Bulk limits of the truncated circular beta ensembles

Introduction

The size $n$ circular beta ensemble with parameter $\beta> 0$ is the distribution of $n$ points $\{e^{i\theta_j}, 1 \le j \le n\}$ on the unit circle with joint probability density

where $Z_{n,\beta}$ is an explicit normalizing constant (see e.g. Section 2.8 of [8]). When $\beta= 2$, (1) recovers the joint eigenvalue density of a uniformly chosen $n \times n$ unitary matrix, known as the size $n$ Circular Unitary Ensemble (CUE). By utilizing the algebraic structure of the joint intensities, Dyson, Gaudin, and Mehta derived the local scaling limit of CUE in the 1960s. They showed that the determinantal structure is preserved in the limit, and the limit process, known as the $\operatorname{Sine}_2$ point process, is a translation invariant determinantal point process on $\mathbb{R}$. In [4] Chhaibi, Najnudel and Nikeghbali studied the local scaling limit of the (normalized) characteristic polynomials of CUE, and constructed a random analytic function $\zeta_2$ (called the stochastic zeta function) so that its zero set has the same distribution as the $\operatorname{Sine}_2$ point process.

When $\beta= 1, 4$, the joint density (1) corresponds to certain random unitary matrices invariant under symmetric/self-dual transformations. For general $\beta> 0$, there is no similar invariant matrix model. In [16] Killip and Nenciu (motivated by the results of [6]) constructed a family of sparse random CMV (named after Cantero, Moral, and Velázquez [1]) matrix models $\{\operatorname{Circ}_{n,\beta}, n \ge1\}$ in terms of the Verblunsky coefficients so that the resulting matrix has a joint eigenvalue density given by (1). See § 2 for an overview of the CMV construction. In [17] Killip and Stoiciu showed that under the same scaling as in the unitary case, the circular beta ensemble has a point process limit, called the Sine$_\beta$ point process. They showed that the Sine$_\beta$ process can be characterized via its counting function using a system of coupled stochastic differential equations. Using the random operator framework, in a series of works [26, 27, 28] Valkó and Virág showed the operator level convergence of the circular beta ensembles with an explicit rate of convergence, and proved that the normalized characteristic polynomials of the circular beta ensembles converge to the random analytic functions $\zeta_\beta(\cdot)$ with zero set given by the Sine$_\beta$ process. See also [6, 7, 23, 25, 22] for a development of the matrix models and local scaling limits for beta ensembles on the real line.

In this paper, we focus on the rank-one truncation of the circular beta ensembles obtained by removing the first row and column of the matrix model $\operatorname{Circ}_{n,\beta}$. Random unitary matrices can be viewed as evolution operators of closed quantum systems. From this perspective, truncations of unitary matrices obtained by removing a fixed number of rows and columns, naturally arise as models for open quantum systems in the regime of weak non-Hermiticity. We refer to the surveys [12, 9] and the references therein for a detailed discussion of rank-one perturbations of random matrix ensembles.

The main goal of this project is to understand the scaling limit of the Blaschke product of the (rank-one) truncated circular beta ensembles in the bulk regime, thereby answering a question posed in [20]. Before presenting the main result, we begin with a review of the known results in the $\beta=2$ case. Throughout the paper, for a square unitary matrix $\mathbf{M}$ we denote by $\operatorname{Trunc}(\mathbf{M})$ the matrix obtained by removing the first row and column from $\mathbf{M}$. Since the truncation map corresponds to a projection, the eigenvalues of the resulting matrix $\operatorname{Trunc}(\mathbf{M})$ have modulus less than or equal to one.

Consider the case when $\mathbf{M}_{n+1}$ is a uniformly chosen $(n+1)\times(n+1)$ unitary matrix with respect to the Haar measure on the unitary group $U(n+1)$. In [29] Życzkowski and Sommers proved that the joint eigenvalue density of $\operatorname{Trunc}(\mathbf{M}_{n+1})$ is given by

with respect to the Lebesgue measure in the unit disk. Note that Życzkowski and Sommers [29] also computed the joint eigenvalue distribution for the general finite rank truncation of CUE.

From the Vandermonde structure in (2), one can show that the point process distributed according to (2) is a determinantal point process with kernel function given by

with respect to the Lebesgue measure on the unit disk. Taking the $n\to\infty$ limit of the kernel function (3) leads to the classical Bergman kernel on the unit disk

The Bergman kernel is related to the study of random analytic functions as follows. Let $\{\xi_n, n \ge0\}$ be a sequence of i.i.d. standard complex Gaussian variables. Throughout the paper, a complex Gaussian variable $\xi$ is called standard if

We define the random Gaussian analytic function (GAF) formally as the random series

In the seminal work [21], Peres and Virág showed that the zero set of $f_{\mathrm{GAF}}(\cdot)$ is a determinantal point process with kernel function given by $K_{\mathrm{Bergman}}(z,w)$.

Putting these results together implies that the eigenvalues of the truncated CUE converge in distribution without any additional scaling to the zeros of $f_{\mathrm{GAF}}(\cdot)$. We call this the bulk scaling limit of the truncated CUE. This convergence was later strengthened in Krishnapur [18] to the convergence of the Blaschke product of the truncated CUE.

Note that [18] also provides a generalization of the work [21] by connecting the limit of the eigenvalues of the finite-rank truncation of CUE with the singular points of a matrix-valued Gaussian analytic function.

In [15] Killip and Kozhan studied the truncated circular beta ensemble obtained by removing the first row and column of the matrix models $\operatorname{Circ}_{n+1,\beta}$ of Killip-Nenciu [16]. They constructed a sparse matrix model with spectrum distributed as that of $\operatorname{Trunc}(\operatorname{Circ}_{n+1,\beta})$, and showed that the joint eigenvalue density of $\operatorname{Trunc}(\operatorname{Circ}_{n+1,\beta})$ with respect to the Lebesgue measure on the unit disk is given by

We call the resulting distribution the size $n$ truncated circular beta ensemble. When $\beta= 2$, equation (6) recovers the joint eigenvalue density (2). However, for general $\beta> 0$ the truncated circular beta ensemble does not have a nice algebraic structure.

In [20] Li and Valkó derived the edge scaling limit of the normalized characteristic polynomials and the point process limit of the eigenvalues near the point $1$ on the boundary of the spectrum. They characterized the limiting point process as the zero set of certain random analytic function, which is related to the stochastic zeta function $\zeta_\beta(\cdot)$ constructed in [28]. Their approach relies on the matrix model representation of the full and truncated circular beta ensembles [16, 15], and the random operator framework for circular ensembles developed by Valkó and Virág [26, 27, 28].

In [20], Li and Valkó also raised the question (Problem 1 in Section 8.2) of whether the characteristic polynomial of the truncated circular beta ensemble has a nice scaling limit in the bulk regime of the spectrum. Our main result fully answers this question and can be viewed as a beta-generalization of Theorem 5 of Krishnapur [18] (see Theorem A for the statement) in the rank-one truncation regime.

Recall the Gaussian analytic function $f_{\mathrm{GAF}}(\cdot)$ defined in (5).

Since $\beta$ appears as a multiplicative constant in (7), we obtain the bulk scaling limit of the truncated circular beta ensemble as follows.

Corollary 2, together with the work of Peres and Virág [21], implies that the limiting point process is determinantal with $K_{\mathrm{Bergman}}(\cdot,\cdot)$ with respect to the Lebesgue measure on $\mathbb{D}$, and is independent of $\beta$. More precisely, the $k$-th joint intensity of the limiting process is given by

Using the determinantal structure, one can compute the asymptotics of the hole probability as follows. Let $\Lambda_n$ be distributed as the size $n$ truncated circular beta ensemble, and for $0 < r < 1$ let $\mathbb{D}_r = \{z \in\mathbb{C} : |z| < r\}$, then we have

As $r \uparrow1$, the hole probability tends to $\exp\left(-\frac{r^2+o(1)}{12(1-r)}\right)$, see Corollaries 5.1.7 and 5.1.8 in [13] for more details.

On the other hand, the finite size truncated circular beta ensemble distributed according to (6) has no known determinantal structure for general $\beta\ne2$. We prove that the linear statistics of compactly supported functions of the truncated circular beta ensemble are asymptotically independent from the so-called reversed characteristic polynomial, see Section 4 and Proposition 19 for more details.

Our approach also applies to the rank-one multiplicative perturbation of the circular beta ensemble. For a square matrix $\mathbf{M}$, consider its multiplicative perturbation

Note that when $r=0$ the eigenvalues of $\mathbf{M}^{[r]}$ consist exactly of the eigenvalues of the truncated matrix $\operatorname{Trunc}(\mathbf{M})$ plus an additional eigenvalue at $0$, whereas when $r=1$ we simply have $\mathbf{M}^{[r]}=\mathbf{M}$. Hence, as $r$ changes from $0$ to $1$, the perturbed model $\mathbf{M}^{[r]}$ can be viewed as an interpolation between the truncated and unperturbed model. The joint eigenvalue distribution of $\mathbf{M}^{[r]}$ in the case when $\mathbf{M}$ is a random Haar unitary matrix was derived in [11]. In [10], Forrester and Ipsen studied the $n \to\infty$ limit of the rank-one multiplicative perturbation of the CUE in the regime when $r=r_n=O(n^{-1/2})$. They characterized the limiting point process as the zeros of a certain random Laurent series, and calculated the one- and two-point correlations of the limiting process.

For general $\beta\ne2$, one may define the rank-one multiplicative perturbation of the circular beta ensemble as the spectrum of $\operatorname{Circ}_{n,\beta}^{[r]}$ via (8). The joint eigenvalue distribution of $\operatorname{Circ}_{n,\beta}^{[r]}$ was derived in Killip-Kozhan [15]. Using the perturbed matrix model of Killip and Kozhan, and a similar argument as in the proof of Theorem 1, we obtain the scaling limit of the Blaschke product of the rank-one multiplicative perturbation of the circular beta ensemble in the mesoscopic regime $r_n=O(n^{-1/2})$.

In particular, the eigenvalues of $\operatorname{Circ}_{n,\beta}^{[r_n]}$ converge weakly to the zeros of the random series $r+z f_{\mathrm{GAF}}(z)$. This extends the result of [10] to general $\beta>0.

Outline of the paper: In § 2, we provide a brief introduction of the theory of orthogonal polynomials on the unit circle (OPUC) and the CMV construction for finitely supported probability measures on the unit circle. § 3 includes the proofs of Theorems 1 and 3 on the scaling limit of the Blaschke products of the rank-one truncated and multiplicatively perturbed circular beta ensembles in the bulk regime. § 4 discusses the asymptotic independence of the linear statistics of the truncated circular beta ensembles and the reversed characteristic polynomials.

Acknowledgment: YL is grateful to Benedek Valkó for valuable discussions. YL and JN thank the Institut Mittag-Leffler for its hospitality, where part of this work originated during the program Random Matrices and Scaling Limits in Fall 2024.

AI use statement: ChatGPT was used to proofread and assist with language editing of an earlier version of this paper.

Background

This section presents a brief introduction to the OPUC theory and the CMV matrix constructions for probability measures on the unit circle, as well as a review of the matrix models for the full and truncated circular beta ensembles.

OPUC theory and CMV matrix

We briefly review the necessary background for the OPUC theory and the CMV algorithm. We refer the reader to [24] for a detailed discussion of the subject.

Let $\mu$ be a probability measure supported on $n$ distinct points $e^{\mathrm{i}\theta_j}$, $1 \le j \le n$ on $\partial\mathbb{D}$. For $0 \le k \le n-1$, the $k$-th orthogonal polynomial is defined as the unique monic polynomial of degree $k$ that is orthogonal to $1,z,\ldots,z^{k-1}$ in $L^2(\mathrm{d}\mu)$. We extend this definition to $\Phi_n(z) = \prod_{j=1}^{n}(z-e^{\mathrm{i}\theta_j})$, which can be viewed as the characteristic polynomial of the measure $\mu$. Let $\Phi_k^*(z) := z^k\overline{\Phi_k(1/\bar{z})}$ be the reversed polynomial of $\Phi_k$ for $0 \le k \le n$, then the Szegő recursion states that (see e.g. Section 1.5, vol. 1 of [24]):

Here the constants $\alpha_k$, $0 \le k \le n-1$ are called the Verblunsky coefficients of $\mu$. They satisfy the property that $\alpha_k \in\mathbb{D}$ for $0 \le k \le n-2$ and $\alpha_{n-1} \in\partial\mathbb{D}$. In fact, the map between probability measures supported on $n$ points on $\partial\mathbb{D}$ and the sequence of Verblunsky coefficients $(\alpha_0,\ldots,\alpha_{n-1}) \in\mathbb{D}^{n-1} \times\partial\mathbb{D}$ is a homeomorphism, see e.g. Theorem 1.7.11 of [24]. Note that a probability measure on the unit circle that is not finitely supported has an infinite sequence of Verblunsky coefficients lying within the unit disk.

Using the Verblunsky coefficients, we are able to construct the so-called CMV matrix corresponding to $\mu$ as follows. Definition 4. Let $\mu_n$ be a probability measure supported on $n$ distinct points on the unit circle and let $\alpha_k$, $0 \le k \le n-1$ be its Verblunsky coefficients obtained via the Szegő recursion (10). Define

and set $\Xi_{-1}=(1)$ and $\Xi_{n-1}=(\overline{\alpha_{n-1}})$ to be $1\times1$ matrices. Then we define the CMV matrix corresponding to $\mu_n$ or $\{\alpha_k,0\le k\le n-1\}$ as

where $\mathcal{L}$ and $\mathcal{M}$ are $n\times n$ block-diagonal matrices

Note that $C(\alpha_0,\ldots,\alpha_{n-1})$ is unitary if and only if $\lvert\alpha_{n-1}\rvert=1$. To deal with the rank-one truncation and multiplicative perturbation of random unitary matrices, we also allow $\alpha_{n-1}\in\mathbb{D}$, in which case the resulting matrix $C(\alpha_0,\alpha_1,\ldots,\alpha_{n-1})$ becomes sub-unitary.

In the following proposition, we summarize the connection between CMV matrices, probability measures on $\partial\mathbb{D}$, and the orthogonal polynomials associated with these probability measures.

To describe the matrix models for rank-one truncations and multiplicative perturbations of unitary ensembles, we introduce a simple transformation on the sequence of Verblunsky coefficients and the corresponding CMV matrix below.

Definition 6. Suppose $\mu$ is a finitely supported probability measure on $\partial\mathbb{D}$. Let $(\alpha_0,\ldots,\alpha_{n-1})\in\mathbb{D}^{n-1}\times\partial\mathbb{D}$ and $C$ be the corresponding Verblunsky coefficients and the CMV matrix with respect to $\mu$. We define the reversed version of the Verblunsky coefficients as

Then we denote by $\widetilde{\mu}$ and $\widetilde{C}$ the probability measure and CMV matrix corresponding to the sequence $\widetilde{\alpha}_k$, $0\le k\le n-1$. We call $\widetilde{\mu}$ and $\widetilde{C}$ the reversed version of $\mu$ and $C$, respectively.

In Killip-Nenciu [16], it was shown in Proposition B.2 that $C$ and $\widetilde{C}$ have the same eigenvalues. In [15] Killip and Kozhan observed that the truncated and multiplicative perturbation of a CMV matrix is unitarily equivalent to a sub-unitary CMV matrix model.

Matrix models for the full and truncated circular beta ensembles

In this subsection, we explicitly describe the matrix models for the circular beta ensembles constructed by Killip and Nenciu [16], as well as their rank-one truncated and multiplicatively perturbed version introduced by Killip and Kozhan [15].

Recall the definition of the size $n$ circular beta ensemble with density function (1). We define the size $n$ Killip-Nenciu measure as a random probability measure on $\partial\mathbb{D}$ as

where the support is distributed as the size $n$ circular beta ensemble, and the weights $\pi_j$, $1 \le j \le n$ are chosen according to $\operatorname{Dirichlet}(\beta/2,\ldots,\beta/2)$ distribution, independently of the support. The joint distribution of the Verblunsky coefficients with respect to $\mu_{n,\beta}^{\mathrm{KN}}$ was computed in [16].

Recall the reversed version of a sequence of Verblunsky coefficients (13). By the independence and rotational invariance of $\alpha_k$, $0 \le k \le n-1$, we have $\widetilde{\alpha}_k$, $0 \le k \le n-1$ are independent, rotationally invariant such that $|\widetilde{\alpha}_k|^2 \sim\operatorname{Beta}\left(1,\frac{\beta}{2}(k+1)\right)$ for $0 \le k \le n-2$ and $\widetilde{\alpha}_{n-1} \sim\operatorname{Unif}(\partial\mathbb{D})$. Note that the spectral weights of the $\widetilde{\alpha}_k$, $0 \le k \le n-1$ are no longer Dirichlet-distributed. The advantage of the reversed ordering is that now the Verblunsky coefficients can be coupled in the same probability space. See Assumption 11 below for more discussions. Together with Proposition 7, this yields the following result of Killip and Kozhan [15].

Let $\eta\sim\operatorname{Unif}(\partial\mathbb{D})$ be independent of $\alpha_k$, $k \ge0$. For fixed $r \in[-1,1]$, the sub-unitary CMV matrix $C(\alpha_0,\alpha_1,\ldots,\alpha_{n-2},r\eta)$ has the same joint eigenvalue distribution as the multiplicatively perturbed model matrix $\operatorname{Circ}^{[r]}_{n,\beta}$.

Limits of the Blaschke products

This section establishes the scaling limit of the Blaschke product associated with the truncated and multiplicatively perturbed circular beta ensembles. We first show the weak convergence of the Taylor coefficients of the Blaschke products in § 3.1, and then upgrade this to uniform convergence on compacts in law by proving the necessary tightness result in § 3.2. These results yield proofs of Theorem 1 and Corollary 2. Using a similar argument, we complete the proof of Theorem 3 in § 3.3.

Weak convergence of the Taylor coefficients

Recall the Blaschke product defined in Theorem 1. By Propositions 5 and 7, we have

Since the zeros of $\Phi_n^*(z)$ lie outside of the unit disk, the Blaschke product $B_n(z)$ is well defined for $z \in\mathbb{D}$ and $\{z \in\mathbb{D}: B_n(z) = 0\} = \{z \in\mathbb{D}: \Phi_n(z) = 0\}$. Expanding $B_n(z)$ in a Taylor series around $z = 0$, we write

The goal of this section is to prove the following statement.

The proof of Proposition 10 relies on two main ingredients: a martingale structure of the logarithm of the reversed orthogonal polynomials $\Phi_n^*$, and a Gaussian approximation for the Verblunsky coefficients. Throughout the paper, we work with the following probability space.

and set $\alpha_{-1} \equiv-1$. Let $\eta\sim\operatorname{Unif}(\partial\mathbb{D})$ be independent of $\{\alpha_j, j \ge0\}$. In this probability space, we introduce the filtration

and define the CMV matrices

By Proposition 9, the eigenvalues of $C_n$ and $C_n^{[r]}$, $r \in[-1,1]$ are distributed as the size $n$ truncated and multiplicatively perturbed circular beta ensemble, respectively. As already observed in [2, 3], the probability space defined in Assumption 11 provides a natural coupling of full and truncated circular beta ensembles of all dimensions.

Define $\log\Phi_k^*$ as the unique holomorphic logarithm on $\mathbb{D}$ such that $\log\Phi_k^*(0)=0$. By the Szegő recursion (10), for $k \ge0$, $z \in\mathbb{D}$ we have

where we take the principal branch of the logarithm in the second term of the equality. By definition, $\Phi_k$ and $\Phi_k^*$ are $\mathcal{F}_k$-measurable, and $\alpha_k$ is independent of $\mathcal{F}_k$. Using the rotational invariance of $\alpha_k$, the integrability of $\log(1-|\alpha_k|)$ and the fact that $|zB_k(z)|<1$ for $z\in\mathbb{D}$, one can check that $\{\log\Phi_k^*(z),z\in\mathbb{D}\}$ is a $\mathcal{F}_k$-martingale. In [14], Jiang and Matsumoto computed the moments of traces of circular beta ensembles and established Central Limit Theorems for linear statistics of polynomials. (See also [19] for a direct proof of the CLTs by the first author.) By controlling the error between $\log\Phi_n^*$ and the logarithm of the reversed characteristic polynomial of the full C$\beta$E, and by establishing suitable integrability properties of $\log\Phi_k^*$, Chhaibi and Najnudel [3] characterized the limit $\log\Phi_\infty^*$ as follows.

Note that Proposition 3.1 of [3] is stated for $\beta>2$ due to its connection with Gaussian Multiplicative Chaos on $\partial\mathbb{D}$. Applying the arguments used in the proof of Proposition 3.1 of [3] with Proposition 3.6 of [3] and the CLTs for circular beta ensembles established in [14] yields the general convergence result for $\beta>0$.

We now turn to the Gaussian approximation of the Verblunsky coefficients. The next result follows directly from the rotational invariance and a CLT type approximation for a Beta distributed random variable.

Indeed, using the fact that $|\xi|^2$ has a standard exponential distribution, one can construct an explicit coupling between the Verblunsky coefficients defined in Assumption 11 and an i.i.d. sequence of standard complex Gaussian variables, see Remark 14 below. However, this coupling does not substantially improve the convergence of the Blaschke product. For our purpose, the weak convergence stated in Fact 13 is enough.

We are now ready to prove Proposition 10.

Proof of Proposition 10. By applying Szegő recursion (10) repeatedly, we have

where we used $\alpha_{-1}=-1$ by convention and set $\Phi_{-1}=\Phi^*_{-1}=1$ for the ease of notation. Therefore, we have

Recall that $b_k^{(n)}$ denotes the Taylor coefficient of $B_n$ of degree $k$ at zero. Fix $\ell\ge0$, for any $0<r<1$ and $k\le\ell<n$, by the residue theorem we have

Note that for $j>k$ the integrand on the right hand side of (18) is holomorphic, hence the corresponding integral is zero. Proposition 12 implies that $\Phi^*_n$ converges to $\Phi^*_\infty$ uniformly on compacts in $\mathbb{D}$: moreover, $\Phi^*_\infty$ does not vanish in $\mathbb{D}$. Using this convergence, the fact that the zeros of $\Phi^*_k$, $k\ge0$ lie outside the unit disk, and Cauchy’s integral formula we obtain that

Recall the distribution of $\alpha_k$ as in Assumption 11, by Fact 13 we have

jointly in distribution as $n\to\infty$, where $\xi_i$, $0\le i\le\ell$ are i.i.d. standard complex Gaussian variables. Applying Slutsky’s theorem with (19) and (20) completes the proof of the statement. □

We conclude this section with a brief discussion of an explicit coupling between the Verblunsky coefficients and the i.i.d. complex Gaussian variables.

Tightness results

Proposition 10 establishes the weak convergence of the Taylor coefficients in the power series. In general, this result alone does not imply the convergence of zeros. For instance, the sequence $f_n(z)=2^{n^2}z^n$ only converges at zero (see e.g. [18, 13] for more discussions). To complete the proof of Theorem 1, we need to show weak convergence of $\sqrt{n}B_n(\cdot)$ with respect to the topology of uniform convergence on compacts in $\mathbb{D}$.

We first introduce some notation. Denote by $\mathcal{H}=\mathcal{H}(\mathbb{D})$ the set of holomorphic functions on the unit disk. Define $d_{\mathcal{H}}$ to be the following metric on $\mathcal{H}$: for every $f,g\in\mathcal{H}$, we define

Note that $(\mathcal{H},d_{\mathcal{H}})$ is a complete and separable metric space. The following statement gives the required tightness.

Proof. Recall the Szegő recursion (10), we have

Using the fact that

we get

For $0<r<1$ recall the norm $\|\cdot\|_r$ defined in (21). By iterating the bound (22) we have

Recall that $|\alpha_k|^2\sim\operatorname{Beta}(1,\beta_k)$, we have $\mathbb{E}[|\alpha_k|^2]=\frac{1}{1+\beta_k}$ with $\beta_k=\frac{\beta}{2}(k+1)$. By Minkowski’s inequality we have

where $c_\beta$ is a constant depending only on $\beta$. By splitting the summation at $j=\lfloor n/2\rfloor$, we can find a deterministic constant $c>0$ such that

Since $nr^{n/2}\leq\frac{1}{1-r}$, we may enlarge $c_\beta$ (if necessary) to conclude that

uniformly in $n$. We claim that this uniform bound also implies the tightness of the sequence $\sqrt{n}B_n,n\geq1$. Indeed, for any $\varepsilon>0$, we set $M_{m,\varepsilon}=\sqrt{\frac{c_\beta2^{3m}}{\varepsilon}}$ and consider the compact set $\mathcal{H}_\varepsilon:=\bigcap_{m\geq1}\{f\in\mathcal{H}:\|f\|_{r_m}\leq M_{m,\varepsilon}\}$. By Markov’s inequality, we have

Taking the union bound in $m$ gives that $\mathbb{P}(\sqrt{n}B_n\notin\mathcal{H}_\varepsilon)\leq\varepsilon$ for all $n$. This completes the proof of the statement.

We are now ready to prove Theorem 1.

Proof of Theorem 1. Proposition 10 proves the weak convergence of the Taylor coefficients in the power series, and identifies any subsequential limit. Proposition 15 gives the necessary tightness to conclude the convergence in distribution in $\mathcal{H}$, with respect to the topology of uniform convergence on compact sets of $\mathbb{D}$. Combining these two results completes the proof.

To prove Corollary 2, we invoke the following fact from complex analysis.

Proof. The deterministic statement follows from Hurwitz’s theorem (see e.g. Theorem VII.2.5 in [5]). The probabilistic statement then follows from the continuous mapping theorem.

Proof of Corollary 2. Since $B_n(z)$, $n \ge1$ and $f_{\mathrm{GAF}}(z)$ are a.s. not identically zero, the statement follows from Theorem 1 and Fact 16.

Limits of the multiplicatively perturbed circular beta ensemble

Recall the rank-one multiplicative perturbation of a unitary matrix $\mathbf{M}$ defined in (8). In [10], Forrester and Ipsen studied the asymptotic behavior of the zeros of $\mathbf{M}_n^{[r_n]}$, where $\mathbf{M}_n$ is a size $n$ CUE (corresponding to the $\beta= 2$ case) and $r_n = O(n^{-1/2})$. Their result can be summarized as follows.

The function $f_{\mathrm{GAF},r}(\cdot)$ can be viewed as the Gaussian analytic function $f_{\mathrm{GAF}}(\cdot)$ conditioned on its value at 0. Following the result of Forrester and Ipsen, it is natural to study the asymptotic behavior of zeros of the rank-one multiplicative perturbation of the circular beta ensemble for general $\beta> 0$. To this end, we derive the scaling limit of the normalized Blaschke product of the perturbed model using an argument similar to the proof of Theorem 1. This convergence is strong enough to extend Proposition 17 to general $\beta> 0$.

Proof of Theorem 3. Fix $r \in\mathbb{R}$ and we assume $n$ is large enough such that $|r_n| = \left|\sqrt{\frac{2}{\beta n}}r\right| < 1$. Let $\alpha_k$, $k \ge0$ and $\eta$ be defined as in Assumption 11. By Propositions 7 and 9, the eigenvalues of $\operatorname{Circ}_{n,\beta}^{[r_n]}$ have the same joint distribution as the (sub-unitary) CMV matrix $\mathcal{C}(\alpha_0,\ldots,\alpha_{n-2},r_n\eta)$.

Let $\Phi_k^{[r_n]}$, $0 \le k \le n$ be the monic orthogonal polynomials with respect to the sequence of Verblunsky coefficients $(\alpha_0,\alpha_1,\ldots,\alpha_{n-2},r_n\eta)$, and let $(\Phi_k^{[r_n]})^*$ be the associated reversed polynomials. By the Szegő recursion (10), these polynomials do not depend on $r$ for $0 \le k \le n-1$, and we may therefore suppress the $r$-dependence in the notation for these functions. Consequently, Theorem 1 and Proposition 12 remain valid for the sequences $B_{n-1}$ and $(\Phi_{n-1}^{[r_n]})^*=\Phi_{n-1}^*$.

Applying Szegő recursion (10) to $\Phi_n^{[r_n]}$ and $(\Phi_n^{[r_n]})^*$, we have

hence

Since $r_n=O(n^{-1/2})$ and $|B_{n-1}(z)|<1$ for $z\in\mathbb{D}$, the denominator converges to 1 uniformly on compact sets of $\mathbb{D}$ (see also Theorem 1.7.4 of [24]). By Theorem 1, Slutsky’s theorem and the independence of $\eta$ from $B_{n-1}$, we conclude that

in distribution as $n\to\infty$ uniformly on compact sets of $\mathbb{D}$. Since $zf_{\mathrm{GAF}}(z)-r\bar{\eta}\overset{d}{=}r\eta+\sum_{j=1}^{\infty}\xi_jz^j$, we conclude the proof of Theorem 3.

Repeating the proof of Corollary 2 and applying Theorem 3, we conclude that the zeros of the multiplicatively perturbed C$\beta$E converge to those of the random series $zf_{\mathrm{GAF}}(z)-r\bar{\eta}$, independent of $\beta$. Using the fact that the Gaussian variables are rotation invariant, the limiting zero set has the same distribution as the zero set of the conditional Gaussian series $f_{\mathrm{GAF},r}(\cdot)$ as claimed. $\square$

Asymptotic independence of $\Phi_n^*$ and the bulk zeros

Let $\Lambda_n=\{z_1^{(n)},z_2^{(n)},\ldots,z_n^{(n)}\}$ be distributed according to the size $n$ truncated C$\beta$E and set $\Lambda=\{z\in\mathbb{D}:f_{\mathrm{GAF}}(z)=0\}$. Recall the reversed characteristic polynomial $\Phi_n^*$ of $\Lambda_n$. The goal of this section is to prove the asymptotic independence of $\Phi_n^*$ and the linear statistics of $\Lambda_n$ of compactly supported test functions. Heuristically speaking, the reversed polynomial $\Phi_k^*$ is determined by the first $k$ Verblunsky coefficients, and is close to its limit $\Phi_\infty^*$ when $k$ is sufficiently large (see Proposition 12). In contrast, the bulk zeros, namely those lying inside a compact subset of $\mathbb{D}$, are mainly determined by the first few terms in the series expansion of $\sqrt{n}B_n$. As shown in Proposition 10, these Taylor coefficients depend on the Verblunsky coefficients in reverse order.

Before we present the result, let us introduce a sequence of auxiliary functions

The function $B_n^{(m)}$ depends on $\alpha_m,\alpha_{m+1},\cdots,\alpha_{n-1}$, and is independent of $\mathcal{F}_m$. By the inequality $|w-\overline{z}|\leq|1-wz|$, it is direct to check that $|B_n^{(m)}|\leq1$ on $\mathbb{D}$. Recall the metric $d_{\mathcal{H}}$ defined in (21), we obtain the following bound on $B_n-B_n^{(m)}$.

Proof. By definition, we have

Since $|B_n(z)|\leq1,|B_n^{(m)}(z)|\leq1$ for $z\in\mathbb{D}$, we get the difference $B_{n+1}-B_{n+1}^{(m)}$ is a multiple of $z$. By induction, one can check that $B_n-B_n^{(m)}$ is a multiple of $z^{n-m}$, i.e., it has a zero of order at least $n-m$ at $z=0$. This implies that the first $n-m$ Taylor coefficients of $B_n$ and $B_n^{(m)}$ agree with each other. Repeating the proof of the Schwarz lemma proves the first claim. More precisely, let $g(z)=z^{-(n-m)}(B_n(z)-B_n^{(m)}(z))$. For $|z|=r<1$, by the maximum principle we have $|g(z)|\leq2r^{-(n-m)}$. Letting $r\uparrow1$ yields that $|B_n(z)-B_n^{(m)}(z)|\leq2|z|^{n-m}$ as claimed.

We now prove the second claim. For $n>m$ we have

By splitting the integral at $r_0=(2\sqrt{n})^{-1/(n-m)}$, we get the upper bound

where $c$ is an absolute constant independent of $n,m$. Sending $n\to\infty$ completes the proof. \quad \square

For a continuous function $g$ with compact support inside the unit disk, we denote by $L_g(\Lambda_n):=\sum_{z\in\Lambda_n}g(z)$ the linear statistics of $\Lambda_n$. The next result proves the asymptotic independence of $(\Phi_n^*,L_g(\Lambda_n))\in\mathcal{H}\times\mathbb{C}$.

Proof. By Corollary 2 and Proposition 12, we have the marginal convergence $L_g(\Lambda_n) \Rightarrow L_g(\Lambda)$ and $\Phi_n^* \to\Phi_\infty^*$ a.s. uniformly on compacts as $n \to\infty$. It remains to prove the joint convergence.

For any fixed $m$, recall the auxiliary function $B_n^{(m)}$ defined in (24). For $n > m$ we set $\Lambda_n^{(m)} = \{z \in\mathbb{D}: B_n^{(m)}(z) = 0\}$, and $L_g(\Lambda_n^{(m)}) = \sum_{z \in\Lambda_n^{(m)}} g(z)$. By definition, we have $\Lambda_n^{(m)}$ and $L_g(\Lambda_n^{(m)})$ are independent of $\mathcal{F}_m$.

Let $F:\mathcal{H}(\mathbb{D}) \to\mathbb{C}$ and $G:\mathbb{C} \to\mathbb{C}$ be bounded continuous functions. By the triangle inequality we have

where in (26) we used the fact that $\Phi_m^*$ is independent of $\Lambda_n^{(m)}$. By Theorem 1 and Proposition 18, we have

in law as $n\to\infty$, with respect to the topology of uniform convergence on compact sets of $\mathbb{D}$. By Fact 16 again, we have $(L_g(\Lambda_n),L_g(\Lambda_n^{(m)}))\Rightarrow(L_g(\Lambda),L_g(\Lambda))$, and $\Lambda_n^{(m)}\Rightarrow\Lambda$ as $n\to\infty$ in the vague topology on locally finite point measures on $\mathbb{D}$. Hence, for any fixed $m$ we have

as $n\to\infty$. Therefore, we have

Sending $m\to\infty$ and using the a.s. uniformly on compacts convergence of $\Phi_m^*\to\Phi_\infty^*$ in Proposition 12 completes the proof. □

Note that by applying the argument used in the proof of Proposition 19 to functionals of $(\Phi_n^*,\sqrt{n}B_n)$, one obtains the joint convergence

with respect to the compact-open topology, where the limiting functions $\Phi_\infty^*$ and $f_{\mathrm{GAF}}$ are independent. Using the fact that multiplication is continuous with respect to the compact-open topology, one may further deduce that

in law as $n \to\infty$, with respect to the topology of uniform convergence on compact sets of $\mathbb{D}$. Here $\zeta_j$, $j \ge1$ and $\xi_j$, $j \ge0$ are two sequences of i.i.d. standard complex Gaussian variables and are independent of each other.

We conclude the paper with a discussion of the possible extensions of our results.

Remark 20. Our results can be generalized to a large class of sequences of Verblunsky coefficients. Following the approach used in Propositions 10 and 15, we require a scaling limit (in distribution) and suitable moment bounds for the Verblunsky coefficients, together with the uniform-on-compact convergence of $\log\Phi_n^*$ to a limit $\log\Phi_\infty^*$. In particular, we believe that our methods can be adapted to study the bulk limit of the rank-one truncations of the circular Jacobi beta ensemble and real orthogonal beta ensemble.

In [20], Li and Valkó also raised the question of the scaling limit of multi-rank truncation of circular ensembles. We expect the bulk limit to be a beta-generalization of Krishnapur’s result on the matrix-valued Gaussian analytic functions [18]. We leave these extensions for future work.

Yun Li, School of Mathematical Sciences, Nankai University, China. Email: [email protected]

Mingchang Liu, Academy for Multidisciplinary Studies, Capital Normal University, China Email: [email protected]

Joseph Najnudel, School of Mathematics, University of Bristol, United Kingdom Email: [email protected]