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On the singular sector of the Hermitian random matrix model
in the large N limit

B. Konopelchenko 1, L. Mart́ınez Alonso2 and E. Medina3

1 Dipartimento di Fisica, Universitá del Salento and Sezione INFN

73100 Lecce, Italy
2 Departamento de F́ısica Teórica II, Universidad Complutense

E28040 Madrid, Spain
3 Departamento de Matemáticas, Universidad de Cádiz

E11510 Puerto Real, Cádiz, Spain

January 3, 2014

Abstract

The singular sector of zero genus case for the Hermitian random matrix model in the large N
limit is analyzed. It is proved that the singular sector of the hodograph solutions for the underlying
dispersionless Toda hierarchy and the singular sector of the 1-layer Benney (classical long wave
equation) hierarchy are deeply connected. This property is due to the fact that the hodograph
equations for both hierarchies describe the critical points of solutions of Euler-Poisson-Darboux
equations E(a, a), with a = −1/2 for the dToda hierarchy and a = 1/2 for the 1-layer Benney
hierarchy.

Key words: Integrable systems. Hodograph equations. Random matrix
models. Euler-Poisson-Darboux equation.
PACS number: 02.30.Ik.

1

http://arxiv.org/abs/1005.4773v1


1 Introduction

The Hermitian matrix (H) model is nowadays the paradigmatic model in the theory of random
matrices ( see e.g. [1, 2]). The partition function for this model is

ZN =

∫

dH exp
(

− N

T
tr(

∑

k≥1

tk H
k)
)

, (1)

where T > 0 is a real parameter which represents the temperature and the integration is performed
on the space of N×N Hermitian matrices. The large N limit of theH model is of particular interest
since it exhibits many important universality properties. It turns out that most of the applications
of the H model [1, 2] arise after regularizing the large N -limit solutions at their singular points
(double-scaling limit method). In this sense the analysis and characterization of the singular sector
of the H model is of great relevance.

The simplest situation corresponds to the so called zero genus case in which the support of
the eigenvalue density reduces to a single interval [β1, β2] as N → ∞. Moreover, it is well-known
( see e.g.[3, 4]) that the endpoints β := (β1, β2) evolve with the temperature T and the coupling
constants tk according to the dispersionless Toda (dToda) hierarchy . The first members of this
hierarchy are the dToda equation vTT = (log v)t1t1 or equivalently the dToda system

ut1 = −vT , vt1 = −v uT , (2)

and the 1-layer Benney (B) system

ut2 = 2 (uut1 + vt1), vt2 = 2 (u v)t1 . (3)

These systems represent themselves two distinguished examples of integrable 2-component hydro-
dynamical type systems (see e.g. [5]). The dToda equation is the 1+1-dimensional version of the
Boyer-Finley equation from the general relativity [6]. It also arises in various problems of fluid
mechanics (see e.g. [7, 8, 9]). The B system describes long waves in shallow water with free sur-
face in a gravitational field [10]. It represents the dispersionless limit of the celebrated nonlinear
Schrödinger equation [11]. Recently, the B system became a crucial ingredient in the analysis of
the universality of critical behaviour for nonlinear equations [12]. In general, the B system (3) is
an excellent laboratory for study properties of integrable hydrodynamical type systems.

Thus, the analysis of the singular sector for the H model in the large N limit reduces to the
corresponding analysis for the dToda hierarchy. Recently the authors [13] have provided a method
for studying the hodograph solutions and their singular sectors (gradient catastrophe points) of
the family of coupled KdV hierarchies dcKdVm (m ≥ 1), where the case m = 2 corresponds to
the B hierarchy. The study of these singular sectors was already initiated in [14]. In the method
of [13] the hodograph solutions are described by the critical points of scalar functions that satisfy
Euler-Poisson-Darboux (EPD) equations [15]. This property simplifies drastically the analysis and
classification of the singular sectors .

In the present paper we show that the results of [13] can be extended to the dToda hierarchy
thus providing us with complete analysis of the singular sector of H model for the zero genus case.
Moreover, we demonstrate that there is deep similarity between the properties of the sets of the
hodograph solutions for the dToda and B hierarchies.

Our main observation is that in both cases the hodograph solutions represent the critical points

∂W

∂βi
= 0, i = 1, 2, (4)

2



of a scalar function W which depends linearly on the coordinates t = (t0, t1, t2, . . .), where t is
the set of coupling constants for the H model (t0 = T ) and the set of flow parameters in the case
of the B hierarchy. Moreover, these functions satisfy the Euler-Poisson-Darboux equation E(a, a)
[15]

(β1 − β2)
∂2W

∂β1 ∂β2
= a

( ∂W

∂β1
− ∂W

∂β2

)

, a =



















1

2
for WB

−1

2
for WH .

(5)

Both functions are related according to a transformation [15] which maps solutions of E(−1/2,−1/2)
into solutions E(1/2, 1/2)

WB = 4
∂2WH

∂β1 ∂β2
− tH,1,

where the parameters of both models are identified according to

tB,n = (n+ 2) tH,n+2, n ≥ 0.

The equation (5) and its multidimensional version are well known for a long time in the classical
geometry [15]. Its relevance to the theory of Whitham equations has been demonstrated recently in
the papers [16]-[17]. The observation made in [13] that the hodograph equations for the B system
(3) have the form (4)-(5) seems to be new, though results close to this have been provided in the
papers [12, 18].

The paper is organized as follows. In Section 2 we show how the hodograph equations of the
B system and the H models are the equations for the critical points of certain solutions of Euler-
Poisson-Darboux equations. In Section 3 we use the Euler-Poisson-Darboux equations to formulate
a common description of the singular sectors for theses hodograph equations. Our results are
summarized into three Propositions which exhibit the deep connection arising between the singular
sectors of both models. Moreover, the corresponding singular classes are also characterized in terms
of the behaviour near λ = βi (i = 1, 2) of the S-function (B system) and the eigenvalue density ρ
(H model). Section 4 is devoted to a method for the explicit determination of singular classes by
means of constraints for the t-parameters. Finally, some illustrative examples are provided.

2 Hodograph equations and the Euler-Poisson-Darboux

equations

B hierarchy

The B system (3) is a member of a dispersionless integrable hierarchy which describe deformations
of the curve (see e.g. [14, 19]).

p2 = (λ− β1) (λ− β2). (6)

The flows
β(t), t = (x := t0, t1, t2, . . .)

are characterized by the following condition: There exists a family of functions S(λ, t,β) satisfying

∂S(λ, t,β(t))

∂tn
= Ωn(λ,β(t)), n ≥ 0. (7)

3



where

Ωn(λ,β) =
( λn+1

√

(λ− β1) (λ− β2)

)

⊕

√

(λ− β1) (λ− β2). (8)

Functions S which satisfy (7) are referred to as action functions in the theory of dispersionless
integrable systems (see e.g. [20]). Notice that for n = 0 Eq.(7) reads

p =
∂S

∂x
,

so that the system (7) is equivalent to

∂p

∂tn
= ∂xΩn, (9)

and, in terms of β, it can be rewritten in the hydrodynamical form

∂βi
∂tn

=
(

Ωn(λ,β)
)

⊕

∣

∣

∣

λ=βi

∂x βi, i = 1, 2. (10)

The t1-flow of this hierarchy is the B system (3) (t := t1), which in terms of β reads


















∂t β1 =
1

2
(3β1 + β2)β1 x,

∂t β2 =
1

2
(3β2 + β1)β2 x.

(11)

It was proved in [13] that the system (4) for the function

WB(t,β) :=

∮

γ

dλ

2 i π

λVB(λ, t)
√

(λ− β1) (λ− β2)
, (12)

where
VB(λ, t) =

∑

n≥0

λn tn,

is a system of hodograph equations for the B hierarchy. Moreover, the action function for the
corresponding solutions is given by

S(λ, t,β) =
∑

n≥0

tnΩn(λ,β) = hB(λ, t,β)
√

(λ− β1)(λ− β2). (13)

where

hB(λ, t,β) :=
( λVB(λ, t)
√

(λ− β1)(λ− β2)

)

⊕
.

Obviously, WB satisfies the Euler-Poisson-Darboux equation E(1
2
, 1
2
). Written explicitly, WB rep-

resents itself the series

WB =
x

2
(β1 + β2) +

t1
8
(3β2

1 + 2β1β2 + 3β2
2) +

t2
16

(

5β3
1 + 3β2

1β2 + 3β1β
2
2 + 5β3

2

)

+
t3
128

(35β4
1 + 20β3

1β2 + 18β2
1β

2
2 + 20β1β

3
2 + 35β4

2 ) + · · · . (14)

4



The hodograph equations (4) with tn = 0 for n ≥ 4 take the form



















8x+ 4t1(3β1 + β2) + 3t2
(

5β2
1 + 2β1β2 + β2

2

)

+
t3
8
(140β3

1 + 60β2
1β2 + 36β1β

2
2 + 20β2

2) = 0,

8x+ 4t1(β1 + 3β2) + 3t2
(

β2
1 + 2β1β2 + 5β2

2

)

+
t3
8
(140β3

2 + 60β2
2β1 + 36β2β

2
1 + 20β2

1) = 0.

(15)
Detailed analysis of equations (15) will be performed in section 4.

dToda hierarchy

It is known (see e.g. [1]) that as N → ∞ the asymptotic density of eigenvalues {λ1, . . . , λN} of the
H model in the zero genus case concentrates on a single interval [β1, β2] and is given by [1]

ρ(λ, t,β) :=
hH(λ, t,β)

2π i T

√

(λ− β1)(λ− β2),

where

hH(λ, t,β) :=
( VH,λ(λ, t)
√

(λ− β1)(λ− β2)

)

⊕
.

Here ⊕ denotes the projection on strictly positive powers of λ. Moreover, the endpoints β1 and β2
of the eigenvalue support are determined by the equations

∮

γ

dλ

2 i π

∂λVH(λ, t)
√

(λ− β1) (λ− β2)
= 0,

∮

γ

dλ

2 i π

λ ∂λVH(λ, t)
√

(λ− β1) (λ− β2)
= 2T, (16)

where
VH(λ, t) :=

∑

n≥1

λn tn,

and γ denotes a large positively oriented circle |λ| = R. It is immediate to see that these equations
are equivalent to the system (4) for the critical points of the function

WH(t,β) := T (β1 + β2) +

∮

γ

dλ

2 i π

√

(λ− β1) (λ− β2) ∂λVH(λ, t), (17)

which satisfies the Euler-Poisson-Darboux equation E(−1

2
,−1

2
). It is also easy to check that the

functions WB and WH are related according to

WB = 4
∂2WH

∂β1 ∂β2
− tH,1, tB,n = (n + 2) tH,n+2, n ≥ 0. (18)

The first terms of the function WH are

WH = T (β1 + β2)−
t1
8
(β1 − β2)

2 − t2
8
(β1 + β2)(β1 − β2)

2 − 3t3
128

(

5β2
1 + 6β2β1 + 5β2

2

)

(β1 − β2)
2

− t4
64

(

7β3
1 + 9β2β

2
1 + 9β2

2β1 + 7β3
2

)

(β1 − β2)
2 + · · · ,

5



The corresponding equations (4) for tn = 0, ∀n ≥ 4 take the form



















T +
t1
4
(β2 − β1) +

t2
8

(

−3β2
1 + 2β2β1 + β2

2

)

− 3t3
32

(

5β3
1 − 3β2β

2
1 − β2

2β1 − β3
2

)

= 0,

T +
t1
4
(β1 − β2) +

t2
8

(

β2
1 + 2β2β1 − 3β2

2

)

+
3t3
32

(

β3
1 + β2β

2
1 + 3β2

2β1 − 5β3
2

)

= 0.

It is known that the hodograph system (16) provides solutions of the dispersionless Toda hier-
archy (see e.g. [3, 4]). Indeed, for ”times” T and t1 one gets



















∂t1 β1 = −1

4
(β1 − β2)β1T ,

∂t1 β2 =
1

4
(β1 − β2)β2T ,

(19)

which in terms of variables

u =
1

2
(β1 + β2), v =

1

16
(β1 − β2)

2, (20)

becomes the dToda system (2). Moreover, for times t1, t2 the hodograph equations (16) imply the
Benney system (3). If we use the dependent variables (u, v) and introduce the function

f(t, u, v) := WH(t, u, v) − 2T u+ 2 t1 v + 4 t2 u v, (21)

then the hodograph equations read

{

2T − 4 v t2 + ∂u f = 0,

−2 t1 − 4u t2 + ∂v f = 0,
(22)

which, under trivial rescalings and the substitution (u, v) 7→ (v, u), coincides with the hyperbolic
version of hodograph system (2.11) of [12]. In particular, from the EPD equation (5) it follows
easily that

∂2 WH

∂ u2
− v

∂2 WH

∂ v2
=

∂2 f

∂ u2
− v

∂2 f

∂ v2
= 0. (23)

3 Characterization of singular sectors

Using (4) and (5) we may now analyze the structure of singular sectors of the B system and the
H model in a unified way. Thus, let us denote by M the set of solutions (t,β) (β1 6= β2) of the
hodograph equations (4). There is a partition of M into a regular and a singular sector

M = Mreg ∪Msing,

such that given (t,β) ∈ M

(t,β) ∈ Mreg if det
(∂2 W (t,β)

∂ βi ∂ βj

)

6= 0, (t,β) ∈ Msing if det
(∂2 W (t,β)

∂ βi ∂ βj

)

= 0.

6



The elements of Mreg, correspond to the case when the system (4) is uniquely solvable and hence,

it defines a unique solution β(t). The singular sector Msing contains the degenerate critical points
of the function W on which the implicit solutions β(t) of the hodograph equations exhibit gradient
catastrophe behaviour.

The Euler-Poisson-Darboux equation is of great help to analyze the structure ofMsing. Indeed,
if (t,β) ∈ M, as a consequence of (5) it is clear that

∂2 W

∂ β1 ∂ β2
= 0. (24)

Consequently

det
( ∂2 W

∂ βi ∂ βj

)

=
∂2W

∂ β2
1

· ∂
2W

∂ β2
2

. (25)

Thus, we have

Proposition 1. Given (t,β) ∈ M then

1. (t,β) ∈ Mreg if and only if ∂2 W
∂ β2

1

6= 0 and ∂2 W
∂ β2

2

6= 0.

2. (t,β) ∈ Msing if and only at least one of the derivatives ∂2 W
∂ β2

1

, ∂2 W
∂ β2

2

, vanishes.

Furthermore, using (5) it follows at once that at any point (t,β) ∈ M all mixed derivatives
∂i
β1
∂j
β2
W can be expressed in terms of linear combination of diagonal derivatives ∂n

β1
W and ∂m

β2
W .

Let us now define Msing
n1,n2

as the set of points (t,β) ∈ M such that

∂ni+2W

∂βni+2

i

6= 0;
∂kW

∂βk
i

= 0, ∀1 ≤ k ≤ ni + 1, (i = 1, 2). (26)

It is clear that
Msing

n1,n2

⋂

Msing
n′

1
,n′

2

= ∅, for (n1, n2) 6= (n′
1, n

′
2)

Then, it follows at once that

Proposition 2. The singular sector satisfies

Msing =
⋃

n1+n2≥1

Msing
n1,n2

. (27)

According to (18) and using (5) we have

WB = −tH,1 −
2

β1 − β2

(∂WH

∂β1
− ∂ WH

∂β2

)

, tB,n = (n + 2) tH,n+2, n ≥ 0.

It is then immediate to conclude that

Proposition 3. The singular sectors of the H model and the Benney system, with

tB,n = (n+ 2) tH,n+2 for n ≥ 0, satisfy














Msing
H,1,1 ⊂ Mreg

B ,

Msing
H,n1+1,n2+1

⊂ Msing
B,n1,n2

for n1 ≥ 1, n2 ≥ 1.

(28)

7



Singular sectors in the H model

In the applications of the H model in quantum gravity [1] the singular sectors are described in
terms of the behaviour of the eigenvalue density ρ(λ) at the endpoints (β1, β2) of its support. We
will now show how this description derives in a natural way from the classification (27) based on

the subsets Msing
H,n1,n2

of the singular sector . To this end let us consider the derivative ∂k+1

β1
WH

with k ≥ 2 , it is obviously proportional to the integral

∮

γ

dλ

2 i π
(λ−β2)

VH,λ/
√

(λ− β1) (λ − β2)

(λ− β1)k
=

∮

γ

dλ

2 i π

(λ− β2)hH(λ)

(λ− β1)k
=

∂k−1

λ

(k − 1)!

(

(λ−β2)hH(λ)
)
∣

∣

∣

λ=β1

,

and a similar result follows for the derivatives ∂k+1

β2
W with k ≥ 2. As a consequence we have

Proposition 4. A point (t,β) ∈ MH belongs to the singularity class Msing
H,n1,n2

if and only if

ρ(λ, t,β) ∼ (λ− βi)
2ni+1

2 as λ → βi, (i = 1, 2) (29)

This property is crucial to establish the relationship between the regularized singularity sectors of
the H model and the minimal conformal models (p, q) with q = 2 [1].

Singular sectors in the B system

In analogy with the H model we may characterize the classes Msing
B,n1,n2

of the singular sector of

the B system in terms of the behaviour of S(λ) at λ = βi (i = 1, 2) . Indeed the derivative ∂k+1

β1
WB

with k ≥ 1 is proportional to the integral

∮

γ

dλ

2 i π
(λ− β2)

λVB/
√

(λ− β1) (λ− β2)

(λ− β1)k+1
=

∮

γ

dλ

2 i π

(λ− β2)hB(λ)

(λ− β1)k+1
=

∂k
λ

k!

(

(λ− β2)hB(λ)
)
∣

∣

∣

λ=β1

,

and a similar result follows for the derivatives ∂k+1

β2
WB with k ≥ 2. As a consequence we have

Proposition 5. A point (t,β) ∈ MB belongs to the singularity class Msing
B,n1,n2

if and only if

S(λ, t,β) ∼ (λ− βi)
2ni+3

2 as λ → βi, (i = 1, 2) (30)

This result shows the existing duality between the eigenvalue density ρ of the H model and the
S function of the B system.

4 Explicit determination of singular sectors in terms

of constrains fk(t) = 0

The singular classes Msing
n1,n2

can be determined by means of systems of n1 + n2 constraints for the

coordinates t. To see this property notice that the points (t,β) of Msing
n1,n2

are characterized by the
equations

∂kW

∂βk
i

= 0, ∀1 ≤ k ≤ ni + 1, i = 1, 2, (31)

8



and
∂ni+2W

∂βni+2

i

6= 0, i = 1, 2. (32)

Furthermore, the jacobian matrix of the the system of two equations

∂ni+1W

∂βni+1

i

= 0, i = 1, 2 (33)

is not singular as

∆ :=

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

∂n1+2 W

∂ βn1+2

1

∂n2+2 W

∂ β1∂ β
n2+1

2

∂n1+2 W

∂ βn1+1

1
∂ β2

∂n2+2 W

∂ βn2+2

2

∣

∣

∣

∣

∣

∣

∣

∣

∣

∣

6= 0. (34)

Indeed, we notice that as a consequence of (5) the derivatives outside the diagonal of ∆ are linear
combinations of the derivatives {∂k

βi
W, 1 ≤ k ≤ ni + 1, i = 1, 2}, so that from (31)-(32) we have

∆ =
∂n1+2W

∂ βn1+2

1

· ∂
n2+2 W

∂ βn2+2

2

6= 0.

Therefore, one can solve (33) and get a solution β(t). Substituting this solution in the remaining
equations (31) gives n1 + n2 constraints of the form

fk(t) = 0, k = 1, . . . , n1 + n2.

It is not difficult to determine the solutions of (31)-(32) when tn = 0, n ≥ 4. Lists of such solutions
are given next.

Example: the H model

Let us consider the H model and assume that tn = 0 if n ≥ 4 . Then it follows that the condition

det(∂βiβj
WH(t,β)) = 0 which characterizes Msing

H reads

32t22 − 24xt3 + 72t2t3(β1 + β2) + 27t23
(

β2
1 + 6β2β1 + β2

2

)

= 0.

There are two generic classes inMsing
H given byMsing

H,1,0 andMsing
H,0,1. Assuming that (t1, t2, t3, β1, β2)

must be real, both classes include two cases

9



Msing
H,1,0 :



























































































1) x =
2t22 − 9

3
√
6T 2 t

4/3
3

6t3
,

β1 = −2t2 +
3
√
36T t

2/3
3

6t3
, β2 =

−2t2 + 3 3
√
36T t

2/3
3

6t3
.

2) x =
4t22 + 9

3
√
6T 2

(

1− i
√
3
)

t
4/3
3

12t3
,

β1 =
−4t2 +

(

1 + i
√
3
)

3
√
36T t

2/3
3

12t3
, β2 =

−4t2 − 3
(

1 + i
√
3
)

3
√
36T t

2/3
3

12t3
,

where we assume t3 < 0 in the second case.

Msing
H,0,1









































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

1) x =
2t22 − 9

3
√
6T 2 t

4/3
3

6t3
,

β1 =
−2t2 + 3 3

√
36T t

2/3
3

6t3
, β2 = −2t2 +

3
√
36T t

2/3
3

6t3
,

2) x =
4t22 + 9

3
√
6T 2

(

1− i
√
3
)

t
4/3
3

12t3
,

β1 =
−4t2 − 3

(

1 + i
√
3
)

3
√
36T t

2/3
3

12t3
, β2 =

−4t2 +
(

1 + i
√
3
)

3
√
36T t

2/3
3

12t3
,

where we assume t3 < 0 in the second case.

Example: the B system

Let us now consider the system of hodograph equations for the B system with tn = 0 for all n ≥ 4.
Now the condition det(∂βiβj

WB(t,β)) = 0 reduces to

32t21 + 96t2(β1 + β2) + 702t23β
2
1β

2
2 + 72

(

3t22 + t1t3
)

β1β2 + 12
(

3t22 + 13t1t3
) (

β2
1 + β2

2

)

+

486t2t3
(

β2β
2
1 + β2

2β1
)

+ 90t2t3
(

β3
1 + β3

2

)

+ 180t23
(

β2β
3
1 + β3

2β1
)

+ 45t23
(

β4
1 + β4

2

)

= 0.

There are two classes in Msing
B,1,0

10



1. x =
−45t3t

3
2 + 180t1t

2
3t2 +

√
15(8t1t3 − 3t22)

√

t2
3

(

3t2
2
− 8t1t3

)

360t3
3

,

β1 = −
5t2t3 +

√
15

√

t2
3

(

3t2
2
− 8t1t3

)

20t2
3

, β2 =
−3t2t3 +

√
15
√

t2
3

(

3t2
2
− 8t1t3

)

12t2
3

,

2. x =
−45t3t

3
2 + 180t1t

2
3t2 −

√
15(8t1t3 − 3t22)

√

t2
3

(

3t2
2
− 8t1t3

)

360t3
3

,

β1 =
−5t2t3 +

√
15

√

t2
3

(

3t2
2
− 8t1t3

)

20t2
3

, β2 = −
3t2t3 +

√
15
√

t2
3

(

3t2
2
− 8t1t3

)

12t2
3

In the same way it follows that Msing
B,0,1 has two classes given by

1. x =
−45t3t

3
2 + 180t1t

2
3t2 −

√
15(8t1t3 − 3t22)

√

t2
3

(

3t2
2
− 8t1t3

)

360t3
3

,

β1 = −
3t2t3 +

√
15

√

t2
3

(

3t2
2
− 8t1t3

)

12t2
3

, β2 =
−5t2t3 +

√
15
√

t2
3

(

3t2
2
− 8t1t3

)

20t2
3

,

2. x =
−45t3t

3
2 + 180t1t

2
3t2 +

√
15(8t1t3 − 3t22)

√

t2
3

(

3t2
2
− 8t1t3

)

360t3
3

,

β1 =
−3t2t3 +

√
15

√

t2
3

(

3t2
2
− 8t1t3

)

12t2
3

, β2 = −
5t2t3 +

√
15
√

t2
3

(

3t2
2
− 8t1t3

)

20t2
3

.

Acknowledgements

The authors wish to thank the Spanish Ministerio de Educación y Ciencia (research project
FIS2008-00200/FIS) for its finantial support. B. K. is thankful to the Departamento de F́ısica
Teórica II for the kind hospitality.

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