The reciprocal Mahler ensembles of random polynomials

The reciprocal Mahler ensembles of random polynomials

Christopher D. Sinclair    Maxim L. Yattselev
Abstract

We consider the roots of uniformly chosen complex and real reciprocal polynomials of degree whose Mahler measure is bounded by a constant. After a change of variables this reduces to a generalization of Ginibre’s complex and real ensembles of random matrices where the weight function (on the eigenvalues of the matrices) is replaced by the exponentiated equilibrium potential of the interval on the real axis in the complex plane. In the complex (real) case the random roots form a determinantal (Pfaffian) point process, and in both cases the empirical measure on roots converges weakly to the arcsine distribution supported on . Outside this region the kernels converge without scaling, implying among other things that there is a positive expected number of outliers away from . These kernels, as well as the scaling limits for the kernels in the bulk and at the endpoints are presented. These kernels appear to be new, and we compare their behavior with related kernels which arise from the (non-reciprocal) Mahler measure ensemble of random polynomials as well as the classical Sine and Bessel kernels.

Keywords: Mahler measure, random polynomials, asymmetric random matrix, Pfaffian point process, universality class

Mathematics Subject Classification 2000: 15B52, 60B20, 60G55, 82B23, 15A15, 11C20

1 Introduction

We study two ensembles of random polynomials/matrices related to Ginibre’s real and complex ensembles of random matrices, but with weight functions which are not derived from ‘classical’ polynomial potentials, but rather from the equilibrium logarithmic potential of the interval on the real axis in the complex plane. This complements our study of similar ensembles formed from the equilibrium logarithmic potential of the unit disk \@cite?,?.

We introduce the complex ensemble first since it is simpler to define. Consider a joint density function of complex random variables identified with given by , where

(1.1)

Here is the weight function, and is a constant necessary to make a probability density. When , gives the joint density of eigenvalues of a matrix chosen randomly from Ginibre’s ensemble of complex matrices with i.i.d. complex standard normal entries \@cite?. When for , then defines the joint density of the roots of random complex polynomials of degree chosen uniformly from the set of such polynomials with Mahler measure at most 1 (Mahler measure is a measure of complexity of polynomials; see Section 1.1 below).

The ensembles (1) can be put in the context of two-dimensional electrostatics by envisioning the random variables as a system of repelling particles confined to the plane, and placed in the presence of an attractive potential which keeps the particles from fleeing to infinity. When such a system is placed in contact with a heat reservoir at a particular temperature, the location of the particles is random, and the joint density of particles is given by for . The connection between eigenvalues of random matrices and particle systems in 2D electrostatics is originally attributed to Dyson \@cite?, and is central in the treatise \@cite?. From this perspective, it makes sense to investigate for different naturally-arising confining potentials.

Real ensembles are different in that the roots/eigenvalues of the real polynomials/matrices are either real or come in complex conjugate pairs. Hence, the joint density for such ensembles is not defined on , but rather on

where the union is over all pairs of non-negative integers such that . For each such pair, we specify a partial joint density given by

(1.2)

where is the vector formed by joining to all the and their complex conjugates, and is the normalization constant given by

(Here and are Lebesgue measure on and respectively). Note that we may assume that , since otherwise we could replace with , without changing the partial joint densities.

In this work we focus on the case , where the confining potential is the scaled logarithmic equilibrium potential of the interval , see Section 1.2. That is,

(1.3)

where we take the principal branch of the square root. We view as a parameter of the system and we will call the point process on whose joint density is given by (1) and (1.3) the complex reciprocal Mahler ensemble. Similarly, the joint densities given by (1.2) and (1.3) define a process on roots of real polynomials, and we will call this process the real reciprocal Mahler process. The reason we call these point processes Mahler ensembles is they can be interpreted as choosing polynomials uniformly at random from starbodies of Mahler measure when viewed as distance functions (in the sense of the geometry of numbers) on coefficient vectors of reciprocal polynomials.

1.1 Mahler Measure

The Mahler measure of a polynomial is given by

This is an example of a height or measure of complexity of polynomials, and arises in number theory when restricted to polynomials with integer (or otherwise algebraic) coefficients. There are many examples of heights (for instance norms of coefficient vectors), but Mahler measure has the attractive feature that it is multiplicative: .

There are many open (and hard) questions revolving around the range of Mahler measure restricted to polynomials with integer coefficients. Perhaps the most famous is Lehmer’s question which asks whether 1 is a limit point in the set of Mahler measures of integer polynomials \@cite?. Since cyclotomic polynomials all have Mahler measure equal to 1, it is clear that 1 is in this set; it is unknown whether there is a ‘next smallest’ Mahler measure, though to date the current champion in this regard provided by Lehmer himself is

whose Mahler measure is approximately .

A reciprocal polynomial is a polynomial whose coefficient vector is palindromic; that is, is a degree reciprocal polynomial if . Clearly if is a root of , then so too is , from which the name ‘reciprocal’ arises. Reciprocal polynomials arise in the number theoretic investigation of Lehmer’s conjecture since, as was shown by Smyth in the early 1970s, the Mahler measure of a non-reciprocal integer polynomial is necessarily greater than approximately 1.3 \@cite?. Thus, many questions regarding the range of Mahler measure reduce to its range when restricted to reciprocal integer polynomials.

Another question regarding the range of Mahler measure concerns estimating the number of integer polynomials (or the number of reciprocal integer polynomials) of fixed degree and Mahler measure bounded by as . Such questions were considered in \@cite? and \@cite?, and the main term in this estimate depends on the volume of the set of real degree polynomials (or real reciprocal polynomials) whose Mahler measure is at most 1. This set is akin to a unit ball, though it is not convex. It is from here that our interest in the zeros of random polynomials chosen uniformly from these sorts of ‘unit balls’ arose. The (non-reciprocal) case was studied in \@cite? and \@cite?, and here we take up the reciprocal case.

In order to be precise we need to specify exactly what we mean by choosing a reciprocal polynomial uniformly from those with Mahler measure at most 1. For we define the -homogeneous Mahler measure by

(1.4)

To treat polynomials with complex and real coefficients simultaneously, we shall write to mean or depending on the considered case. Identifying the coefficient vectors of degree polynomials with , we also view as a function on , and define

These are the degree unit-starbodies for Mahler measure. We can then define the reciprocal unit starbody as the intersection of the subspace of reciprocal polynomials with the . However, as the set of reciprocal polynomials has Lebesgue measure zero in , this is not an optimal definition for the purposes of selecting a polynomial uniformly from this set. In order to overcome this difficulty we need some natural parametrization of the set of reciprocal polynomials.

If is odd, and is reciprocal, then . That is, is always a root of an odd reciprocal polynomial, and is an even degree reciprocal polynomial. Thus, when considering the roots of random reciprocal polynomials, it suffices to study even degree reciprocal polynomials. By declaring that and demanding that be multiplicative, we can extend Mahler measure to the algebra of Laurent polynomials , and we define a reciprocal Laurent polynomial to be one satisfying . Notice that reciprocal Laurent polynomials form a sub-algebra of Laurent polynomials. We can map the set of degree reciprocal polynomials on a set of reciprocal Laurent polynomials via the map , and the -homogeneous Mahler measure is invariant under this map. We will call the image of this map the set of degree reciprocal Laurent polynomials (the leading monomial is , but there are zeros).

Now observe that if is a degree polynomial, then is a degree reciprocal Laurent polynomial and conversely, any reciprocal Laurent polynomial is an algebraic polynomial in . Hence, we define the reciprocal Mahler measure of to be the Mahler measure of . Specifically, let be defined by . As before, if we identify the set of degree polynomials with and define the reciprocal starbodies to be

The real/complex reciprocal Mahler ensemble is the point process on induced by choosing a polynomial uniformly from for or . It was observed by the first author that the joint density function of such a process is given by (1) and (1.3) in the complex case \@cite? and by (1.2) and (1.3) in the real case \@cite? with . Without going into the details we just mention that the factors and in (1) and (1.2), respectively, come from the Jacobian of the change of variables from the coefficients of polynomials to their roots and with as in (1.3) appears because

where we take the principal branch of the square root, which necessarily yields that

(1.5)

1.2 Mahler Measures and Logarithmic Potentials

Mahler measure and the reciprocal Mahler measure can put into the more general framework of multiplicative distance functions formed with respect to a given compact set . Indeed, given a compact set , it is known that the infimum of the logarithmic energies

taken over all probability Borel measures supported on is either infinite ( cannot support a measure with finite logarithmic energy; such sets are called polar) or is finite and is achieved by a unique minimizer, say , which is called the equilibrium distribution on . The logarithmic capacity of is set to be zero in the former case and in the latter. It is further known that the function

is positive and harmonic in and is zero on except perhaps on a polar subset. Assume for convenience that has capacity , i.e., . Then we can define the multiplicative distance function with respect to on algebraic polynomials by

where . When is simply connected, is in fact continuous in and is given by which is continued by to , where is a conformal map of the complement of to the complement of the closed unit disk such that . Then

One can easily check now that the Mahler measure , see (1.4), and the reciprocal Mahler measure , see (1.5), are equal to and , respectively.

1.3 Determinantal and Pfaffian Point Processes

Everything in this section is standard, but we include it for completeness. The interested reader might consult \@cite?,? and \@cite? to get a more detailed explanation of the complex and real cases, respectively.

Given a Borel set and a random vector chosen according to (1), we define the random variable to be the number of points in that belong to . The th intensity or correlation function of the ensemble (1) is a function such that for disjoint Borel sets ,

(1.6)

Correlation functions are the basic objects from which probabilities of interest are computed. Ensembles with joint densities of the form (1) are determinantal, that is, there exists a kernel such that for all ,

(1.7)
Theorem 1.1

The kernel for the ensemble (1) is given by

where are orthonormal polynomials with respect to (w.r.t.) the weight , i.e., .

The situation is a bit more complicated for real ensembles with partial joint densities given by (1.2) since the expected number of real roots is positive. In this case, we define the -intensity or correlation function to be such that for disjoint Borel subsets of the real line and disjoint Borel subsets of the open upper half plane,

(1.8)

Ensembles with partial joint densities given as in (1.2) are Pfaffian: there exists a matrix kernel such that

(1.9)

The formula for the kernel depends on the species (real or complex) of the argument. This kernel takes the form

(1.10)

where of a non-real number or zero is set to be zero, is an orto-kernel , and

(1.11)

where when written on the left, as in , acts on as a function of , and when written on the right it acts on as a function of .

Theorem 1.2

Let be even. Then the orto-kernel for ensemble (1.2) is given by

(1.12)

where polynomials , , are skew-orthonormal, that is, and , w.r.t. the skew-symmetric inner product

(1.13)

Note that the skew-orthogonal polynomials are not uniquely defined since one may replace with without disturbing skew-orthogonality. Moreover, if polynomials and are skew-orthonormal, then so are and . However, neither of these changes alters the expression (1.12) for the orto-kernel .

When is odd, there is a formula for the orto-kernel similar to that given in Theorem 1.2. We anticipate that the scaling limits of the odd case will be the same as those for the even case (reported below), but due to the extra complexity (with little additional gain) we concentrate only on the even case here.

2 Main Results

Henceforth, we always assume that and are such that the limits

(2.1)

exist. Furthermore, we set , . Notice that is well defined in the whole complex plane.

2.1 Orthogonal and Skew-Orthogonal Polynomials

Denote by and the kernels introduced in (1.7) and (1.10), respectively, for as above. It follows from Theorem 1.1 and \@cite?Proposition 2 that

(2.2)

where is the -th monic Chebyshëv polynomial of the second kind for , i.e.,

(2.3)

Similarly, we know from Theorem 1.2 that the orto-kernel is expressible via skew-orthogonal polynomials.

Theorem 2.1

Polynomials skew-orthonormal w.r.t. skew-inner product (1.13) with as above are given by

(2.4)

where is the classical ultraspherical polynomial of degree , i.e., it is orthogonal to all polynomials of smaller degree w.r.t. the weight on having as the leading coefficient.

2.2 Exterior Asymptotics

We start with the asymptotic behavior of the kernels in .

Fig. 1: The unscaled limiting spatial density of complex roots near for the complex ensemble on the left and the real ensemble on the right. Notice the cleft along the real axis for the real ensemble. This is due to the fact that roots repel. When the roots come in complex conjugate pairs, this repulsion introduces a paucity of complex zeros near the real axis.
Theorem 2.2

Assuming (2.1), it holds that

(2.5)

locally uniformly for .

It follows from (2.5) that the limit of is equal to zero when , while

when . Hence, the expected number of zeros of random polynomials in each open subset of is positive and finite in this case, see (1.7).

Theorem 2.3

Let be even. Assuming (2.1), it holds that

(2.6)

locally uniformly for .

Theorem 2.3 indicates that has a non-zero exterior limit only when .

Theorem 2.4

Under the conditions of Theorem 2.3, assume in addition that . Then it holds that

(2.7)

locally uniformly for and , where

for , where is understood to be holomorphic in .

Even though is defined for real arguments only, its partial derivatives naturally extend to complex ones. Notice also that is equal to the right-hand side of (2.6).

2.3 Scaling Limits in the Real Bulk

To find the scaling limits of and on , it is convenient to compute these limits for separately. Notice that whenever .

Proposition 2.5

Given , set . Assuming (2.1), it holds that

locally uniformly for and when and locally uniformly for and when (the limit is zero in this case).

In light of Proposition 2.5, let us write

(2.8)

Then the following theorem holds.

Theorem 2.6

Assuming (2.1), it holds that

(2.9)

locally uniformly for and .

In the real case, analogously to (2.8), let us set

(2.10)

where ( in this case). Then the following theorem holds.

Theorem 2.7

Let be even. Assuming (2.1), it holds that

(2.11)

locally uniformly for and . Furthermore, it holds that

(2.12)

locally uniformly for , , and . Finally, we have that

(2.13)

locally uniformly for and .

Fig. 2: The scaled kernels (2.9) and (2.12) as a function for . The darkest curve is for and is equal to the classical sine kernel. Note that for the real ensemble, when this kernel does not tell us about the local density of real roots, but rather tells us about density of complex roots near the real axis. In this situation, as increases, the attraction of zeros to decreases and the zeros are more likely to drift into the complex plane. This phenomenon is captured by the decrease in amplitude of the kernel.

Notice that knowing limit (2.12) is sufficient for our purposes as by (1.12) as the orto-kernel is antisymmetric. Observe also that

uniformly for when by (2.12) ( is exactly the function needed to compute the expected number of real zeros, see (2.18) further below).

2.4 Scaling Limits at the Real Edge

Since , , and , we report the scaling limits at only.

Proposition 2.8

Assuming (2.1), it holds that

uniformly on compact subsets of when , and uniformly on compact subsets of when (the limit is zero).

In the case of random polynomials with complex coefficients the following theorem holds.

Theorem 2.9

Let be as (2.8). Assuming (2.1), it holds that

(2.14)

uniformly for on compact subsets of .

Recall that the Bessel functions of the first kind are defined by

where we take the principal branch of the ’s power of . Since , the right-hand side of (2.14) specializes to

when and , which is a classical Bessel kernel up to the factor .

Theorem 2.10

Let and be as in (2.10) and be even. Assuming (2.1), it holds that

(2.15)

uniformly for , where . Furthermore, we have that

(2.16)

uniformly for and , where . Finally, it holds that

(2.17)

uniformly for , where .

Fig. 3: The scaled spatial density of complex roots near for the complex ensemble on the left and the real ensemble on the right. Here we see a desire for roots to accumulate near (the origin) with a sharper decrease in the density as we move along the positive -axis (away from the bulk) than along the negative -axis (into the bulk). The difference between the ensembles is starkest in the direction, and we can see the competition between the attraction to caused by the potential and the repulsion from the -axis caused by the repulsion between complex conjugate pairs of roots. These images are produced from (2.14) and (2.16).

Equations (2.16) and (2.17) do not cover the cases and , respectively. Such limits exist and we do derive formulas for them, see (3.36) and (3.37) in Lemma 3.20. Unfortunately, these formulas are much more cumbersome, which is the reason they are not presented here.

2.5 Expected Number of Real Zeros

The zeros of polynomials with real coefficients are either real or come in conjugate symmetric pairs. Hence, one of the interesting questions about such polynomials is the expected number of real zeros. Given a closed set , denote by the number of real roots belonging to of a random degree polynomial chosen from the real reciprocal Mahler ensemble. Then

(2.18)

by (1.8), (1.9), (1.10), and the anti-symmetry of and . Moreover, the following theorem holds.

Theorem 2.11

Let be the number of real roots on of a random degree polynomial chosen from the real reciprocal Mahler ensemble. Then

(2.19)

Furthermore, let be the number of real roots on of the said polynomial. Then

(2.20)

where if there exists such that .

3 Proofs

3.1 Proof of Theorem 2.1

We start by representing polynomials from (2.4) as series in Chebyshëv polynomials.

Lemma 3.1

Let and be given by (2.4). Then it holds that