Convergence of the logarithm of the characteristic polynomial of unitary Brownian motion in Sobolev space

We prove that the convergence of the real and imaginary parts of the logarithm of the characteristic polynomial of unitary Brownian motion toward Gaussian free fields on the cylinder, as the matrix dimension goes to infinity, holds in certain suitable Sobolev spaces, which we believe to be optimal. This is the natural dynamical analogue of the result for a fixed time by Hughes, Keating and O'Connell [1]. A weak kind of convergence is known since the work of Spohn [2], which was widely improved recently by Bourgade and Falconet [3]. In the course of this research we also proved a Wick-type identity, which we include in this paper, as it might be of independent interest.


Introduction
As unitary Brownian motion preserves the Haar measure on the unitary group U (n), to many results of Haar distributed unitary matrices there is a corresponding dynamical result for a unitary Brownian motion U at equilibrium.This is in particular the case for some properties of the eigenvalues, whose dynamics have been studied first by Dyson [4], who computed a stochastic differential equation describing their evolution.In this paper, we intend to achieve such a transition from static to dynamic for the Hughes-Keating-O'Connell theorem on the large n limit of the logarithm log p n of the characteristic polynomial.
Characteristic polynomials of random matrices are fundamental objects in random matrix theory.They are closely related to the theory of log-correlated fields and to Gaussian multiplicative chaos [1,3,5].In the case of Haar-distributed matrices from the classical compact groups, there are also remarkable similarities between the statistics of the characteristic polynomial and those of the Riemann zeta function and other number-theoretic L-functions, which led to a number of very precise conjectures for those L-functions [6,7,8,9,10] -see [11] for a review.
The real and imaginary part of the logarithm log p n of the characteristic polynomial also enters the wide family of linear statistics of the eigenvalues λ 1 , . . ., λ n , that is functions that can be expressed as n i=1 f (λ i ).This family has received much attention already, both in the static [12,13,14,15] and dynamical [2,16] frameworks.Except for [12] in the static case, all these papers assume too much regularity on f to be applicable directly to log p n , for the type of convergence they use is too strong.However, it is still possible to use the result of Spohn [2] to identify the large n limit of log p n and log p n as Gaussian free fields, and prove a weak type of convergence (see Lemma 3.1 below or the remark (i) below Theorem 1 in [2] ).
In a recent paper [3], Bourgade and Falconet gave the first dynamical extension of Fisher-Hartwig asymptotics.Those asymptotics allowed them to give a new proof and improvement of Spohn's result.They also used those asymptotics to prove that |p n | α , for certain α and when properly normalized, converge to a Gaussian multiplicative chaos measure associated to the Gaussian free field h on the cylinder, i.e. informally the exponential of a multiple of h.
The goal of this paper is to specify some Sobolev spaces, which we think to be optimal, in which we prove the convergence of log p n and log p n . 1 This is the natural dynamical version of the corresponding stationary result for Haar-distributed unitary matrices by Hughes, Keating and O'Connell [1], who proved that for any fixed time the logarithm of the characteristic polynomial converges to a generalized Gaussian field on the unit circle.
In the last section, we state and prove an identity that allows to express the second moment of the trace of arbitrary products of a GUE matrix H and an independent CUE matrix U in terms of moments of U only.When the dimension n is large enough, the Diaconis-Shahshahani theorem on moments of traces of unitary matrices [13] allows to then compute this new expression explicitly as a polynomial in n.
Hughes, Keating and O'Connell proved that for any fixed time t ≥ 0, log p n (t, •) converges to a generalized Gaussian field.Their result, reformulated to our setting, is as follows: Theorem 1.1 (Hughes, Keating, O'Connell [1]).For any > 0 and any fixed t ≥ 0, the sequence of random functions (log p n (t, •)) n∈N converges in distribution in H − 0 (S 1 ) to the generalized Gaussian field where A k is a complex Gaussian whose real and imaginary parts are independent centered Gaussians with variance 1/(2k).
It is natural thus to assume that in the dynamic case, i.e. when considering log p n also as a function of t, that the limit (in an appropriate function space) would be given by where A k (•), k ∈ N, are independent complex Ornstein-Uhlenbeck processes started from their stationary distribution, i.e. (up to a linear time change) solutions to the SDEs with A k (0) being a complex Gaussian whose real and imaginary parts are independent Gaussians with variance 1/(2k), and (W k (t)) t≥0 , ( Wk (t)) t≥0 , k ∈ N, denoting real standard Brownian motions.
Our main result proves precisely that (for a definition of the Sobolev spaces H s ([0, T ]) and H − 0 (S 1 ) see Section 2.2): Theorem 1.2 (Main Result).For any s ∈ (0, 1 2 ), > s and T > 0, the sequence of random fields (log p n (•, •)) n∈N converges in distribution in the tensor product of Hilbert spaces H s ([0, T ]) ⊗ H − 0 (S 1 ) to the generalized Gaussian field X in (1).
A calculation shows that the covariance functions of X and X are given by The centered Gaussian fields X and X with such a covariance function have been identified as Gaussian free fields on the infinite cylinder R × R/2πZ in [3, Section 2.2].
Remark 1.3.Theorem 1.2 implies that there is a trade-off between regularity in θ and regularity in t.We believe that the regularity we obtain is optimal, in the sense that for s = 1/2 or = s, X is almost surely not an element of the tensor product of H s ([0, T ]) ⊗ H − 0 (S 1 ) anymore.While the limiting field is rotationally invariant from an infinitesimal point of view, this is not the case for log p n with finite n.In particular, one can exchange the regularity in the variable t with the regularity in the variable θ for the limiting field, but for our proof of convergence to work, the Sobolev regularity − in the variable θ needs to be negative which is not the case for the Sobolev regularity s in the variable t.
Just like in the stationary case, the Gaussian field X can't be defined pointwise as its variance at each point is infinite, but it can still be "exponentiated" to build a Gaussian multiplicative chaos (GMC) measure.When we let h(t, θ) denote the real part of X(t, θ), and denote by h δ (t, θ) a mollification of h, then for γ ∈ (0, 2 √ 2) the random measures exist and are non-trivial, where the limit is in probability w.r.t. the topology of weak convergence of measures on R × R/2πZ, see [17] for a self-contained proof of this fact.Bourgade and Falconet proved that exponentiating log |p n (t, θ)| in this way, and then taking the large n limit, gives the same limiting measure as when first taking the large n limit to obtain the Gaussian free field h, and then exponentiating it.Their result is the dynamical analogue to Webb's result for fixed t and the measures being on the unit circle [16], and its precise statement is as follows: where the convergence is in distribution in the space of Radon measures on the infinite cylinder R × R/2πZ, equipped with the topology of weak convergence.
Our main result complements their asymptotics in that it shows in which Sobolev spaces the convergence of the underlying fields log |p n | and log p n to the Gaussian free field h holds.
Further, Theorem 1.2 is related to the below result by Spohn, which we also use in our proof.For real-valued functions f ∈ H 3/2+ 0 (S 1 , R), > 0, Spohn considered linear statistics of the eigenvalues e iθ1(t) , ..., e iθn(t) of unitary Brownian motion (in fact he more generally considered interacting particles on the unit circle with different repulsion strengths): Theorem 1.5 (Spohn [2]).For any > 0, as n → ∞, ξ n (t, f ) converges to a stationary solution of the SDE where dW is a white noise given by and where the convergence is in distribution in C(R, H −3/2− (S 1 , R)), endowed with the topology of locally uniform convergence.The stationary distribution is given by a Gaussian with covariance Here, −∂ 2 θ f is simply the function whose j th Fourier coefficient is |j| times the j th Fourier coefficient of f .This result shows in particular that the k th Fourier coefficient of ln p n converges toward A k √ k (see Lemma 3.1 below).Further, during the proof of Theorem 1.2, we will need the following result from Bourgade and Falconet [3, Corollary 3.2]: Corollary 1.6 (Bourgade, Falconet).Let (z 1 (t), ..., z n (t)) t≥0 denote the eigenvalue process of unitary Brownian motion, started at Haar measure, and denote sgn 2 Mathematical Preliminaries

Unitary Brownian motion
Brownian motion (U n (t)) t≥0 on the unitary group U (n) is the diffusion governed by the stochastic differential equation with (B n (t)) t≥0 denoting a Brownian motion on the space of skew-Hermitian matrices.That is where B(k) , k = 1, ..., n 2 , are independent one-dimensional standard Brownian motions, and where the matrices X k , k = 1, ..., n 2 , are an orthonormal basis of the real vector space of skew-Hermitian matrices w.r.t. the scalar product A, B := n Tr(AB * ).One such basis is given by the matrices Unitary Brownian motion is usually defined using a different normalisation, i.e. satisfying the SDE d Ũn (t) = Ũn (t) dB n (t) − 1  2 Ũn (t).With this normalisation the generator is given by one half times the Laplacian on U (n), which is the usual definition of Brownian motion on a Riemannian manifold.The relation between the two normalisations is Ũn (2t) = U n (t).
In this paper we always consider unitary Brownian motion started from Haar measure on U (n), which is its stationary distribution.Thus U n (t) is Haar distributed for all t ≥ 0.

Sobolov spaces and their Tensor Product
Consider the space of square integrable C-valued functions on the unit circle, with vanishing mean: For s ≥ 0, we define H s 0 (S 1 ) as the restriction of L 2 0 (S 1 ) w.r.t. the functions for which the inner product f, g s = k∈Z |k| 2s f k g k is finite.For s ≤ 0, we define H s 0 (S 1 ) as the completion of L 2 0 (S 1 ) w.r.t.this scalar product.Note that H s 0 (S 1 ), •, • s is a Hilbert space for all s ∈ R. For s ≥ 0 it is a subspace of H 0 0 (S 1 ) = L 2 0 (S 1 ), i.e. the space of square-integrable functions with zero mean, while for s < 0, H s 0 (S 1 ) can be interpreted as the dual space of H −s 0 (S 1 ), i.e. as a space of generalized functions defined up to additive constant.
Remark 2.2.For the fact that the fractional Sobolev spaces defined through Fourier series or through the Slobodeckij norm agree, the reader can consult e.g.[18].
For s > 0 and > 0 we let H s ([0, T ]) ⊗ H − 0 (S 1 ) denote the tensor product of Hilbert spaces H s ([0, T ]) and H − 0 (S 1 ).Since the inner product on that space is determined by first when F and G are linear combinations of pure tensor products, and then for all F, G ∈ H s ([0, T ]) ⊗ H − 0 (S 1 ) by density and continuity.
3 Proof of the main result Theorem 1.2 The proof strategy is as in the stationary case in [1]: we treat (log p n ) n∈N as a sequence in H s ([0, T ]) ⊗ H − 0 (S 1 ), and show that if any of its subsequences has a limit then that limit has to be X.We do this by showing that the finite-dimensional distributions of (log p n ) n∈N , i.e. the distributions of finite sets of Fourier coefficients at a finite number of times, converge to those of X.We then show that the set (log p n ) n∈N is tight in H s ([0, T ])⊗H − 0 (S 1 ).Since H s ([0, T ])⊗H − 0 (S 1 ) is complete and separable, Prokhorov's theorem implies that the closure of (log p n ) n∈N is sequentially compact w.r.t. the topology of weak convergence.In particular this means that every subsequence of (log p n ) n∈N has a weak limit H s ([0, T ]) ⊗ H − 0 (S 1 ).Since any such limit has to be X it follows that the whole sequence (log p n ) n∈N must converge weakly to X.
Let e k : θ → e ikθ .Then, using the notations of Theorem 1.5, log(p n ) k (t) = ξ n t, e k k .Thus Spohn's theorem, combined with the continuous mapping theorem with the appropriate continuous map Combining the real and imaginary part of e k , we obtain that the SDE for ξ(•, where B k is a complex Brownian motion, i.e. a process whose real and imaginary parts are independent standard Brownian motions.Besides, the Brownian motions (B k ) k≥0 are independent, so that (A k ) k≥1 and ξ •, e k k k≥1 are equal in distribution, which concludes the proof.
We proceed to show tightness of (log We let 0 < s < such that 0 < s < s < < , and choose for a C δ depending on δ.By Lemma 3.2 below we see that ), and by Lemma 3.3 below we see that sup n∈N E || log p n ||2 s ,− < ∞.Thus, when choosing C δ large enough, we see that which shows tightness of log p n and thus together with Lemma 3.1 proves our Theorm 1.2.
Lemma 3.3.For all s ∈ 0, 1 2 and all > s, it holds that Proof: We see that For the first summand it holds that (with k ∧ n denoting min{k, n}) For the second summand it holds that: dt. ( For the second summand in (3) we get is finite as soon as s < .
For the first sum in (3) we use Corollary 1.6, which implies that for all k ≥ 1 with k ∨ n := max{k, n}.Using this, and the fact that sinh x ≥ x and 1/ sinh x ≥ 1/x − x/6 for all x > 0, we see that for t < k −1 : Thus we see that when s < 1/2, the first sum in (3) is bounded by T ∞ k=1 k −1−2 +2s , which is finite for s < .This finishes the proof.
Further we define the pairing ρ ∈ C 4j as See Example 4.2 for a list of the pairings π, π and ρ, for j = 2, and Figure 4 for their depiction.
Note that ρ and all pairings π pair even numbers with odd numbers, thus πρ maps even numbers to even numbers and odd numbers to odd numbers.Using the pairing π, the even numbers i 2 , i 4 , ...i 4j determine all the odd ones.Thus we see that i1,...,i4j π∈C2j By repeatedly applying πρ to {2, 4, ..., 4j}, we get a partition of {2, 4, ..., 4j} into orbits.The set of these orbits we denote by O πρ .We see that  Putting together (4), ( 5), ( 6) and ( 7), we have proven the following proposition: where the last equality holds for large enough n by Theorem 4.3.