The characteristic polynomial of a random unitary matrix and Gaussian multiplicative chaos - The $L^2$-phase

We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.


Introduction
Studying the eigenvalues of the Circular Unitary Ensemble (CUE) -that is Haar distributed random unitary matrices -is a classical problem in random matrix theory [18].Recently it has gotten a lot of attention due to the conjectured relationship between the Riemann ζ-function on the critical line t → 1  2 + it [27] and characteristic polynomials of large random matricesnamely it is believed that statistical properties of the ζ function on the critical line are related to the corresponding properties of the characteristic polynomial of a large random matrix.The goal of this note is to study the behavior of the absolute value of the characteristic polynomial of a large CUE matrix.
As we are dealing with unitary matrices, the eigenvalues lie on the unit circle, so to study the eigenvalues of a CUE matrix, it is natural to study its characteristic polynomial restricted to the unit circle.This object has indeed been studied and there are several types of results.For example, in [27], a central limit theorem is proved: if one normalizes the logarithm of the characteristic polynomial in a suitable deterministic way at any fixed point, it converges in law to a Gaussian random variable.In [7] the law of the characteristic polynomial at a given point was given a rather simple representation in terms of a product of independent random variables.In [11] it was proven that on a microscopic scale (the scale of the separation of the eigenvalues) the characteristic polynomial converges to a random analytic function after a suitable normalization.The result that is the strongest motivation for this work is that in [24] (which builds on the work in [14]).
Here it is proven that on a global scale (namely without any scaling), the logarithm of the absolute value of the characteristic polynomial converges (without any further normalization or shifts) to a random distribution.
While this is an interesting theorem in itself, it has proven fruitful to study objects that are formally exponentials of such distributions.This has given stronger information about the geometric structure of such fields.
Exponentiating Gaussian fields with a logarithmic singularity is the content of the theory of Gaussian Multiplicative Chaos (GMC).It is a theory of random multifractal objects (the objects being measures in the case of real fields and distributions in the case of complex fields) going back to [25].For an excellent review see [34] and for work on the complex case see [29,3].In addition to defining these exponentials, properties such as the multifractal spectrum, modulus of continuity and tail estimates have been studied in the literature [34,5].These objects are also known to be related to the extreme value statistics of the underlying fields [8,31,32].
GMC has recently received a lot of attention from a variety of directions.From the point of view of random geometry, these objects can be seen as the 'quantum area measures' of Liouville Quantum Gravity where they have been used to shed light on the so called KPZ relation [28,15,33] which relates the dimension of fractals in the random geometry to their dimension in Euclidean geometry.GMC has also been used to construct random conformally invariant planar curves through conformal welding [1,38].Other applications involve disordered systems of a particle in a random potential [10] and models in finance [2].Also the general theory of GMC itself has been studied and extended [37,36].
The main goal of this paper is to prove a conjecture that is implied in [22]: after a suitable normalization, the characteristic polynomial (and certain powers of it) converges in law to a GMC measure (or GMC distribution for complex powers).In [22] this connection between GMC and the CUE was used to study (non-rigorously) the maximum of the absolute value of the characteristic polynomial and the ζ-function on the critical line.This connection between GMC and random matrix theory suggests many interesting questions such as how do the known geometric properties of GMC objects (such as the KPZ-relation or modulus of continuity) relate to eigenvalue statistics and vice versa: can results from random matrix theory be used to prove new results for GMC objects.Also from the point of view of GMC this is an example of a rather curious regularization to the measure and underlying field: it is neither Gaussian nor a martingale.This is thus an example of a type of universality result and demonstrates what kind of things one should expect of a regularization of the field so that the GMC construction yields a limiting object.Finally we mention that a secondary goal of this article is to arouse further interest in GMC in the random matrix community and vice versa through the connection proved here.

Acknowledgements:
The author wishes to thank Antti Kupiainen, Eero Saksman, Yan Fyodorov and Nicholas Simm for useful discussions.

Definitions, main result and strategy of proof
In this section we go over the relevant objects and notation as well as discuss loosely the strategy of the proof of our main result.We consider a Haar distributed random unitary matrix and study its characteristic polynomial restricted to the unit circle, namely (1) p where the θ k ∈ [0, 2π] are random variables (the eigenvalues of a Haar distributed n × n random unitary) distributed according to As it is known from [24] that log |p n (θ)| converges to a random distribution (that can't be realized as a random function) it is reasonable to consider the convergence of |p n (θ)| β in a space of distributions as well (though it will actually turn out that for real β we have convergence in a space of measures).For complex β, we actually expect slightly stronger convergence -see the discussion about open problems at the end.Recall that (3) is a separable Hilbert space whose inner product is given by For s > 0, the action of elements of H −s on functions in H s is defined through the orthogonality of Fourier modes: We are interested in proving that for suitable values of β ∈ C and any s > 1  2 , the random distributions M β n ∈ H −s (the densities are of course random functions, but this is the space we can prove convergence in so we prefer to think of them as distributions) converge in law to the random distribution (6) µ β =: e βX : ∈ H −s .
Here ( 7) where Z −k = Z * k and (Z k ) ∞ k=1 are i.i.d.standard complex Gaussians.This is a random distribution living in H −ǫ for any ǫ > 0. It is a centered Gaussian field with covariance E(X(θ)X(θ ′ )) = − log |e iθ −e iθ ′ | 2 .Physicists might call this the massless two-dimensional scalar field restricted to the unit circle.
The actual definition of µ β is in the appendix, where there is a short review of Gaussian multiplicative chaos.
Our main theorem is the following: n understood as a measure converges weakly in law to the measure µ β .
The restriction that Re(β) must not be too negative should not be surprising: for Re(β has a non-integrable singularity.Our main tool is estimating second moments so essentially this translates into a non-integrable singularity for Re(β) ≤ − 1  4 .For further discussion about this, see the section about open problems.The restriction that |β| 2 < 1 2 is crucial to our proof though we expect that the convergence holds in a wider domain (again, see the section on open problems).We also expect that convergence also holds in a slightly stronger form.We discuss this more later.
Our approach is based on the following identity (which essentially follows from det(e A ) = e Tr(A) and log |z| = Re log z): where U n is any unitary with eigenvalues (e iθ j ) j .The result of [24] motivating this work comes from the result in [14], where it is proven that finite collections of 1 √ k TrU k n for positive k converge in law to i.i.d.standard complex Gaussians.This motivates the following approximation of where We also define One can check easily that these are independent of θ due to the translation invariance of the law of (e iθ j ) j Our strategy will be to argue that for fixed k, M β n,k converges in law (as n → ∞) to the random distribution (again the density is a random function but we interpret it as something living in If we are able to control the approximation M β n − M β n,k well enough, the convergence of M β n,k to µ β k and that of µ β k to µ β will be enough to ensure convergence of M β n to µ β .Our strategy for controlling M β n,k − M β n is calculating the variance of its Fourier coefficients.This can be done through powerful results on asymptotics of Toeplitz determinants (some of whose analysis has been carried out only quite recently).We now proceed to the details.

Estimating Fourier Coefficients
To make the statements at the end of the previous section more concrete, let us consider the Fourier coefficients of the distributions: define and ( 14) Our goal will be to estimate ).We expand this as T 1 + T 2 − T 3 − T 3 , where The expectations here can be written in terms of Toeplitz determinants.By the Heine-Szegö formula (see e.g.[9]) .
So each T i is an integral of D n−1 (σ i ).Studying the asymptotic behavior of Toeplitz determinants is a classical problem going back to Szegö.In particular, the strong Szegö theorem can be used to study asymptotics of Toeplitz determinants whose symbol is suitably regular and does not have zeroes (see e.g.[39]).In our case, σ 2 and σ 3 do have zeroes of the form (2 − 2 cos(θ − φ i )) β .This type of Toeplitz determinants have also been studied extensively.An asymptotic form for them was conjectured by Lenard [30] (and a more general conjecture by Fisher and Hartwig [20]) and then proved on different levels of generality in e.g.[40,6,19,13].Let us state the theorem in the generality that applies to our case (namely that proven in [40]).
where θ r = θ s for s = r, Re(α r ) > − 1 2 , τ is sufficiently smooth, non-zero, has zero winding and log τ has a Fourier expansion k∈Z z k e ikθ .Then where G is the Barnes G-function.
So to estimate T i in terms of these results will depend on whether or not we can interchange the order of integration and passing to the limit in the Toeplitz determinant.This is especially non-trivial in the case of T 2 , where the limiting integrand has a singularity.Fortunately, this case has been studied recently [12].The following theorem was a corollary of the powerful theorems proved in the article.
Theorem 3 (Claeys and Krasovsky: Theorem 1.15).Consider the symbol If we could extend to t 1 = π, this integral would be precisely that one appearing in T 2 for β = β = α (due to translation invariance of the T 2 integral, one can get rid of one of the φ i integrals) and l = 0.The extension to t 1 = π is of no issue: we have by the translation invariance of the law of (θ i ).Thus t 1 = π 2 gives us the asymptotic behavior of the integral in T 2 for real β and l = 0.
The extension of this result to complex β and non-zero l is in fact more or less immediate: one uses Theorem 1.8. of [12] instead of Theorem 1.5 and the proof of Theorem 1.15 carries over pretty much word by word (one uses a slightly indented integration contour in the integrals along the negative imaginary axis, but this changes nothing in the proof).The presence of e ilθ plays no role in the proof -in fact, the convergence is uniform in l.Also setting t 1 = π 2 gives us what we need as we can split the integral for T 2 into two parts of similar form.The restriction to real β and l = 0 was simply done to address a conjecture in [22].One thing that is non-trivial is making sure that there is uniform convergence in the area where the singularity is not relevant.As noted in [12] for the real case, this does follow from [13] (or [40]) though it is not explicitly stated in the relevant theorems, so it requires some work for the reader to check this.One finally obtains Corollary 4. For Re(β) > − 1 4 and |β| 2 < 1 2 , where o(1) is uniform in l.
The terms T 1 and T 3 don't require nearly as involved machinery.In fact, when suitably normalized, the integrands converge uniformly (with respect to the φ-variables).Everything boils down to the control we have over functions of the form (20) θ → e ±β k l=1 1 l e ±il(θ−φ) .It is uniformly bounded (uniformity over both φ and θ but not k) as are arbitrary derivatives of it as well.So in particular it is in H ∞ .This gives one strong enough control to prove for example a uniform version of the strong Szegö theorem (see e.g. the proof in section III of [40] or the proof of the strong Szegö theorem -Theorem 4 -in [9]).Alternatively one could prove uniform L 2 bounds for the integrand of T 1 through random matrix theory arguments and then use the convergence in distribution of finite combinations of the TrU j n to independent Gaussians to justify convergence of T 1 in L 2 -though this approach amounts to essentially the proof of a uniform version of the strong Szegö theorem with the methods of [9].
T 3 is slightly more complicated, but again uniform convergence is due to the strong bounds we have on the function above.For an interested reader (who like the author is a non-expert) we refer to [19] for perhaps the most direct way to the details on this part.As a guide to the details, we first note that in our case, the relevant symbols are (in the notation of [19]) (21) b(e iθ ) = e −β 0<|j|≤k 1 We stress that these are all in H ∞ which implies that in the notation of [19], b ∈ F ℓ 2,2 ρ 1 ,ρ 2 for any ρ 1 , ρ 2 ∈ R. Thus the only restriction for the convergence is 2β / ∈ Z − .Starting from the end of the proof, the final and crucial estimate for uniform convergence is on page 254 of [19].Here one sees that the necessary ingredient is a uniform bound on the trace class norm of the operator A which is defined in formulas ( 50)-(53) on page 249.Going through the proof of theorem 5.1 (starting on page 249) one sees that the important thing is an estimate on a suitable Hilbert-Schmidt norm of the Hankel operators associated to the symbols b + and b− and bounds on the operator norm of Toeplitz operators associated to the symbols b −1 + and b −1 − .Finally, from Propositions 4.2 and 4.5 this boils down to bounds on the H p norm of the different functions related to b, but as noted, they are all in H ∞ and for any p, their H p norms are uniformly bounded in φ 1 , φ 2 .This implies the uniform convergence needed for T 3 .
Collecting the results for T 1 and T 3 , we have Proposition 5.For any β ∈ C, where the o(1) terms are uniform over l.
Finally, we need the asymptotic behavior of E n,k (β) and E n (β).The first one can be calculated with the strong Szegö theorem and the second one is a Selberg integral (whose asymptotic behavior was studied in [27]).
) and Putting everything together and noting that T 3 is real (due to the parity of cos) we arrive at Proposition 7.For Re(β) > − 1 4 and |β| 2 < 1 2 , we have We will prove our main theorem by making use of the following approximation result (see [26] Theorem 4.28 for a proof).
In our case, the metric space is H −s with the metric given by its inner product.M β n,k plays the role of η k n , M β n the role of ξ n , µ β k the role of η k and ξ the role of µ β .
We are now ready to prove our main result.
Proof of Theorem 1. From Proposition 7 (making use of the uniform convergence and the fact that for each n, the sum is finite for s > 1  2 ), we have We note that the integral here is bounded by (31) 2π 0 e ilφ |1 − e iφ | −2|β| 2 dφ 2π which is a Selberg integral and can be evaluated as Thus by the dominated convergence theorem (applied to the sum over l), where we used the remark after Proposition 7. Going back, we conclude that (33) lim The convergence of M β n,k to µ β k follows from the results in [14]: one can realize (( 1 ) n and the corresponding Gaussians on the same probability space in such a way that the convergence is almost sure.This implies almost sure uniform convergence of the densities which implies almost sure uniform convergence of the Fourier coefficients.This in turn implies the almost sure convergence in H −s .On the other hand, µ β was defined as the almost sure limit of µ β k .Thus Proposition 8 implies that M β n converges to µ β in law.
For the case of real β, we note that weak convergence in law is equivalent to for each continuous f .Trigonometric polynomials are dense in the space of continuous functions so it is enough to check this on all trigonometric polynomials which boils down to checking that arbitrary finite linear combinations of Fourier coefficients converge in law.This follows immediately from our argument.

Open questions and discussion
Our main theorem leaves one with many questions.From the point of view of multiplicative chaos, perhaps the first one that comes to mind is what happens for other β? Compared to GMC, it seems reasonable that there should be a subcritical phase where one can bound some moments of order p > 1 and use this to prove that there is convergence and the limit is non-trivial.Indeed, something we didn't point out is that all of the objects considered here are random (weakly) analytic functions of β in a suitable neighborhood of the origin.µ β is known to be analytic precisely in the domain where it has a moment of order p for some p > 1.If one were able to bound moments of M β n , this should imply compactness and one could deduce that limits of subsequences are unique due to analyticity (and that they equal µ β in the L 2 phase).Beyond the subcritical phase, one runs into the critical curves ( [29,16,17]), the freezing transition [21,32] and the extreme value statistics of the underlying field.There is still certainly much to understand on the random matrix side of this issue.
As mentioned earlier, a stronger convergence than that of H −s might take place.In [3], one-dimensional complex GMC objects are considered and it is shown that even for complex β, the functions corresponding to t → t 0 M β n (θ) dθ 2π converge in the space of continuous functions.It might be that this is the case for us as well, but such a result seems to require something more than what we have proven.
Another question that might be interesting from the Toeplitz determinant side of the problem: how relevant is the Re (β) > − 1  4 condition.Do the singularities in the integrand cancel with singularities in E n (β) and can one analytically continue to a larger domain?The results in [19] suggest something like this.
Another natural extension of this model would be to consider the characteristic polynomial itself instead of just its absolute value.In [24] they proved that also the imaginary part of the logarithm of the characteristic polynomial converges to a random distribution of similar type.Indeed, the approach taken here would seem to work almost as it is in studying the characteristic polynomial itself.Only the limiting object would be a more complicated GMC object.
Another natural question is what about other ensembles?For example, in [23], it is shown that in the GUE, the logarithm of the characteristic polynomial converges to another Gaussian field with a logarithmic singularity in its covariance.In [35] it is shown that for the Ginibre ensemble, the logarithm of the characteristic polynomial converges to something related to the whole plane massless scalar field.It is not exactly clear what their limiting object is as they test on functions of the form ∆f so one does not see the zero-mode of the field, but still it seems that such structures appear in various questions of random matrix theory.
As hinted in the introduction, the connection between random matrix theory and Gaussian multiplicative chaos opens up questions.What does the KPZ formula, modulus of continuity, tail estimates or approximate stochastic scale invariance mean for the eigenvalues?
Finally, we mention a curious conjecture in [21].Here it is noted that for real β with |β| < 1, the total mass of the measure µ β , i.e. µ β (1) should be distributed as , where Y is an exponentially distributed random variable.Such a result would be nice as it would demonstrate explicitly some of the expected universal properties of GMC measures.

Appendix: Gaussian Multiplicative Chaos
Gaussian multiplicative chaos is classically a theory of random multifractal measures introduced by Kahane in [25] with the purpose of studying measures which can be interpreted as exponentials of Gaussian distributions (generalized functions).Such measures can be studied in quite an abstract setting (measures on locally compact metric spaces) though it has proven fruitful to specialize to the situation of R d .Also the type of random distribution that is in a sense the most interesting is one with a logarithmic singularity in its covariance.We shall go over the basic construction of Kahane, though for a proper introduction to the field, one should consult [34].
Let us consider a covariance K which can be written as where log + (x) = max(0, log x), T > 0 and g is continuous and bounded.Assume further that K is of σ-positive type, i.e. it can be written as where K k is a continuous, non-negative and positive definite covariance.One then defines the centered Gaussian field X n with covariance n k=1 K k and for β ∈ R, the associated random measure (37) µ n,β (dx) = e βXn(x) E(e βXn(x) ) dx =: e βXn(x) : dx, where we have used the notation of normal (or Wick) ordering: for a Gaussian random variable (both real and complex) V , : e V : denotes e V E(e V ) .The reason for assuming such a decomposition for the covariance is that it gives µ n,β the structure of a (measure valued) martingale which gives one a rich theory for limits to work with.
We summarize some of Kahane's theorems here: Theorem 9.For real β, µ n,β converges weakly almost surely in the space of Radon measures to a limit µ β (which we will occasionally write as : e βX :) and the law of µ β is independent of the choice of the family (K k ).The limiting measure is non-trivial for β 2 < 2d (d being the dimension of the underlying Euclidean space) and it is the zero measure for β 2 ≥ 2d.
The condition that the covariances K k must be positive is only to ensure that the law of the limit is independent of the choice of the family (K k ).One still has convergence if one gives up positivity.Kahane also studied questions such as continuity of these measures and existence of moments of e.g. the measure of a bounded open set.
Recently it has also been noticed that for β 2 ≥ 2d, non-trivial measures can be obtained by multiplying the measure by a suitable deterministic factor ( [16,17,4,32]).Another direction that has been taken in studying these types of objects is studying either complex values of β or complex Gaussian fields (see e.g.[29,3]).In this case, one does not expect the limiting objects to be complex measures (i.e. to have finite total variation) so a natural setting for studying them is spaces of distributions.
Let us now proceed to discuss the case relevant to us.The field we are interested in is (38) X(θ) = where Z −k = Z * k and (Z k ) ∞ k=1 are i.i.d.standard complex Gaussians.This series does not define a random function but it does converge in H −ǫ so one can treat it as a random distribution.The covariance of the field is E(X(θ)X(θ ′ )) = − log |e iθ − e iθ ′ | 2 .We will consider an approximation to this field and covariance by truncating the Fourier series (though other possibilities exist -see [1]).Let (39) X k (θ) = The corresponding covariance is For real β, µ β k (defined as before) converges weakly almost surely by Kahane's theorem and is non-trivial for β 2 < 1 (as we had the power 2 inside the logarithm).Kahane's theorem does not guarantee uniqueness so we don't know this measure is the same as in [1] immediately, but this should not be too hard to prove.
For complex β, we shall study µ β k in the space H −s for some s > 1 2 and focus on the (simple) case of |β| 2 < 1 2 .We note that µ β k is a H −s valued martingale.Moreover, for |β as k → ∞.Thus the martingale is bounded in L 2 and converges in L 2 and almost surely.We call the limit µ β =: e βX :.

Proposition 8 .
Let ξ, ξ n , η k and η k n be random elements in a metric space (S, ρ) such that η k n d → η k as n → ∞ for fixed k and also η k d → ξ.Then ξ n d → ξ holds under the further condition (30) lim k→∞ lim sup n→∞