The Hermitian two matrix model with an even quartic potential
Abstract
We consider the two matrix model with an even quartic potential and an even polynomial potential . The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices . The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a matrix valued RiemannHilbert problem that characterizes the correlation kernel for the eigenvalues of . Our results generalize earlier results for the case , where the external field on the third measure was not present.
Contents
 1 Introduction and statement of results
 2 Preliminaries and the proof of Lemma 1.2
 3 Proof of Theorem 1.1
 4 A Riemann surface
 5 Pearcey integrals and the first transformation of the RH problem
 6 Second transformation
 7 Opening of lenses
 8 Global parametrix
 9 Local parametrices and final transformation
1 Introduction and statement of results
1.1 Hermitian two matrix model
The Hermitian twomatrix model is a probability measure of the form
(1.1) 
defined on the space of pairs of Hermitian matrices. The constant in (1.1) is a normalization constant, is the coupling constant and is the flat Lebesgue measure on the space of pairs of Hermitian matrices. In (1.1), and are the potentials of the matrix model. In this paper, we assume to be a general even polynomial and we take to be the even quartic polynomial
(1.2) 
Without loss of generality we may (and do) assume that
(1.3) 
We are interested in describing the eigenvalues of in the large limit.
1.2 Background
The twomatrix model (1.1) with polynomial potentials and was introduced in [59, 70] as a model for quantum gravity and string theory. The interest is in the double scaling limit for critical potentials. It is generally believed that the twomatrix model is able to describe all conformal minimal models, whereas the onematrix model is limited to minimal models [30, 41, 48]. In [61] the twomatrix model was proposed for the study of the Ising model on a random surface, where the logarithm of the partition function (i.e., the normalizing constant in (1.1)) is expected to be the generating function in the enumeration of graphs on surfaces. For more information and background on the physical interest we refer to the the surveys [39, 40], and more recent physical papers [9, 49, 51, 52]
The two matrix model have a very rich integrable structure that is connected to biorthogonal polynomials, isomonodromy deformations, RiemannHilbert problems and integrable equations, see e.g. [2, 10, 12, 13, 14, 46, 50, 60, 66]. This is the basis of the mathematical treatment of the two matrix model, see also the survey [11].
The eigenvalues of the matrices and in the twomatrix model are a determinantal point process with correlation kernels that are expressed in terms of biorthogonal polynomials. These are two families of monic polynomials and , where has degree and has degree , satisfying the condition
(1.4) 
The polynomials are welldefined by (1.4) and have simple and real zeros [46]. Moreover, the zeros of and , and those of and are interlacing [43].
The kernels are expressed in terms of these biorthogonal polynomials and their transformed functions
as follows:
(1.5)  
(1.6)  
(1.7)  
(1.8) 
Then Eynard and Mehta [50, 72], see also [24, 37, 71], showed that the joint probability density function for the eigenvalues of and of is given by
and the marginal densities take the form
(1.9) 
In particular, by taking , so that we average over the eigenvalues of , we find that the eigenvalues of are a determinantal point process with kernel , see (1.5). This kernel is constructed out of the biorthogonal family and and the associated determinantal point process is an example of a biorthogonal ensemble [23]. It is also an example of a multiple orthogonal polynomial ensemble in the sense of [62].
In order to describe the behavior of the eigenvalues in the large limit, one needs to control the kernels (1.5)–(1.8) as . Due to special recurrence relations satisfied by the biorthogonal polynomials, there exist ChristoffelDarbouxtype formulas that express the term sums (1.5) and (1.8) into a finite number (independent of ) of biorthogonal polynomials and transformed functions, see [13]. This paper also gives differential equations and a remarkable duality between spectral curves, see also [12, 14].
A RiemannHilbert problem for biorthogonal polynomials was first formulated in [46]. The RiemannHilbert problem in [46] is of size but it is nonlocal and one has not been able to apply an asymptotic analysis to it. Local RiemannHilbert problems were formulated in [14, 60, 66], but these RiemannHilbert problems are of larger size, depending on the degrees of the potentials and . The formulation of a local RiemannHilbert problem for biorthogonal polynomials, however, opens up the way for the application of the DeiftZhou [35] steepest descent method, which was applied very successfully to the RiemannHilbert problem for orthogonal polynomials, see [18, 31, 33, 34] and many later papers.
In [45] the DeiftZhou steepest descent method was indeed applied to the RiemannHilbert problem from [66] for the case where is given by (1.2) with . It gave a precise asymptotic analysis of the kernel as , leading in particular to the local universality results that are wellknown in onematrix models [33]. The analysis in [45] was restricted to the genus zero case. The extension to higher genus was done in [75].
1.3 Vector equilibrium problem
As already stated, it is the purpose of the present paper to extend the results of [45, 75] to the case of general .
An important role in the analysis in [45] is played by a vector equilibrium problem that characterizes the limiting mean density for the eigenvalues of (and also gives the limiting zero distribution of the biorthogonal polynomials ). One of the main contributions of the present paper is the formulation of the appropriate generalization to general . We refer to the standard reference [78] for notions of logarithmic potential theory and equilibrium problems with external fields.
1.3.1 Case
Let us first recall the vector equilibrium problem from [45], which involves the minimization of an energy functional over three measures. For a measure on we define the logarithmic energy
and for two measures and we define the mutual energy
The energy functional in [45] then takes the form
(1.10) 
and the vector equilibrium problem is to minimize (1.10) among all measures , and such that

the measures have finite logarithmic energy;

is a measure on with ;

is a measure on with ;

is a measure on with ;

where is the unbounded measure with density
(1.11) on the imaginary axis.
A main feature of the vector equilibrium problem is that it involves an external field acting on the first measure as well as an upper constraint (1.11) acting on the second measure. Note that an upper constraint arises typically in the asymptotic analysis of discrete orthogonal polynomials, see e.g. [7, 22, 42, 68, 77]. The interaction between the measures in (1.10) is of the Nikishin type where consecutive measures attract each other, but there is no direct interaction between measures and if . The notion of a Nikishin system originated in works on HermitePadé rational approximation, see [5, 56, 62, 76]. Vector equilibrium problems also played a role in the recent papers [8, 15, 17, 65] that are related to random matrix theory and [6, 43, 44, 69, 84] that are related to recurrence relations.
1.3.2 General
For general , the relevant energy functional takes the form
(1.12) 
where and are certain external fields acting on and , respectively. The vector equilibrium problem is to minimize among all measures , , , such that

the measures have finite logarithmic energy;

is a measure on with ;

is a measure on with ;

is a measure on with ;

where is a certain measure on the imaginary axis.
Comparing with (1.10) we see that there is an external field acting on the third measure as well. The vector equilibrium problem depends on the input data , , and that will be described next. Recall that is an even polynomial and that is the quartic polynomial given by (1.2).
External field :
The external field that acts on is defined by
(1.13) 
The minimum is attained at a value for which , that is
(1.14) 
For , this value of is uniquely defined by (1.14). For there can be more than one real solution of (1.14). The relevant value is the one that has the same sign as (since , see (1.3)). It is uniquely defined, except for .
External field :
The external field that acts on is not present if . Thus
(1.15) 
For , the external field is nonzero only for where
(1.16) 
For those , the equation (1.14) has three real solutions , , which we take to be ordered such that
Thus the global minimum of is attained at , and this global minimum played a role in the definition (1.13) of . The function has another local minimum at and a local maximum at , and these are used in the definition of . We define by
(1.17) 
Thus is the difference between the local maximum and the other local minimum of , which indeed exist if and only if , where is given by (1.16). In particular for .
The constraint :
To describe the measure that acts as a constraint on , we consider the equation
(1.18) 
There is always a solution on the imaginary axis. The other two solutions are either on the imaginary axis as well, or they are off the imaginary axis, and lie symmetrically with respect to the imaginary axis. We define
(1.19) 
where is the solution of (1.18) with largest real part. We then have for the support of ,
(1.20) 
where
(1.21) 
This completes the description of the vector equilibrium problem for general . It is easy to check that for it reduces to the vector equilibrium described before.
1.4 Solution of vector equilibrium problem
Our first main theorem deals with the solution of the vector equilibrium problem. We use to denote the support of a measure . The logarithmic potential of is the function
(1.22) 
which is a harmonic function on and superharmonic on .
Theorem 1.1.
The above vector equilibrium problem has a unique minimizer that satisfies the following.

There is a constant such that
(1.23) If or then
(1.24) for some and , and on each of the intervals in there is a density
(1.25) and is nonnegative and real analytic on .

We have
(1.26) and there is a constant such that
(1.27) Moreover, has a density
(1.28) that is positive and real analytic on . If , then vanishes as a square root at . If , then , where is given by (1.21).

We have
(1.29) and there is a constant such that
(1.30) Moreover, has a density
(1.31) that is positive and real analytic on . If , then . If , then where is given by (1.16). If , then vanishes as a square root at .

All three measures are symmetric with respect to , so that for we have for every Borel set .
In part (a) of the theorem it is stated that is a finite union of intervals under the condition that and are disjoint. If this condition is not satisfied then we are in one of the singular cases that will be discussed in Section 1.5 below. However, the condition is not necessary as will be explained in Remark 4.9 below. We chose to include the condition in Theorem 1.1 since the focus of the present paper is on the regular cases.
The conditions (1.23), (1.26), and (1.29) are the EulerLagrange variational conditions associated with the vector equilibrium problem. We note the strict inequalities in (1.26) and (1.29). These are consequences of special properties of the constraint and the external field that are listed in parts (b) and (c) of the following lemma.
Lemma 1.2.
The following hold.

Let be a measure on such that . If then is real analytic on .

The density (see (1.19)) is an increasing function for .

Let . Let be a measure on of finite logarithmic energy such that . Then is a decreasing and convex function on .
A major role in what follows will be played by functions defined on a compact foursheeted Riemann surface that we will introduce in Section 4. The sheets are connected along the supports , and of the minimizing measures for the vector equilibrium problem. The main result of Section 4 is Proposition 4.8 which says that the function defined by
on the first sheet has an extension to a globally meromorphic function on the full Riemann surface. This very special property is due to the special forms of the external fields and and the constraint , which interact in a very precise way.
1.5 Classification into cases
According to Theorem 1.1 the structure of the supports is the same for as it was for in [45], that is, and for some . The supports determine the underlying Riemann surface, and so the case is very similar to the case . There are no phase transitions in case , except for the possible closing or opening of gaps in the support of . These type of transitions already occur in the onematrix model.
For , however, certain new phenomena occur which come from the fact that the external field on (defined in (1.17)) has its maximum at and therefore tends to move away from . As a result there are cases where is no longer the full real axis, but a strict subset (1.30) with .
In addition, it is also possible that in (1.27) such that is the full imaginary axis and the constraint is not active. These new phenomena already occur for the simplest case
for which explicit calculations were done in [43] based on the coefficients in the recurrence relations satisfied by the biorthogonal polynomials. These calculations lead to the phase diagram shown in Figure 1.1 which is taken from [43]. There are four phases corresponding to the following four cases that are determined by the fact whether is in the support of the measures , , or not:
 Case I:

, , and ,
 Case II:

, , and ,
 Case III:

, , and ,
 Case IV:

, , and .
The four cases correspond to regular behavior of the supports at . There is another regular situation (which does not occur for ), namely
 Case V:

, , and .
The five cases determine the cut structure of the Riemann surface and we will use this classification throughout the paper.
Singular behavior occurs when two consecutive supports intersect at .
 Singular supports I:

, ,
 Singular supports II:

, .
There is a multisingular case, when all three supports meet at :
 Singular supports III:

.
Besides a singular cut structure for the Riemann surface, we can also have a singular behavior of the first measure . These singular cases also appear in the usual equilibrium problem for the onematrix model, see [33], and they are as follows.
 Singular interior point for :

The density of vanishes at an interior point of .
 Singular endpoint for :

The density of vanishes to higher order than square root at an endpoint of .
 Singular exterior point for :

Equality holds in the variational inequality in (1.23) at a point .
The measures and cannot have singular endpoints, singular exterior points, or singular interior points, except at . Singular interior points of these measures at are as follows.
 Singular interior point for :

The density of vanishes at .
 Singular interior point for :

The density of vanishes at .
While there is great interest in the singular cases we restrict the analysis in this paper to the regular cases, for which we give the following precise definition.
Definition 1.3.
The triplet is regular if the supports of the minimizers from the vector equilibrium problem satisfy
(1.32) 
and if in addition, the measure has no singular interior points, singular endpoints, or singular exterior points, and the measures and do not have a singular interior point at .
The condition (1.32) may be reformulated as
1.6 Limiting mean eigenvalue distribution
The measure is the limiting mean eigenvalue distribution of the matrix in the twomatrix model as . In this paper we prove this only for regular cases. To prove it for singular cases, one would have to analyze the nature of the singular behavior which is beyond the scope of what we want to do in this paper.
Theorem 1.4.
Suppose is regular. Let be the first component of the minimizer of the vector equilibrium problem. Then is the limiting mean distribution of the eigenvalues of as with .
We recall that the eigenvalues of after averaging over are a determinantal point process on with a kernel as given in (1.5). The statement of Theorem 1.4 comes down to the statement that
(1.33) 
where is the density of the measure .
The restriction to is for convenience only, since it simplifies the expressions in the steepest descent analysis of the RiemannHilbert problem that we are going to do.
The existence of the limiting mean eigenvalue distribution was proved by Guionnet [57] in much more general context. She characterized the minimizer by a completely different variational problem, and also connects it with a large deviation principle. It would be very interesting to see the connection with our vector equilibrium problem. A related question would be to ask if it is possible to establish a large deviation principle with the energy functional (1.12) as a good rate function.
We are going to prove (1.33) by applying the DeiftZhou steepest descent analysis to the RiemannHilbert problem (1.36) below. Without too much extra effort we can also obtain the usual universal local scaling limits that are typical for unitary random matrix ensembles. Namely, if then the scaling limit is the sine kernel
while if is an end point of then the scaling is the Airy kernel, i.e., for some , we have
with if and if for some . Recall that we are in the regular case so that the density of vanishes as a square root at . The proofs of these local scaling limits will be omitted here, as they are very similar to the proofs in [45].
1.7 About the proof of Theorem 1.4
The first step in the proof of Theorem 1.4 is the setup of the RiemannHilbert (RH) problem for biorthogonal polynomial and its connection with the correlation kernel . We use the RH problem of [66] which we briefly recall.
The RH problem of [66] is based on the observation that the polynomial that is characterized by the biorthogonality conditions (1.4) can alternatively be characterized by the conditions (we assume is quartic and is a multiple of three)
(1.34) 
which involves three varying (i.e., dependent) weight functions
(1.35) 
The conditions (1.34) are known as multiple orthogonality conditions of type II, see e.g. [4, 63, 76, 82].
A RH problem for multiple orthogonal polynomials was given by Van Assche, Geronimo and Kuijlaars in [83] as an extension of the wellknown RH problem for orthogonal polynomials of Fokas, Its, and Kitaev [55]. For the multiple orthogonality (1.34) the RH problem is of size and it asks for satisfying
(1.36) 
The RH problem has a unique solution. The first row of is given in terms of the biorthogonal polynomial as follows
and the other rows are built out of certain polynomials of degree in a similar way, see [66, 83] for details.
Multiple orthogonal polynomials have a ChristoffelDarboux formula [29] which implies that the correlation kernel (1.5) can be rewritten in the integrable form
for certain functions , for , and in fact it has the following representation
(1.37) 
for , in terms of the solution of the RH problem (1.36), see [29].
The proof of Theorem 1.4 is an involved and lengthy steepest descent analysis of the RH problem 1.36 in which the vector equilibrium problem is used in an essential way. This is similar to [45] which deals with the case . Certain complications arise because the formulas for the external field and the constraint in the vector equilibrium problem are less explicit as in the case . This not only complicates the analysis of the vector equilibrium problem in Sections 2 and 3, but it will continue to play a role via the functions defined in Section 2.2 and defined in Section 4.4 throughout the paper.
We also note that the analysis in [45] was restricted to the onecut case, which leads to an underlying Riemann surface of genus . This restriction was removed in [75]. The problem in the higher genus case is in the construction of the global parametrix. In Section 8 we give a selfcontained account that is based on the ideas developed in [75] and [67], which we think is of independent interest.
We also wish to stress that in Case IV the Riemann surface always has genus , even if consists of one interval, see (4.16) below. This phenomenon did not appear for .
1.8 Singular cases
Although we do not treat the singular cases in this paper we wish to make some comments about the possible critical behaviors that we see in the twomatrix model with the quartic potential .
As already discussed in Section 1.5 the singular behavior is associated with either a singular behavior in the measures , , or , or a singular behavior in the supports. The singular behavior in the measure also appears in the onematrix model that is described by orthogonal polynomials. It is known that the critical behavior at a singular interior point where the density vanishes quadratically is described by the HastingsMcLeod solution of the Painlevé II equation, see [19, 27, 79]. This Painlevé II transition is the canonical mechanism by which a gap opens up in the support in the onematrix model.
The critical behavior at a singular endpoint where the density vanishes with exponent is described by a special solution of the Painlevé I equation (the second member of the Painlevé I hierarchy), see [28]. The critical behavior at a singular exterior point is described by Hermite functions [16, 26, 74] and this describes an opening of a new band of eigenvalues (birth of a cut).
We see these critical behaviors also in the twomatrix model with an even quartic . In particular, the opening of a gap at in the support of is a Painlevé II transition. In our classification of regular cases, this is a transition from Case I to Case II, or a transition from Case IV to Case V. In the phase diagram of Figure 1.1 for , this transition is on the part of the parabola , with .
A Painlevé II transition also appears when either or has a density that vanishes quadratically at . Then is a singular interior point and again a gap can open but now in the support of the measures “that are on the other sheets” and have no direct probabilistic meaning. If the density of vanishes at then the transition is from Case III to Case V. If the density of vanishes at then the transition is from Case I to Case IV or from Case II to Case V. In the phase diagram of Figure 1.1 the transition from Case I to Case IV takes place on the part of the parabola , with .
The cases of singular supports represent critical phenomena that do not appear in the onematrix model. What we called Singular Supports I in Section 1.5 corresponds to a transition from Case III to Case IV. This is a transition when the gap around in the support of closes and simultaneously the gap in the support of opens up (or vice versa). On the level of the Riemann surface it means that the two branch points on the real line that are the endpoints of the gap in come together at , and then split again to become a pair of complex conjugate branch points. These branch points are then on the imaginary axis and are the endpoints of . A transition of this type does not change the genus of the Riemann surface.
This type of transition was observed first in the context of random matrices with external source and nonintersecting Brownian motions, see [3, 21, 25, 81],i where it was described in terms of Pearcey integrals. The Pearcey transition is a second mechanism by which a gap in the support may open up (or close). As it involves three sheets of the Riemann surface it cannot take place in the onematrix model which is connected to a twosheeted Riemann surface.
The case of Singular Supports II gives a transition from Case II to Case III. This is a situation where the gap in the support of closes and simultaneously the gap in opens. This also typically corresponds to a Pearcey transition, but it does not involve the first sheet of the Riemann surface, which means that this transition is not visible in the eigenvalue distribution of the random matrix. In the phase diagram of Figure 1.1 the Pearcey transitions are on the curve , .
The case of Singular Supports III represents a new critical phenomenon. Here the supports of all three measures , and are closed at . In Figure 1.1 this is the case at the multicritical point and where the Painlevé transitions and Pearcey transitions come together. One may approaches the multicritical point from the Case III region, where there is a gap around in the supports of both and , while the support of is the full imaginary axis. At the multicritical point the supports of and close simultaneously, while also the support of opens up, which results in a transition from Case III to Case I.
We conjecture that the case of Singular Supports III is of similar nature as was studied very recently [1, 38] for a critcal case of nonintersecting Brownian motions (or random walks) with two starting and two ending points. By finetuning the starting and ending points one may create a situation where two groups of nonintersecting Brownian motions fill out two ellipses which are tangent to each other at one point. Our conjecture is that the local eigenvalue behavior around in the multicritical case is the same as that for the nonintersecting Brownian motions at the point of tangency. The conjecture is supported by preliminary calculations that suggest that the local parametrix of [38] can also be used if one tries to extend the RH analysis of the present paper to the multicritical situation.