Large deviations for the empirical measure of random polynomials: revisit of the Zeitouni-Zelditch theorem.
This article revisits the work by Ofer Zeitouni and Steve Zelditch on large deviations for the empirical measures of random orthogonal polynomials with i.i.d. Gaussian complex coefficients, and extends this result to real Gaussian coefficients. This article does not require any knowledge in geometry. For clarity, we focus on two classical cases: Kac polynomials and elliptic polynomials.
- 1 Introduction
- 2 Definitions and notations
3 Large deviations for Kac polynomials
- 3.1 Step 1: Distribution of the roots
- 3.2 Step 2: Large deviations on the unit sphere
- 3.3 Step 3: Proof of the Large Deviations Principles
- 3.4 Step 4: Going back on the plane
- 4 Large deviations for Elliptic polynomials
- 5 General result of Zeitouni and Zelditch
- 6 Acknowledgements
We study three different models of random polynomials, orthogonal polynomials, Kac polynomials, and elliptic polynomial, the two last being examples of orthogonal polynomials. The coefficients are i.i.d. random variables which can be either:
|Complex Gaussian coefficients||where|
|Real Gaussian coefficients|
We will always refer to those two possibilities as the complex and real case. In all the article, we assume that the ’s are independent. Given a basis of we consider the random polynomials:
In order to study the zeros of the random polynomials we introduce their empirical measure:
We will focus on three classes of random polynomials corresponding to different choices of the polynomials ’s:
|Orthogonal polynomials||orthonormal family in|
The study of the zeros of random polynomials started with articles by Kac [kac], Littlewood and Offord [littlehoodofford], Hammersley [hammersley] which focused on the number of real zeros. The literature about random polynomials is vast, we refer to the book by Bharucha-Reid and Sambandham [bharucha] and the article by Tao and Vu [taovu] for a nice account of the classical results. The study of the complex roots was initiated by Polya, Sparo and Sur. Recently, the minimal condition to obtain the convergence of was given by Kabluchko and Zaporozhets [kabluchkozapopoly].
The purpose of this article is to revisit the article of Zeitouni and Zelditch [zeitounizelditch]. They prove that the empirical measures of the zeros of random orthonormal polynomials with respect to a scalar product in and complex Gaussian coefficients satisfy a large deviation principle in the projective space . Here we revisit their proof of their theorem in an elementary way although the techniques used are mainly a reformulation of their work. Using a compactification technique based on inverse stereographic projection, we prove a large deviation principle for the push-forward problem on a sphere of and then obtain the result in . The proof adapts to the case of real coefficients, which allows us to extend the theorem. The compactification technique was first introduced by Zelditch in [zeitounizelditch], and discovered again independently by Hardy [hardy] in order to prove a large deviation principle for Coulomb gases with weakly confining potential. The compactification method was also used by Bloom in [bloom] in a more general framework.
Large deviations for empirical measures of random polynomials are only known for Gaussian complex coefficients [zeitounizelditch], which is the subject of the present work, and for exponential coefficients in the Kac case studied by Ghosh and Zeitouni in [ghoshzeitouni]. All these cases rely on the ability to compute the law of the roots of . These results should be compared with their equivalent in random matrix theory: the Ginibre ensemble, real or complex. Many authors used the link between Coulomb gases and eigenvalues of random matrices to obtain large deviation principles as in Ben Arous and Guionnet [benarousguionnet], Ben Arous and Zeitouni [benarouszeitouni], Hiai and Petz [hiaipetz]. In a more general setup, large deviation principle for empirical measures of a Coulomb gas are valid. See for example [chafai] for a similar result in any dimension with general repulsion, Hardy [hardy], or Bloom [bloom].
Given a probability measure and a continuous function , we consider the scalar products on :
Let be an orthonormal basis for this scalar product, we define
We call the support of . We assume that its compactification by inverse stereographic projection is non-thin at all the points of its closure. This notion comes from potential theory and is detailed in [ransford1995potential, p. 78]. We can understand it as the requirement that the support of is not too degenerated. For instance, if the support of is connected and has more than one point, it is non-thin at all its points [ransford1995potential, Thoerem 3.8.3 p 79]. It also holds if it has a finite number of connected components with more than one point. On the other hand, a polar set is thin at every point. We define the Berstein-Markov property which was introduced in [zeitounizelditch]. This property is the key of the proof of the large deviations upper bound.
Definition 1.1 (Bernstein-Markov property).
We say that the couple satisfies the Bernstein-Markov property if, for every , there exists a constant such that, for any and for any polynomial we have:
where is the support of the measure .
We define the Hamiltonian:
In the complex case, the distribution of the roots of is given by:
where is the Lebesgue measure on , and is a constant. is the inverse of a temperature, so we can see as a cooling scheme. In this article, we will always consider:
which corresponds to the distribution of the roots of random polynomials. We can see as a system of particles in interaction. The term
corresponds to a repulsion between the particles and is compensated by the confinement
This model is very close to the classical Coulomb gas model, where the confinement takes the simpler form , which does not involve any interaction between the particles. This non-interaction property can be seen as linearity with respect to the empirical measure via the relation:
The confinement term associated to the Hamiltonian (5) is more complicated, but can still be compared to a classical potential thanks to the Jensen inequality.
The study of the real case is interesting only if the polynomials ’s are real. In the real case, the distribution of the roots is not absolutely continuous with respect to the Lebesgue measure of as the probability to have a real root is positive. This distribution is given by the following mixture:
where we defined, and being the Lebesgue measures on and ,
and where are constants. The first particles are on the real line and with pairs of complex numbers and their conjugates. In the complex case, all the results of this article are valid for any sequence satisfying
In this article, the term weak topology corresponds to the topology of convergence in distribution, which is the weak topology associated to continuous and bounded test functions. This topology is associated to the Bounded Lipschitz metric defined as:
where the surpremum is taken over functions bounded by and -Lipschitz.
Theorem 1.2 (Large deviation principle for complex orthogonal polynomials).
Theorem 1.3 (Large deviation principle for real orthogonal polynomials).
Let be the empirical measure of the gas (7). If the couple satisfies the Bernstein-Markov property, then satisfies a large deviation principle in with the weak topology, speed and good rate function:
This means that for any Borel set we have:
Those last two theorems imply that, in both cases, almost surely:
where is the unique minizer of function . This is a consequence of the Borel-Cantelli Lemma used with the sets . The minimizer is the equilibrium measure of the support of , see [zeitounizelditch, Lemma 30].
The most important example of orthonormal polynomials are Kac polynomials:
The canonical basis is orthonormal with respect to the scalar product on :
where is the uniform measure on , the unit circle of . The sequence converges almost surely weakly towards the measure . Although this result is quite ancient, we can deduced it from (9). We define the Hamiltonian:
In the complex case, the distribution of the roots is given by the Gibbs measure:
In the real case, the distribution of the roots is the mixture:
where the are constants.
Theorem 1.4 (Large deviations for complex Kac polynomials).
Let be the empirical measure of the gas (13). Let us define :
When is finite, this function can be simplified to:
The random sequence satisfies a large deviation principle in with the weak topology and with speed and good rate function . For any Borel set we have:
Theorem 1.5 (Large deviations for real Kac polynomials).
Let be the empirical measure of the gas (14), then the random sequence satisfies a large deviation principle in for the weak topology with speed and good rate function where:
This means that for any Borel set we have:
We will see how the study of Kac polynomials can be adapted to prove a large deviation principle for the empirical measure associated to the roots of polynomials of the form
The polynomials are orthonormal for the scalar product on :
As multiplying a polynomial by a constant does not change the zeros, the factor is omitted. It is known that the random sequence converges almost surely weakly towards
which is called the complex Cauchy measure222or Fubini-Study measure. It can be seen as a consequence of (9). We define the Hamiltonian:
and the Gibbs measure associated to the distribution of the roots in the complex case:
In the real case, the roots form a mixture of Coulomb gases distributed with respect to:
where the are constants.
Theorem 1.6 (Large deviation principle for complex elliptic polynomials).
Let be the empirical measure of the gas (17). Let us define :
When , we can write:
satisfies a large deviation principle in with the weak topology and with speed and good rate function . This means that for any Borel set we have:
Theorem 1.7 (Large deviation principle for real elliptic polynomials).
Let be the empirical measure of the gas (18). satisfies a large deviation principle in with the weak topology, speed and good rate function:
This means that for any Borel set we have:
Outline of the article
We will give a full proof of the results for Kac polynomials, and then we will show how to adapt the proof for elliptic and orthogonal polynomials. The proofs of the previous theorems are similar, and will follow these steps:
Compute the distribution of the roots on ;
Use of inverse stereographic projection to push-forward every object on , the sphere in centered on and of radius ;
Prove a large deviation principle in ;
Use of contraction principle to obtain the large deviation principle in .
We use inverse stereographic projection because, as is compact, a weak large deviation principle is equivalent to a full large deviation principle, without proving exponential tightness. [dembozeitouni, Lemma 1.2.18]
In Section 2 we introduce the objects that will be studied in the article. In Section 3 we give a dettailed proof of the result for Kac polynomials following the steps given above. Section 4 is about elliptic polynomials. As the proof is nearly the same, we focus on what should be changed to import the proof from the previous section. In Section 5 we prove the general result that was originally proved by Zeitouni and Zelditch in [zeitounizelditch] and we extend it for real Gaussian coefficients.
In contrast, the article [zeitounizelditch] has a more geometric and intrisinc approach. The scalar product (3) is related to a notion of curvature on . The zeros are seen as elements of and the rate function is expressed in terms of Green function and Green energy associated to this geometric setup.
2 Definitions and notations
We give some definitions that will be useful in the article.
Definition 2.1 (Logarithmic potential, logarithmic energy).
We call the logarithmic potential of a measure the function:
We also define the logarithmic energy of a measure :
and defined by
where is the unit circle in .
Definition 2.2 (Discrete logarithmic energy.).
Let then we write:
where stands for the off-diagonal integral .
We will use the same notation for measures on or on .
Let us define now the inverse stereographic projection that will be the key tool in this article.
Definition 2.3 (Inverse stereographic projection).
Let be the sphere in of center and radius . We call the point the north pole. Let the inverse stereographic projection
We have the following relations, valid for any and in :
where if , is its Euclidean norm and when is a complex number, is its modulus. The same notation holds for the norm in and the norm in .
The first relation can be found in [ashnovinger, lemma ], and the second relation is obtained from the first one by squaring, taking the limit as tends to infinity and using the Pythagorean theorem.
To avoid confusions between what lies in and what lies in , we will only use the letters for complex numbers and the letters for vectors in .
Definition 2.4 (Push-forward of the objects on the sphere).
We define the push-forward by of the empirical measure:
Let , we call its logarithmic potential on the sphere the function:
takes its values in . We define on the function:
The function is called logarithmic potential. It inherits its name from as it is the analog formula on the sphere. The name logarithmic potential is not really appropriate as this notion is already defined on the sphere in potential theory, but it is convenient as the formulas are the same.
3 Large deviations for Kac polynomials
3.1 Step 1: Distribution of the roots
Theorem 3.1 (Distribution of the roots in for the complex case).
Let , the law of is absolutely continuous with respect to the Lebesgue measure on with density:
where is a normalizing constant.
Let . Then the transformation
has Jacobian determinant [houghtkris, Lemma 1.1.1]. We compute the law of the random vector . The density of the law of is
We consider now the change of variables:
whose Jacobian determinant is , as . Hence, the law of is absolutely continuous with respect to the Lebesgue measure on . We want to rewrite the density of the random vector with the new variables . We notice that if then:
where is the uniform probability measure on the unit circle of . This relation comes from the fact that the canonical basis of is orthonormal for the scalar product (11). The density of the law of is:
We only have to integrate in the variable to obtain the law of . ∎
Theorem 3.2 (Distribution of the roots in the real case.).
The distribution of the random vector of the roots of in the real case is given by:
This law can be re-written as:
The density in the real case is nearly the same as in the complex case, except for the factor in the exponent. In the complex case, the vector of the zeros is an element of while in the real case, we can see the zeros as an element of .
As the random vector has a joint distribution:
we can use Zaporozhets’ computation [zaporozhets] in order to express the distribution of . We use again the relation (22) in order to simplify the expression of the distribution of the roots and we obtain:
We integrate with respect to , which ends the proof. ∎
Remark 3.3 (Symmetries of the problem).
It is easy to check that the law of the zeros is invariant under rotation as a Gaussian vector is invariance by rotation. The distribution of the zeros is also invariant under the mapping . This comes from the fact that has the same distribution as , but if we call the zeros of , then the zeros of are .
Proposition 3.4 (Uniform control of for ).
Let given in 3.2, we have:
which implies that
Proof of Proposition 3.4.
Using the triangular inequality and bounding by gives:
The upper bound is uniform in and tends to as goes to infinity, which proves the result. ∎
This control over the constants is very important. This is the reason why we are able to prove large deviations in the real case. For general , we cannot prove lage deviations without assuming that those limits exist and are equal, not necessarily to zero. This will become clearer in Section 3.3.4. See [benarouszeitouni] and [ghoshzeitouni] for similar results.
3.2 Step 2: Large deviations on the unit sphere
In order to prove the large deviation principles, we are going to use a compactification method introduced in [hardy]. When the potential does not grow faster than a logarithm at infinity, the standard proofs of large deviations principles do not hold. More precisely, exponential tightness of the sequence of measures cannot be proved using the standard techniques presented in [benarousguionnet], [benarouszeitouni], [hiaipetz]. The gas we are studying is also weakly confining as the confinement term grows at infinity like in each variable.
Using the inverse stereographic projection (2.3) we will push the problem on the sphere in . As the sphere is a compact set, it is sufficient to prove a weak large deviation principle instead of a full one.
Remark 3.5 (Push Forward).
In this article, we will use the notation for the push-forward of the measure by the function .
Definition 3.6 (Measure on ).
We call the push-forward of the Lebesgue measure of on by and the push-forward of the Lebesgue measure on by , where is seen as a subspace of . We will use the notation:
Proposition 3.7 (Pushing the complex case on the sphere).
Let be the zeros of in the complex case, then the law of is absolutely continuous with respect to the push forward by of the Lebesgue measure on with density:
We call the finite measure:
This law can be written in the form:
Remark 3.8 (Identification of the uniform measure on ).
The measure on is proportional to the uniform measure on the sphere. Indeed if we push forward this measure by the stereographic projection we obtain the measure which is proportional to the complex Cauchy measure, which is known to be the projection of the uniform measure on the sphere.