iζ2 i(s+2q2) 4ζq+2ir
Ψ(ζ;s) = − − Ψ(ζ;s) (1.19)
dζ 4ζq 2ir 4iζ2+i(s+2q)
(cid:18) − (cid:19)
and
∂ iζ q
Ψ(ζ;s) = − Ψ(ζ;s). (1.20)
∂s q iζ
(cid:18) (cid:19)
The equations (1.19)-(1.20) for the ψ-functions correspond to the ones used
by Flaschka and Newell [24] and we will also use those in what follows. The
Φ
vector 1 is the unique solution of (1.19) with asymptotics
Φ
2
(cid:18) (cid:19)
ei(43ζ3+sζ) Φ1(ζ;s) = 1 +O ζ−1 (1.21)
Φ (ζ;s) 0
2
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
as ζ uniformly in ε argζ π ε for any ε > 0.
→ ∞ ≤ ≤ −
Before discussing our results, we like to point out an integral formula for
the kernel Kcrit(u,v;s). If we take a derivative of (1.16) with respect to s
and use (1.13) and (1.15), we get after some calculations
d 1
Kcrit(u,v;s) = Φ1(u;s)Φ1(v;s)+Φ2(u;s)Φ2(v;s) .
ds π
(cid:2) (cid:3)
Using the Deift/Zhou steepest-descent method for s as done in [23],
→ −∞
one can show that Kcrit(u,v;s) 0 as s , so that we get
→ → −∞
1 s
Kcrit(u,v;s) = Φ1(u;σ)Φ1(v;σ)+Φ2(u;σ)Φ2(v;σ) dσ. (1.22)
π
−∞
Z
(cid:2) (cid:3)
5
Since Φ1(ζ;s) and Φ2(ζ;s) are real for real ζ, formula (1.22) clearly shows
that Kcrit(u,u;s) > 0, as it should be.
2 Statement of results
It is the aim of this paper to show that the kernel Kcrit(u,v;s) is a universal
limit. Whenever the limiting mean eigenvalue density ψ vanishes quadrat-
V
ically at an interior point, the correlation kernel K has a double scaling
n,N
limit given by (1.12).
In our Theorem 2.1 below, we use the equilibrium measure ω of a com-
S
pact set S R. This is the unique probability measure on S that minimizes
⊂
the logarithmic energy
1
I(µ) = log dµ(s)dµ(t) (2.1)
s t
ZZ | − |
among all Borel probability measures µ on S. If S is a single interval [a,b],
then ω has a density w given by
S S
1
w (x) = , x (a,b).
S
π (b x)(x a) ∈
− −
If S is a finite union of disjopint intervals, say S = n [a ,b ] with a < b <
j=1 j j j j
a . Then ω has density
j+1 S
S
n
p(x)
w (x) = | | , x (a ,b ), (2.2)
S j j
π n (b x)(x a ) ∈
j=1| j − − j | j[=1
q
Q
where p(x) is a monic polynomial of degree n 1 with exactly one zero in
−
each of the gaps (b ,a ), j = 1,...,n 1, see e.g. [37, Lemma 4.4.1]. Note
j j+1
−
that (2.2) has an extension to an analytic function in C (R So), where
\ \
So = (a ,b ), which is a fact that we will use in what follows.
j j j
The following is our main result.
S
Theorem 2.1 Let V be real analytic on R such that lim V(x) = + .
x→±∞ log(x2+1) ∞
Let ψ be the density of the equilibrium measure in the external field, and let
V
x∗ be an interior point of S = supp(ψ ) which is such that
V V
∗ ′ ∗ ′′ ∗
ψ (x ) = ψ (x ) = 0, ψ (x ) > 0.
V V V
6
Let n,N such that the limit
→ ∞
n
lim n2/3 1 = L
n,N→∞ N −
(cid:16) (cid:17)
exists with L R. Let K be the correlation kernel (1.2) for the eigenvalues
n,N
∈
of the random matrix model(1.1). Then there exist constants c > 0 and s R
∈
such that
1 u v
lim K x∗ + ,x∗ + = Kcrit(u,v;s) (2.3)
n,N→∞ (cn)1/3 n,N (cn)1/3 (cn)1/3
(cid:18) (cid:19)
uniformly for u,v in compact subsets of R.
Explicit formulas for the constants c and s are
′′ ∗
πψ (x )
c = V (2.4)
8
and
π
∗
s = L w (x ), (2.5)
c1/3 SV
where w is the density of the equilibrium measure of S .
SV V
As noted before, Bleher and Its [10] proved (2.3) for the case of a critical
quartic V. See [11] for a rigorous expansion of the free energy in this critical
case.
Remark 2.2 The random matrix model (1.1) may be generalized to include
a spectral singularity at the origin
Z−1 detM 2αexp( N TrV(M))dM, α > 1/2. (2.6)
n,N| | − −
If ψ (0) > 0 and n = N , then the scaled limit of the correlationkernels
V
→ ∞
is a Bessel kernel which involves Bessel functions of order α 1, see [2, 33].
± 2
In the multicritical case where ψ vanishes quadratically at 0, an analogue of
V
Theorem 2.1 is valid. In work in progress [15], we are considering the double
scaling limit of (2.6) and we show that the limiting kernels are expressed
in terms of the ψ-functions associated with a special solution of the general
Painlev´e II equation
q′′ = sq +2q3 α.
−
The main ingredients in the proof of Theorem 2.1 are equilibrium mea-
sures and Riemann-Hilbert problems. We give some comments on both.
7
0.6 0.6 0.6
t=0.8 t=1 t=1.3
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
–2 0 2 –2 0 2 –2 0 2
Figure 1: The density of µ for V(x) = x4 x2 with t equal to 0.8, 1 and 1.3,
t 4 −
respectively.
Equilibrium measures. Recall that the equilibrium measure in external
field V minimizes (1.4). We need to know how the equilibrium measure µ
V
in the external field changes as a result of a change in V. The particular
modification we consider here is
1
V = V, t > 0,
t
t
so that V = V. Let us put
1
µ = µ .
t Vt
Then it is known that tµ and S are increasing as a function of t, see e.g.
t t
[16, 36, 38]. We also have the Buyarov-Rakhmanov formula [14]
1 t
µ = ω dτ, (2.7)
t t supp(µτ)
Z0
which expresses the equilibrium measure in the external field as an average
of equilibrium measures of sets. A consequence of (2.7) is that
d
(tµ ) = ω , (2.8)
dt t SV
(cid:12)t=1
(cid:12)
which partly explains why the equili(cid:12)brium measure ω plays a role in the
(cid:12) SV
formula (2.5) for s.
∗
For the case of interest in this paper we have that ψ vanishes at x = x .
V
∗ ∗
Then for t > 1, there is a positive density at x , while for t < 1, x is out
of the support of µ . For t slightly less than 1, there is a gap in supp(µ ),
t t
see Figure 1. An asymptotic analysis based on the equilibrium measure µ
t
would require a discussion of the two different situations t > 1 and t < 1, as
is done in [3].
8
0.6 0.6 0.6
t=0.8 t=1 t=1.3
0.4 0.4 0.4
0.2 0.2 0.2
0 0 0
–2 0 2 –2 0 2 –2 0 2
Figure 2: The density of ν for V(x) = x4 x2 with t equal to 0.8, 1 and 1.3,
t 4 −
respectively (compare with Figure 1).
Therefore we found it convenient to introduce a modification of the equi-
librium problem in external field, which will enable us to treat both cases
simultaneously. The modification we make is that we do not require the mea-
∗
sure to be non-negative in a neighborhood of the point x . For a sufficiently
small δ > 0, we consider the problem to minimize
0
1 1
I (ν) = log dν(x)dν(y)+ V(x)dν(x) (2.9)
Vt x y t
ZZ | − | Z
among all signed measures ν = ν+ ν− on R, where ν± are nonnegative
−
measures, such that
− ∗ ∗
dν = 1, and supp(ν ) [x δ ,x +δ ]. (2.10)
0 0
⊂ −
Z
We denote the minimizer by ν and we let S = supp(ν ).
t t t
Then ν = µ for t 1, but for t < 1 there is a clear distinction between
t t
≥
ν and µ , see Figure 2. However we still have the analogue of (2.8) (as we
t t
prove)
d
(tν ) = ω , (2.11)
dt t SV
(cid:12)t=1
(cid:12)
What’smore,wealsohave(2.11)att(cid:12)helevelofdensities, thatis, ifψt denotes
(cid:12) ∗
the density of ν , and if x is an interior point of S (in particular if x = x ),
t V
then
d
(tψ (x)) = w (x), (2.12)
dt t SV
(cid:12)t=1
(cid:12)
where w is the density of the equilibrium measure of S , which is what
SV (cid:12) V
(cid:12)
we need for the proof of Theorem 2.1. Note that we do not have (2.12) for
∗
x = x if ψ is the density of µ .
t t
9
Riemann-Hilbert problem. The second main tool for the proof of The-
orem 2.1 is the characterization of orthonormal polynomials by means of a
Riemann-Hilbert problem, due to Fokas, Its, and Kitaev [25], and the sub-
sequent asymptotic analysis of the Riemann-Hilbert problem by means of
the Deift/Zhou steepest descent analysis of Riemann-Hilbert problems, in-
troduced in [22], and further developed in [21, 19, 20], and other more recent
papers. The Deift/Zhou steepest descent analysis of the Riemann-Hilbert
problem consists of a sequence of explicit transformations, which result in
a Riemann-Hilbert problem that is explicitly solvable in terms of Neumann
series. (In fact, in this paper, we only need the first term of this series.) The
∗
critical point x needs special attention.
Of particular interest for us is the paper [3] by Baik, Deift, and Johansson
on the length of the longest increasing subsequence of a random permutation
of 1,2,...,n . These authors show that the fluctuations of this random
{ }
variable are distributed according to the Tracy-Widom distribution [39] in
the limit as n . One of the technical tools in this important paper is the
→ ∞
asymptotic analysis of a Riemann-Hilbert problem on the unit circle which
is related to an equilibrium measure (also on the unit circle) whose density
vanishes at the point 1. This situation is comparable to ours. The authors
−
of [3], see also subsequent papers [4, 6, 7, 5], construct a local parametrix
near 1 withtheaidoftheψ-functionsassociated withthe Hastings-McLeod
−
solution of Painlev´e II. These ψ-functions satisfy a model Riemann-Hilbert
problem and the local parametrix is constructed by appropriately mapping
the model Riemann-Hilbert problem onto a neighborhood of 1 so that it
−
satisfies certain desired jump properties approximately.
We follow the approach of [3] but we introduce a modification in the con-
struction of the local parametrix so that it has the desired jump properties
exactly, in contrast to [3] where the desired jump properties only hold ap-
proximately. The fact that we have the exact jump properties simplifies the
arguments considerably and we feel that this is also a main contribution of
the present paper.
Outline of the rest of the paper. In Section 3 we collect the necessary
facts about equilibrium measures. In particular we study the modified equi-
librium problem with external field in some detail. In Section 4 we discuss
the Riemann-Hilbert problem satisfied by the ψ-functions. Here we follow
[24]. Then in Section 5 we state the Riemann-Hilbert problem for orthogonal
10
