Modulus of continuity of controlled Loewner–Kufarev equations and random matrices

First we introduce the two tau-functions which appeared either as the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\tau $$\end{document}τ-function of the integrable hierarchy governing the Riemann mapping of Jordan curves or in conformal field theory and the universal Grassmannian. Then we discuss various aspects of their interrelation. Subsequently, we establish a novel connection between free probability, growth models and integrable systems, in particular for second order freeness, and summarise it in a dictionary. This extends the previous link between conformal maps and large N-matrix integrals to (higher) order free probability. Within this context of dynamically evolving contours, we determine a class of driving functions for controlled Loewner–Kufarev equations, which enables us to give a continuity estimate for the solution of such an equation when embedded into the Segal–Wilson Grassmannian.


Introduction
The class of univalent functions is an extraordinarily rich mathematical object within the field of complex variables, with deep and surprising connections with, e.g. conformal field theory (CFT), random matrix theory and integrable systems, cf. [6][7][8][9]15,19,21,23], just to name the most important ones in our context. So, for D := {z ∈ C||z| < 1}, the open unit disc, with boundary the unit circle, i.e. S 1 = ∂D, let The set C of all such Jordan contours encircling the origin forms an infinite dimensional manifold [10,19].
It has been shown by Kirillov and Juriev [9] that there exists a canonical bijection S reg ∼ = Diff + (S 1 )/S 1 which endows Diff + (S 1 )/S 1 with the structure of an infinite-dimensional complex manifold. Geometrically, π : C → S reg is a fibre bundle, with fibre R * + , which is a consequence of the Riemann mapping theorem. There exist two continua of global sections σ r i : S reg → C, r i > 0, i = 1, 2, such that the leaves C r i := σ r i (S reg ) stratify C, i.e. C = r 1 >0 C r 1 = r 2 >0 C r 2 , either according to the conformal radius r 1 > 0, or alternatively, the interior area r 2 > 0, as in [19].
Krichever, Marshakov, Mineev-Weintstein, Wiegmann and Zabrodin [10,14,23], motivated by the Hele-Shaw problem, cf. [15] and the monograph by Gustafsson'et al. [5], introduced a new set of co-ordinates, the so-called harmonic moments of the interior and exterior domain, with respect to a family of harmonic functions. Namely, for {z −k } k∈Z ≥0 the interior harmonic moments are given by: witht k denoting the complex conjugate of t k , are local co-ordinates on the manifold (moduli space) of smooth closed contours C, cf. [19].
The other set of (natural) co-ordinates is given by the coefficients of the normalised Riemann mapping. So, we have two different sets of co-ordinates for C, as shown below: The first co-ordinate respects the conformal radius The first co-ordinate respects the interior area Hence, the tangent space to C permits also two descriptions, namely, as [4,7,10,19] Der The ∂ t k can be determined either by specific boundary variations of the domain, which do only change one harmonic moment at the time, cf. [10, Formula (2.11)], or in terms of the Faber polynomials [19]. Combining results in [10] with our considerations, the relation for the different tangent vectors is given by and ∂ n is the normal derivative on the boundary ∂ D c with respect to ξ ∈ ∂ D c , and G 0 (x, ξ) is the Dirichlet Green function associated to the Dirichlet problem in D c .
A key result, which expresses the Riemann mapping in terms of the τ -function, is the following Theorem 1.2 ([8,19]) Let g :Ĉ \ D →Ĉ \ D be the conformal map, normalised by g(∞) = ∞ and g (∞) > 0. Then the following formula holds: This formula would be a key to describe the solution to the conformal welding problem (c.f. [22]) associated to Malliavin's canonic diffusion [13] within the framework of Loewner-Kufarev equation, which would be a future work.
Another interpretation of the τ -function, given by Takhtajan [19,Corollary 3.10], is that it is a Kähler potential of a Hermitian metric on C a , a > 0. Kirillov and Juriev [9], defined a two-parameter family (h, c) of Kähler potentials K h,c on the determinant line bundle Det * over the (Sato)-Segal-Wilson Grassmannian, where h is the highest-weight and c the central charge of the CFT. For h = 0 and c = 1, i.e. the free Boson, one has [6,9] for the metric where λ is the co-ordinate in the fibre over the schlicht function f . In terms of the Grunsky matrix Z f , associated to an f ∈ S reg , the Kähler potential for h = 0, c = 1 is given by The following diagram summarises our discussion so-far: where Z is the Grunsky matrix, cf. [9] and H * 0 the charge 0 sector of the boson Fock space [7, p.279]. For the second square from the left, we should note that the Krichever mapping does not distinguish S reg and C algebro-geometrically. Namely, the Krichever embedding of a Riemann mapping uses only the negative part of the Grunsky coefficients b −m,−n , m, n = 1, 2, . . . but not b 0,0 . One finds from the defining equation of the Grunsky coefficients that b 0,0 is the only entry of the Grunsky matrix which depends on the conformal radius. Consequently, Krichver's embedding forgets about the conformal radius. But in order to keep track of the modulus of the derivative of the normalised Riemann mapping, we put that information into the determinant line bundle. This is the mapping C → Det * | M .
The structure of the rest of the paper is as follows: In Sect. 2 we establish a relation between the theory of Laplacian growth models, and their integrable structure, with a class of random matrices and second order free probability. We succinctly summarise it in a dictionary. In Sect. 3 we consider controlled Loewner-Kufarev equations and recall the necessary facts. Then, in Sect. 4, we give several estimates for the Grunsky coefficients associated to solutions to a controlled Loewner-Kufarev equation. Proofs of several estimates which need results from [1] are relegated to Appendix A. Finally, we prove Theorem 3.3 in Sect. 5.

Integrability and higher order free probability
Another motivation in the works of Marshakov et al. was the close connection random matrix theory has with (Laplacian) growth models and integrable hierarchies. Takebe, Teo and Marshakov discussed the geometric meaning of the eigenvalue distribution in the large N limit of normal random matrices in conjunction with the one variable reduction via the Loewner equation [18]. In [1] we established and discussed a relation between CFT and free probability theory. Here we briefly present a novel connection between integrable hierarchies, large N limits of Gaussian random matrices and second (higher) order free probability [3]. First, consider, cf. [19, p. 42 where G(z) is the first order Cauchy transform.
Derivative of harmonic moments, ∂v m ∂t n for m, n ≥ 1 The second order limit moments, α m,n for m, n ≥ 1 Riemann mapping, G : D c → D c The first order Cauchy transform, the Green function associated with the ∂-Laplacian 0 = − 1 2 * ∂ * ∂ and G DBC is the Green function associated to 0 on D c [19,Theorem 3.9] The second order Cauchy transform, This suggests us to make a dictionary translating the language from integrable systems and free probability. Table 1 is an attempt to list objects in these fields sharing the same algebraic relations.
From the above it follows now, that general higher order free (local) cumulants are given by Ward identities of n-point functions of the twisted Boson field over arbitrary Riemann surfaces.

Controlled Loewner-Kufarev equations
The connection with the Loewner equation, the class of schlicht functions and integrable systems was established by Takebe et al. [18]. They showed that both the chordal and the radial Loewner equation give consistency conditions of such integrable hierarchies. A particularly important class of such consistency conditions can be obtained from specific control functions.
In the previous paper [1], the authors introduced the notion of a solution to the controlled Loewner-Kufarev equation (see [ are given continuous functions of bounded variation, called the driving functions. We define In the current paper, we determine a class of driving functions for which we establish the continuity of the solution, as a curve embedded in the (Sato)-Segal-Wilson Grassmannian, with respect to time. For this reason, we introduce the following class of controlled Loewner-Kufarev equations.
Let us first recall from the monograph by Lyons is called a control function if it satisfies super-additivity: for all 0 ≤ s ≤ u ≤ t, and vanishes on the diagonal, i.e. ω(t, t) = 0, for all t ∈ [0, T ].

Now, let V be a Banach space and
for a control function ω : T → R + . Then X is called a Lipschitz path controlled by ω.
We have the following Definition 3.2 Let ω be a control function. The driven Loewner-Kufarev equation 1 (3.1) is controlled by ω if for any n ∈ N, p = 1, . . . , n and i 1 , . . . , i p ∈ N with i 1 + · · · + i p = n, we have e nx 0 (t) and e nx 0 (t) Henceforth, we will refer to Eq. A natural question to be asked is, how a control function as driving function determines a control function for (3.1). We will give one of the answers in Corollary 4.5.
Let H = L 2 (S 1 , C) be the Hilbert space of all square-integrable complex-valued functions on the unit circle S 1 , and we denote by Gr := Gr(H ) the Segal-Wilson Grassmannian (see [1,Definition 3.1] or [17,Sect. 2]). Any bounded univalent function Note that f extends to a continuous function on D by Caratheodory's Extension Theorem for holomorphic functions.
Let H 1/2 = H 1/2 (S 1 ) be the Sobolev space on S 1 endowed with the inner product given by h, g H 1/2 = h 0 g 0 + n∈Z |n|h n g n for h = n∈Z h n z n , g = n∈Z g n z n ∈ H 1/2 . Assume that f extends to a holomorphic function on an open neighbourhood of D. Then span {1} ∪ {Q n • f • (1/z)| S 1 } n≥1 ⊂ H 1/2 and we consider the orthogonal projection Then, as we prove with Murayama in [2], to every properly bounded control function ω there exists a unique solution to the Loewner-Kufarev equation which is univalent on the unit disc and can be holomorphically extended across the unit circle. Our main result is for every 0 ≤ s < t ≤ T , where • op is the operator norm.
Thus we obtained a continuity result with respect to the time-variable of the solution embedded into the Grassmannian in which the modulus of continuity is measured by the control function ω.

Controlling Loewner-Kufarev equation by its driving function
We shall begin with a prominent example of a control function as follows.

Definition 4.2
Let ω be a control function. We say that a continuous function y : As is well known, introducing control functions makes our calculations stable as follows.
Since the control function is nonnegative and super-additive, it holds that and hence we get the above inequality.
Hence the assertion is immediate.
(iii) Let 0 ≤ s ≤ t ≤ T be arbitrary. Then we have e nx 0 (t) 0≤u 1 <···<u n ≤t dy 1 (u 1 ) · · · dy n (u n ) − e nx 0 (s) 0≤u 1 <···<u n ≤s Since it holds that |e nx 0 (s) − e nx 0 (t) | ≤ nω 0 (s, t)e nω 0 (0,t) , and by using (ii), the above quantity is bounded by We shall remark here that the control functions form a convex cone, namely, (a) the sum of two control functions is a control function, (b) any control function multiplied by a positive real constant is again a control function. Therefore, the quantities in Proposition 4.4-(ii, iii) are estimated by using a single control function. Therefore, the following is immediate. The following is a consequence of [1, Theorem 2.8] and will be proved in Sect. A.1.

Some analytic aspects of Grunsky coefficients
Let S, S ⊂ Z be countably infinite subsets, and A = (a i, j ) i∈S, j∈S be an S × Smatrix. For each sequence x = (x j ) j∈S of complex numbers, we define a sequence j∈S a i j x j if it converges for all i ∈ S. We will still denote T A x by Ax when it is defined.
Let 2 (S) be the Hilbert space consisting of all sequences a = (a i ) i∈S such that The associated norm will be denoted by • 2 .
For each s ∈ R, the space This can be rephrased with our notation as follows:   each a = (. . . , a −3 , a −2 , a −1 ) ∈ 2 (−N), we have by Theorem 4.7, (ii) Since the Grunsky matrix (b m,n ) m,n≤−1 is symmetric: b m,n = b n,m for all m, n ≤ −1, the assertion is proved similarly to (i).
(iii) The injectivity is clear since the adjoint operator of B is B * . Then the second assertion is also clear since 1 + B B * is self-adjoint.

Remark 4.9
The semi-infinite matrix defined by B 1 := ( √ mnb −m,−n ) m,n∈N is called the Grunsky operator, and then the Grunsky inequality (Theorem 4.7) shows that B 1 is a bounded operator on 2 (N) with operator norm ≤ 1. This operator, together with three additional Grunsky operators, are known to play a fundamental role in the study of the geometry of the universal Teichmüller space. For details, cf. the papers by Takhtajan-Teo [20] or Krushkal [11].
In the sequel, we fix a control function ω, and a solution { f t } 0≤t≤T to a Loewner-Kufarev equation controlled by ω. We denote by b m,n (t) for m, n ≤ −1 the Grunsky coefficients associated with f t , and It is clear that the linear operator (1 t ) −1 extends to 2 (N) and the extension will be denoted by A t : 2 (N) → 2 (N). In particular, it is easy to see that A t ≤ 1, holds for the operator norm. We shall exhibit the indices which parametrise our operators in order to help understanding the following: The following is a consequence from [1, Theorem 2.12] and will be proved in Sect. A.2.

Corollary 4.10
Let ω be a control function, and { f t } 0≤t≤T be a solution to the Loewner-Kufarev equation controlled by ω. Let b −m,−n (t), n, m ∈ N be the Grunsky coefficients associated to f t , for 0 ≤ t ≤ T . Then for any 0 ≤ s ≤ t ≤ T and n, m ∈ N with n + m ≥ 3, we have . .
Along the Loewner-Kufarev equation controlled by ω, we obtain the following
Proof (i) By Corollary 4.10-(ii), we have Finally, define = ( m,n ) m∈Z,n∈Z by m,n := √ m δ m,−n + δ m,0 δ 0,n for m ∈ N and n ∈ −N, that is, It is clear that : and is a continuous linear isomorphism.

Proof of Theorem 3.3
Let ω be a control function such that ω(0, T ) < 1 8 , and let { f t } 0≤t≤T be a univalent solution to the Loewner-Kufarev equation controlled by ω.
Recall, that by the results in [2], f t extends to a holomorphic function on an open neighbourhood of D for all t ∈ [0, T ].
We then note that for each t ∈ [0, T ], it holds that Q n (t, f t (1/z))| S 1 ∈ H 1/2 , where Q n (t, w) is the n-th Faber polynomial associated to f t . Therefore we have In particular, we have We fix an inner product on H 1/2 by requiring for h = n∈Z h n z n , g = n∈Z g n z n ∈ H 1/2 , that h, g H 1/2 := h 0 g 0 + ∞ n=1 n(h −n g −n + h n g n ).
Then Recall that for each univalent function f : D → C with f (0) = 0 and an analytic continuation across S 1 , the orthogonal projection H 1/2 → W 1/2 f is denoted by P f . In order to prove Theorem 3.3, we need to calculate the projection operator P f . For this, we shall consider first the following change of basis.
Let w n (z) := Q n • f (z −1 ), for z ∈ S 1 and n ∈ N. Then we have By putting , . . . , for n ≥ 1 and m ≤ −1), the Eq. (5.1) is written in a simpler form as: Consider the change of basis where we note that the matrix on the right-hand side is non-degenerate, with inverse We note that the identity From these, we find that for n ≥ 1, from which, the following is immediate: Denote by B t the associated matrix of Grunsky coefficients of f t .
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