A Vector Equilibrium Problem for Muttalib-Borodin Biorthogonal Ensembles

The Muttalib-Borodin biorthogonal ensemble is a joint density function for $n$ particles on the positive real line that depends on a parameter $\theta$. There is an equilibrium problem that describes the large $n$ behavior. We show that for rational values of $\theta$ there is an equivalent vector equilibrium problem.

1 Introduction and statement of results

The Muttalib-Borodin ensemble
The Muttalib-Borodin biorthogonal ensemble is the following probability density function for n particles on the half line [0, ∞) with θ > 0 and with an n-dependent weight function w(x) = e −nV (x) having enough decay at infinity. The model is named after Muttalib [17] who introduced it as a simplified model for disordered conductors in the metallic regime, and Borodin [4] who obtained profound mathematical results, in particular for Laguerre and Jacobi weights. The model has attracted considerable attention in recent years. Random matrix models whose eigenvalues (or singular values) have the distribution (1.1) were recently given in [6,13]. The model is also related to products of random matrices [13,16].
In the large n limit, the particles have an almost sure limiting measure µ * which is the minimizer of 1 2 log 1 |x − y| dµ(x)dµ(y) + 1 2 log 1 |x θ − y θ | dµ(x)dµ(y) + V (x)dµ(x) (1.2) among all probability measures µ on [0, ∞). This follows from large deviation results for (1.1) and related models that were studied in [3,5,10]. For θ = 1, the functional (1.2) reduces to the usual energy in the presence of an external field [20]. The minimizer for (1.2) was studied in detail by Claeys and Romano [7]. They found sufficient conditions for the minimizer to be supported on an interval [0, a] for some a > 0. Forrester and co-authors [11,12] analyzed the equilibrium problem for (1.2) with special potentials, and found expressions for the minimizers as Fuss-Catalan and Raney distributions, see also [18].
It is the aim of this paper to show that for rational values of θ, say θ = q/r with q, r ∈ N, there is an equivalent minimization problem for a vector of q + r − 1 measures. We expect that the vector equilibrium problem will be useful for subsequent asymptotic analysis. The special role of rational θ also appeared in the already mentioned work [7]. This paper gives finite term recurrence relations for the biorthogonal polynomials associated with (1.1), as well as a Christoffel-Darboux formula for the correlation kernel are given, but only for rational θ.
In order to state our results we introduce the logarithmic energy of a measure µ and the mutual energy of two measures µ and ν. Throughout we use for j ∈ Z, (1.5)

Result for the case θ = 1/r
We first state the result for the case θ = 1/r with r ∈ N.
The measure µ * 0 has compact support, and it is the unique minimizer of the functional (1.2) with θ = 1/r among probability measures on [0, ∞).
The minimization problem for the energy functional (1.6) is an example of a weakly admissible vector equilibrium problem in the sense of Hardy and Kuijlaars, see also below. The other measures µ * 1 , . . . , µ * r−1 have full unbounded support, supp(µ * j ) = ∆ j for j = 1, . . . , r − 1. In fact where Bal denotes the balayage onto ∆ j , see below as well.

Result for rational θ
For general rational θ = q/r with q, r ∈ N, we first make the change of variables x → x 1/q , y → y 1/q in the energy functional (1.2) to obtain where dν(x) = dµ(x 1/q ), and V (x) = V (x 1/q ). Note that q and r play a symmetric role in the energy functional (1.7).
Then there is a unique vector (ν * −q+1 , . . . , ν * −1 , ν * 0 , ν * 1 , . . . , ν * r−1 ) of q + r − 1 measures that minimizes the energy functional among all vectors satisfying for every j = −q + 1, . . . , r − 1, The measure ν * 0 has compact support and it is the unique minimizer of the functional (1.7) among probability measures on [0, ∞), and dµ * 0 (x) = dν * 0 (x q ) is the unique minimizer of (1.2) with θ = q/r. As in Theorem 1.1 the other measures ν * j from Theorem 1.2 have full unbounded support: supp(ν * j ) = ∆ j if j = 0. The energy functional (1.9) withV = 0 and the normalizations m j as in condition ii) appeared in [9,Theorem 2.3] where it describes the limiting eigenvalue distribution of banded Toeplitz matrices. The supports ∆ j of the measures, however, are more general curves in that case. Theorem 1.1 is the special case q = 1 of Theorem 1.2 and it is enough to prove the latter theorem. However, for sake of exposition we chose to state Theorem 1.1 separately as well.

Vector equilibrium problems
The unique existence of a minimizing vector of measures follows from the result of Hardy and Kuijlaars, which we recall here. The general setup of [14] involves the following ingredients.
for every i = 1, . . . , d for which ∆ i is unbounded.

Variational conditions
We use U ν (z) = log 1 |z−y| dν(y), z ∈ C, to denote the logarithmic potential of a measure ν and to denote the spherical potential. The variational conditions for the vector equilibrium problem were not discussed in [14]. The following result is standard for the case d = 1, and its extension to d ≥ 2 is not difficult.
Proof . The functionalJ is strictly convex on the set where it is finite. The strict convexity is c ijĨ (µ i , µ j ) in the functionalJ and it comes down tõ M m j (∆ j ), with strict inequality if µ = ν and bothJ( µ) < +∞,J( ν) < +∞.
Let µ * be as in the lemma. Then for any µ ∈ d j=1 M m j (∆ j ), we find by integrating (2.5) with and so by summing over i, If we integrate (2.5) with respect to µ * i and sum over i, we find an equality and in particularJ( µ * ) < +∞. Now we use (2.6) with ν = µ * , and combine it with (2.7) and (2.8) to find Thus µ * is indeed the minimizer ofJ.

Nikishin interaction and balayage
In the present paper we are dealing with the interaction matrix C = (c ij ) where which is indeed a positive definite matrix, and sets ∆ j that alternate between the positive and negative real axis as in (1.5). The interaction matrix (2.9) is characteristic for Nikishin systems in the theory of Hermite-Padé approximation [19]. See also [1,15] for surveys on the connections with Hermite-Padé approximation and random matrix theory. In Theorem 1.
Hence there exists a unique minimizing vector of measures (ν * −q+1 , . . . , ν * r−1 ) for the energy functionalJ with We have to show that in addition I(ν * i ) < +∞ for all i, and then we can conclude that it is also a minimizing vector for J.
The variational conditions (2.5) from Lemma 2.1 are in this case (2.11) Here we set ν * −q = ν * r = 0 so that (2.10), (2.11) also hold for j = −q + 1 and j = r − 1. Note that 2ν * j and ν * j−1 + ν * j+1 have the same total masses if j = 0. If (2.11) holds then 2ν * j is the balayage measure of ν * j−1 + ν * j+1 onto ∆ j . The balayage measure has full support ∆ j , and equality holds in (2.11) everywhere on ∆ j . In addition, the constant j is zero. So for j = 0 the variational condition is If x → log(1 + |x| 2 ) is integrable with respect to all measures, then (2.10) reduces to for some constant , while (2.12) reduces to which is the more common form for the balayage in logarithmic potential theory, see [20] where the discussion however is restricted to measures with compact support in C.
Our strategy to prove Theorem 1.2 will be to establish the existence of a vector of measures ν * = (ν * −q+1 , . . . , ν * r−1 ) with supp(ν * j ) ⊂ ∆ j , ν * j (∆ j ) = m j , such that the conditions (2.14) and (2.13) are satisfied. The measure ν * 0 will have compact support, and all other measures have full support. The density of ν * j will behave like 3 Proof of Theorem 1.2

An auxiliary result
The following is our main auxiliary result.

Moreover,
and Proof . The minimization of (3.1) under the conditions i), ii), iii) is a weakly admissible vector equilibrium problem for r − 1 measures with total masses m j = 1 − j r , j = 1, . . . , r − 1. We set µ * 0 = δ a , the Dirac point mass at a > 0 and m 0 = 1. Also µ * r = 0 and m r = 0. Then by the discussion in Section 2, there is a unique minimizer (µ * 1 , . . . , µ * r−1 ) for the extended functional. We are going to construct the measures µ * j explicitly. We show that these measures have densities with respect to Lebesgue measures that decay as |x| −1−1/r as |x| → ∞, and that for j = 1, . . . , r − 1, This implies that also also Then by Lemma 2.1 it follows that (µ * 1 , . . . , µ * r−1 ) is the minimizer for the extended functional, and since the function x → log(1 + |x| 2 ) is integrable with respect to each of the measures, it then also follows that it is the minimizer of (3.1). We will see at the end of the proof that (3.2) and (3.3) hold as well.
We use a geometric construction based on the Riemann surface for the mapping w = z 1/r . The Riemann surface has r sheets where R j is connected to R j+1 along ∆ j for j = 1, . . . , r − 1 in the usual crosswise manner. There is one point at infinity that connects all r sheets. Note that R r = C \ (−∞, 0] if r is even, and R r = C \ [0, ∞) if r is odd. The Riemann surface has genus zero and z = w r is a rational parametrization of it. The rational function is meromorphic on the Riemann surface with a simple pole at z = a, a pole of order r − 1 at z = 0, and a zero of order r at z = ∞. We use Ψ j to denote its restriction to R j . Explicitly, we then have with the principal branch of the rth roots. On the other sheets we have the same formula but with different choices of rth roots in z 1−1/r and z 1/r . We always use a 1/r > 0. The cuts ∆ j are oriented from left to right, and we use Ψ j,± (s), s ∈ ∆ j , to denote the limit of Ψ j (z) as z → s with ± Im z > 0. Then we define for j = 1, . . . , r − 1, This defines a real measure on ∆ j since Ψ j,− (s) = Ψ j,+ (s) for s ∈ ∆ j , but a priori it could be a signed measure. Suppose the density vanishes at an interior point s ∈ ∆ j . Then Ψ j,± (s) is real, which implies that Ψ(w) is real for some w ∈ C with w r = s. From the formula (3.4) for Ψ it then easily follows that w is real. However, if w would be real and positive then w r would be on [0, ∞) on the first sheet, and if w would be real and negative then w r would be on R \ ∆ r−1 on the rth sheet. Since w r = s is on one of the cuts, we have a contradiction and we see that the density (3.6) does not vanish at an interior, and therefore has a constant sign. We compute the total masses by contour integration as in [9]. We consider j = 1 first. Then by (3.6) where C is a contour that starts at −∞ and follows the upper side of the cut ∆ 1 = (−∞, 0], and goes back to −∞ on the lower side of the cut. We deform the contour to a big circle |s| = R, and we pick up a residue condition from the pole at s = a. From (3.5) we calculate the residue as with the circle |s| = R oriented counterclockwise. Since for every j = 1, . . . , r − 1, which easily follows from (3.4), we find Now consider 2 ≤ j ≤ r − 1. Then by (3.6) and the fact that Ψ j−1,± = Ψ j,∓ on ∆ j−1 , Again by contour deformation this is where we used (3.7). Together with (3.8) we conclude that dµ * j (s) = 1 − j r , j = 1, . . . , r − 1.
Since the total masses are positive, and the densities of the measures do not change sign, it now also follows that the measures are positive. We introduce the Cauchy transforms of the measures Then by a similar contour integration argument, where now we pick up a residue contribution at s = z, while there is no contribution from infinity, we get and for j = 2, . . . , r − 1, where F r (z) = 0. The identity Ψ j,+ = Ψ j+1,− on ∆ j then leads to for j = 2, . . . , r − 1. By (3.9) and (3.10), the identity (3.11) also holds for j = 1, if we agree that The measures have a density that decays like |s| −1−1/r as |s| → ∞. This easily follows from the definitions (3.4) and (3.6). Then s → log(1 + s 2 ) is integrable for these measures, and the usual logarithmic potentials exist. By Sokhotskii-Plemelj formulas we have Clearly also Then by integrating (3.11) we obtain There is no constant of integration in (3.12) since for each i ∈ {j − 1, j, j + 1} and ∆ j is unbounded. Thus we have reached the identity (3.12) that we aimed for, as discussed in the beginning of the proof. It remains to verify (3.2) and (3.3).
Finally, we recall that by (3.9) and (3.5) which after integration leads to The constant of integration vanishes since both sides behave like (1 − 1/r) log z + o(1) as z → ∞.
Taking real parts we find (3.3).
We next extend Proposition 3.1 from point masses δ a with a > 0 to general measures with compact support on (0, ∞).

Moreover,
Proof . For µ = δ a this was done in Proposition 3.1. Let (µ * 1 (a), . . . , µ * r−1 (a)) be the vector of measures that we obtain from δ a as in Proposition 3.1. Then for a general probability measure µ on (0, ∞) with compact support, we put These are well-defined positive measures satisfying i), ii) and iii) of the proposition. The measures µ * j (a) have a density that decays as |x| −1−1/r as |x| → ∞, and the same will be true for the measures µ * j since µ is compactly supported. Thus the logarithmic potentials exist, and The identity on ∆ j holds for every a > 0 by Proposition 3.1. Integrating this with respect to a and using Fubini's theorem, we obtain As in the proof of Proposition 3.1 this leads to the first identity of (3.13).
If x → log(1 + |x| 2 ) would be integrable with respect to ν * 0 then it would also be the minimizer of I(ν) −Ĩ(ν, ρ) + Ṽ (x) + log 1 + |x| 2 dν(x) which is a usual minimization problem for one measure with an external field that is continuous on (0, ∞) (since V is continuous and ρ is a measure on (−∞, 0]). It is easy to see that x − s > √ 1 + s 2 for x > 1 and s < 0. Thus log x−s √ 1+s 2 dρ(s) > 0 for x > 1, and it follows from (1.8) that which guarantees that (4.4) has a minimizer with compact support. This minimizer also minimizes (4.3) and thus coincides with ν * 0 which thus has compact support. Theorem 1.2 is now fully proved.

A f inal remark
We consider the minimization problem for (1.2) with θ = 1/r. From Theorem 1.1 we obtain the following result that gives conditions that guarantee that the Cauchy transform of the minimizing measure is an algebraic function.
We construct a Riemann surface R with r + 1 sheets R j , j = 0, . . . , r given by R 0 = C \ supp(µ * ), for j = 2, . . . , r − 1, where R 0 is connected to R 1 along supp(µ * ) and R j is connected to R j+1 along ∆ j for j = 1, . . . , r − 1 in the usual crosswise manner. After adding points at infinity we obtain a compact Riemann surface, since supp(µ * ) consists of a finite union of intervals.
Then Ψ is meromorphic on each of sheets (since V is a rational function). Moreover, the variational conditions (5.1) tell us that Ψ extends to a meromorphic function on the full Riemann surface R. Then also V − Ψ is a meromorphic function on R which agrees with F 0 on the zero sheet. Therefore F = F 0 satisfies an algebraic equation of degree r + 1.
Note that µ * minimizes I(µ) + (V − U µ * 1 )dµ among all probability measures µ on [0, ∞), and the external field V − U µ * 1 is real analytic on (0, ∞). If it were also real analytic at 0, then it would follow from results in [8] that µ * is supported on a finite union of intervals. Maybe the methods of [8] can be adapted to the present situation, and then the assumption in Proposition 5.1 about the finite number of intervals would be unnecessary.
The Riemann surface in the proof of Proposition 5.1 has genus 0 if and only if supp(µ * ) = [0, a] for some a > 0. This is the case if V (x) = x, and for more general conditions see [7,Theorem 1.8]. We note that this Riemann surface also appears in the paper of Forrester, Liu and Zinn-Justin [12], see Fig. 1 in that paper.