# Edge limits of truncated circular beta ensembles

Yun Li and Benedek Valkó

###### Abstract

We study the scaling limit of the rank-one truncation of various beta ensemble
generalizations of classical unitary/orthogonal random matrices: the circular
beta ensemble, the real orthogonal beta ensemble, and the circular Jacobi beta
ensemble. We derive the scaling limit of the normalized characteristic
polynomials and the point process limit of the eigenvalues near the point 1.
We also treat multiplicative rank one perturbations of our models. Our
approach relies on a representation of truncated beta ensembles given by
Killip-Kozhan [19], together with the random operator framework developed in
[36, 37, 38] to study scaling limits of beta ensembles.

##  1 Introduction

For the classical unitary and orthogonal random matrix ensembles the point
process scaling limit of the eigenvalues is well understood. The eigenvalues
are on the unit circle, and if one scales the eigenangles appropriately, one
obtains a point process limit on the real line. More recently the scaling
limit of the (normalized) characteristic polynomials of these classical
ensembles has been derived and characterized as well, these limits lead to
random entire functions where the zero set is given by the point process limit
of the eigenvalues.

If we remove the first row and first column of a unitary (or orthogonal)
matrix then the resulting matrix has eigenvalues within the unit disk. It is
natural to ask what one can say about the limits of the eigenvalues and the
characteristic polynomial if one studies the truncated random matrices, and
what connections can be shown between the limit objects of the original and
the truncated models. Our main goal is to study these questions for beta-
generalizations of classical random orthogonal and unitary ensembles. We will
also consider similar questions for multiplicative rank one perturbations of
these models. Non-normal perturbations of classical ensembles have a rich
history, see e.g. the surveys [14] and [9] and the references within.

###  1.1 Haar unitary matrices and their truncations

To start with a concrete example, we first consider the case of Haar unitary
matrices. Let $M_{n}$ be an $n\times n$ uniformly chosen unitary matrix. With
probability one $M_{n}$ has $n$ distinct eigenvalues $e^{i\theta_{k}},1\leq
k\leq n$, all on the unit circle. The joint eigenvalue density is given by

$\frac{1}{Z_{n}}\prod_{1\leq j<k\leq
n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{2},\qquad\theta_{j}\in[-\pi,\pi),$ (1)

where $Z_{n}$ is an explicit normalizing constant (see e.g. [8]). The
distribution given by (1) is called the size $n$ _circular unitary ensemble_.
Because of the appearance of the squared Vandermonde determinant in the
probability density, this ensemble is _determinantal_ ([2, 16]), all finite
dimensional marginal densities can be expressed via determinants built from a
fixed kernel function. (We will provide more detail on the results discussed
within this section in the Appendix.) The point process scaling limits of
finite determinantal ensembles can be derived by studying the corresponding
scaling limits of the determinantal kernel. It is a classical result due to
Gaudin, Mehta, Dyson [2, 26] that if we scale the eigenangles of $M_{n}$ by
$n$ then we get a translation invariant determinantal point process in the
limit. We call this point process the $\operatorname{Sine}_{2}$ process.

In a more recent result, Chhaibi, Najnudel and Nikeghbali [6] studied the
scaling limit of the (normalized) characteristic polynomial

$p_{n}(z):=\frac{\det(I_{n}-zM_{n}^{-1})}{\det(I_{n}-M_{n}^{-1})}=\prod_{j=1}^{n}\frac{1-ze^{-i\theta_{j}}}{1-e^{-i\theta_{j}}}$

of the circular unitary ensemble. They showed that under the scaling of the
Gaudin-Mehta-Dyson theorem one obtains a random entire function $\bm{\zeta}$
(named the _stochastic zeta function_) with zero set given by the
$\operatorname{Sine}_{2}$ process.

For a square matrix $M$ we denote by $M^{\textstyle{\ulcorner}}$ the matrix
obtained by removing the first row and column from $M$. Note that we can write
$M^{\textstyle{\ulcorner}}$ as $\Pi^{{\dagger}}M\Pi$ where $\Pi$ is the
appropriate projection matrix, and † denotes the transpose.

Now consider the truncated version of a uniformly chosen $(n+1)\times(n+1)$
unitary matrix, i.e. $M_{n+1}^{\textstyle{\ulcorner}}$. With probability one
this matrix has eigenvalues in the open unit disk
${\mathbb{D}}=\\{z\in{\mathbb{C}}:|z|<1\\}$. The obtained random matrix has
been studied in the physics literature because of its connection to chaotic
scattering problems (see [12, 18] for further discussion and references). In
[41] Życzkowski and Sommers proved that the joint eigenvalue density of
$M_{n+1}^{\textstyle{\ulcorner}}$ (with respect to the Lebesgue measure in the
unit disk) is given by

$\frac{1}{\pi^{n}}\prod_{1\leq j<k\leq n}|z_{j}-z_{k}|^{2},\qquad
z_{j}\in{\mathbb{D}}.$ (2)

We call this distribution the _truncated circular unitary ensemble_. (Note
that the [41] provides a description for the eigenvalue distribution for
general rank-$k$ truncation as well.) The squared Vandermonde term in (2)
indicates that this is also a determinantal point process. By studying the
determinantal kernel one can show that the point process limit of the
eigenvalues of $M_{n+1}^{\textstyle{\ulcorner}}$ _without any additional
scaling_ leads to a determinantal point process limit in ${\mathbb{D}}$. We
may call the resulting process the _bulk scaling limit_ of the truncated
circular unitary ensemble.

It is natural to ask if this point process can be connected to the zeroes of a
‘nice’ random analytic function, since it is the scaling limit of the zeros of
the characteristic polynomial of $M_{n+1}^{\textstyle{\ulcorner}}$. In Peres-
Virág [28] it was shown that this is indeed the case, the bulk scaling limit
of the eigenvalues of $M_{n+1}^{\textstyle{\ulcorner}}$ has the same
distribution as the zero set of the so-called Gaussian analytic function.

One can treat the eigenvalues of $M_{n+1}^{\textstyle{\ulcorner}}$ as a
perturbation of the original eigenvalues of $M_{n+1}$. Because of this, it is
natural to study the behavior of the eigenvalues of the truncated matrix under
the scaling

$\displaystyle z\mapsto-ni\log z,$ (3)

since this corresponds to the scaling $e^{i\theta}\mapsto n\theta$ that takes
the original (unit length) eigenvalues to the $\operatorname{Sine}_{2}$
process.111Throughout the paper we are considering the branch of logarithm
that is defined on ${\mathbb{C}}\setminus(-\infty,0]$ and satisfies
$\log(1)=0$. See Figure 1 for an illustration. It was shown in [1] that under
this scaling the kernel of the truncated circular unitary ensemble (and hence
the ensemble itself) indeed has a limit. The limiting point process is
determinantal, and it is supported in the open upper half plane
${\mathbb{H}}=\\{z\in{\mathbb{C}}:\Im z>0\\}$. We call this the _(hard) edge
scaling limit_ of the truncated model, since we zoom in near $z=1$.

The point process obtained as the edge limit of the truncated circular
ensembles limit process in [1] appeared before in [11] and [13] as the point
process limit of the rank-one additive anti-Hermitian perturbation for the
Gaussian unitary ensemble under the appropriate scaling.

Figure 1: The picture on the left shows the eigenvalues of a truncated
uniformly chosen $100\times 100$ unitary matrix. The picture on the right
shows the same eigenvalues under the edge scaling (3).

It is natural to ask if one can connect the edge limit of the truncated
circular ensemble to the zero set of a random analytic function, and whether
one can characterize this random function in a natural way. We answer this
question in the affirmative in our main result, Theorem 2 below. We provide a
scaling limit for the normalized characteristic polynomial of the truncated
model under the edge scaling, and describe the limiting random entire
function. In fact, our goal is to study this and related questions in a more
general setting: for the beta-generalizations of the circular unitary and
other random unitary and orthogonal ensembles.

###  1.2 CMV matrices, beta ensembles, and their truncations

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

$\frac{1}{Z_{n,\beta}}\prod_{j<k\leq
n}|e^{i\theta_{j}}-e^{i\theta_{k}}|^{\beta},\qquad\theta_{j}\in[-\pi,\pi).$
(4)

Here $Z_{n,\beta}$ is an explicit normalizing constant, see [8]. When
$\beta=2$ we get the circular unitary ensemble. The cases when $\beta=1$ and
$4$ correspond to symmetric/self-dual random unitary matrices, but for general
$\beta>0$ there is no known invariant random matrix ensemble with the
appropriate joint eigenvalue distribution. Note however that (4) has a natural
interpretation as the Gibbs measure corresponding to a log-gas of $n$-points
restricted to the unit circle and interacting via a logarithmic potential.

In [20] Killip and Nenciu (motivated by the results of [7]) constructed a
family of sparse random unitary matrix models
$\\{\mathsf{Circ}_{n,\beta},n\geq 1\\}$ with joint eigenvalue distribution
given by (4). Their construction is based on the theory of orthogonal
polynomials on the unit circle. We provide here a quick overview of their
approach, the precise statements will be reviewed in Section 3.

Suppose that $\mu$ is a discrete probability measure on the unit circle
$\partial{\mathbb{D}}$ with a finite support of $n$ points. The probability
measure $\mu$ can be encoded with its system of monic orthogonal polynomials.
These polynomials satisfy the so-called Szegő recursion, which can be
parameterized with a finite collection of complex numbers
$\alpha_{0},\dots,\alpha_{n-1}$, called the Verblunsky coefficients. In [5]
Cantero, Moral, and Velasquez provided a construction for a ‘canonical’ sparse
(five-diagonal) $n\times n$ unitary matrix (called the _CMV matrix_)

${\mathcal{C}}={\mathcal{C}}(\alpha_{0},\dots,\alpha_{n-1})$

in terms of the Verblunski coefficients, so that the spectral measure of
${\mathcal{C}}$ with respect to the unit vector
$\mathbf{e}_{1}=(1,0,\dots,0)^{{\dagger}}$ is exactly $\mu$. Moreover, if the
probability measure $\mu$ is the spectral measure of an $n\times n$ unitary
matrix $U$ with respect to $\mathbf{e}_{1}$ then the CMV matrix
${\mathcal{C}}$ corresponding to $\mu$ is unitary equivalent to $U$. Note that
the CMV matrix is the analogue of the tridiagonal (Jacobi) matrix constructed
from the coefficients of the three-term recursion of the orthogonal
polynomials of a finitely supported probability measure on ${\mathbb{R}}$.

Let $M_{n}$ be an $n\times n$ Haar unitary matrix, and consider its spectral
measure $\mu_{n}$ with respect to $\mathbf{e}_{1}$. This is a (random)
probability measure with support given by the circular unitary ensemble (1).
Using unitary invariance one can show that the joint distribution of the
weights of $\mu_{n}$ is given by a particular Dirichlet distribution, and that
the weights are independent of the support of $\mu_{n}$. Moreover, the
Verblunsky coefficients of $\mu_{n}$ are independent random variables, and
their distributions can be computed explicitly. This motivated Killip and
Nenciu in [20] to study the random probability measure
$\mu_{n,\beta}^{\textup{KN}}$ with support given by the circular beta ensemble
(4) and weights chosen independently from a particular ($\beta$-dependent)
Dirichlet distribution. [20] showed that the Verblunsky coefficients of
$\mu_{n,\beta}^{\textup{KN}}$ are still independent, with explicitly given
distributions. The corresponding CMV matrix
$\mathsf{Circ}_{n,\beta}:={\mathcal{C}}$ provides a natural sparse random
unitary matrix with spectrum given by the circular beta ensemble (4). For
$\beta=2$ this matrix is unitary equivalent to the Haar unitary matrix
$M_{n}$, and their spectral measures with respect to $\mathbf{e}_{1}$ have the
same distribution.

In [19] Killip and Kozhan studied how removing the first row and column
changes the spectrum of classical random unitary and orthogonal matrices. An
important observation of [19] (which is crucial for our paper as well) is the
following: if $U$ is an $n\times n$ unitary matrix then the truncated matrix
$U^{\textstyle{\ulcorner}}$ is unitary equivalent to the truncated version of
the CMV matrix ${\mathcal{C}}$ corresponding to $U$, which in turn is unitary
equivalent to an $(n-1)\times(n-1)$ CMV matrix built from a simple
transformation of the Verblunsky coefficients of $U$. This means that if we
know the Verblunsky coefficients of $U$ then we can construct a sparse matrix
whose spectrum is the same as that of $U^{\textstyle{\ulcorner}}$.

This observation allowed [19] to provide a sparse matrix model with spectrum
distributed as (2). Their approach also allowed them to study the matrix
models $\mathsf{Circ}_{n,\beta}$ of [20] with the first row and column
removed. They proved that the joint eigenvalue density of the truncated matrix
$\mathsf{Circ}_{n+1,\beta}^{\textstyle{\ulcorner}}$ is given by

$\frac{\beta^{n}}{(2\pi)^{n}}\prod_{1\leq j,k\leq
n}(1-z_{j}\bar{z}_{k})^{\frac{\beta}{2}-1}\prod_{j<k\leq
n}|z_{j}-z_{k}|^{2},\qquad z_{j}\in{\mathbb{D}}.$ (5)

We call the resulting distribution the size $n$ _truncated circular beta
ensemble_. Note that for $\beta=2$ we recover (2). (We remark that [19] also
provided a log-gas interpretation for (5).) Our goal is to study this ensemble
(together with some other related models) under the edge scaling (3).

The approach of Killip and Nenciu [20] can be extended to provide random
matrix representations of beta-generalizations of other random unitary and
orthogonal ensembles where the joint distribution of the Verblunsky
coefficients can be described explicitly. The results of Killip and Kozhan
[19] then provide a natural random matrix representation of the _truncated_
version of these beta ensembles. Our main results provide descriptions of the
edge scaling limits of these truncated ensembles.

###  1.3 Scaling limits of circular beta ensembles and their truncations

Using the Killip-Nenciu representation Killip and Stoiciu in [21] showed that
under the scaling (3) the circular beta ensemble has a point process limit.
They characterized the limiting point process via its counting function using
a system of stochastic differential equations. This limit process was later
shown to be the same as the $\operatorname{Sine}_{\beta}$ process, the bulk
scaling limit of the Gaussian beta ensemble ([27, 36]). Note that
$\operatorname{Sine}_{\beta}$ is not determinantal for general $\beta$, in
fact there is no known description for its joint intensity functions in the
general case.

In a series of papers [36, 37, 38] Valkó and Virág developed a framework to
study the scaling limits of beta ensembles using Dirac-type differential
operators (see Section 2 for a more detailed discussion). [36] showed that the
spectra of unitary CMV matrices and some of their point process limits
(including the $\operatorname{Sine}_{\beta}$ process) can be represented as
the eigenvalues of random Dirac-type differential operators. A Dirac-type
differential operator can be parametrized by a path in the upper half plane
${\mathbb{H}}:=\\{z:\Im z>0\\}$ together with two boundary points in
$\partial{\mathbb{H}}={\mathbb{R}}\cup\\{\infty\\}$. In the case of a unitary
CMV matrix these parameters can be built from the Verblunsky coefficients.
[37] showed how this representation can be used to prove operator level
convergence of the circular beta ensemble to the $\operatorname{Sine}_{\beta}$
process. The path parameter of the random differential operator corresponding
to $\mathsf{Circ}_{n,\beta}$ is a random walk, which under the appropriate
scaling converges to a time-changed hyperbolic Brownian motion. This process
is the path parameter of the random Dirac operator corresponding to the
$\operatorname{Sine}_{\beta}$ process.

[38] developed a framework to study scaling limits of normalized
characteristic polynomials of beta ensembles. In particular, [38] proved that
the normalized and scaled characteristic polynomials of the
$\mathsf{Circ}_{n,\beta}$ converge to a random entire function
$\bm{\zeta}_{\beta}$ with zero set given by $\operatorname{Sine}_{\beta}$.
(For $\beta=2$ this random entire function is the stochastic zeta function
introduced in [6].) The random function $\bm{\zeta}_{\beta}$ is characterized
via various equivalent ways, in particular as the solution of the following
random shooting problem.

######  Theorem 1 ([38]).

Let $b_{1},b_{2}$ be independent two-sided standard Brownian motion, and $q$
an independent standard Cauchy random variable. Consider the unique strong
solution
${\mathcal{H}}_{\beta}:(-\infty,0]\times{\mathbb{C}}\to{\mathbb{C}}^{2}$ of
the stochastic differential equation

$d{\mathcal{H}}_{\beta}=\begin{pmatrix}0&-db_{1}\\\
0&db_{2}\end{pmatrix}{\mathcal{H}}_{\beta}-z\frac{\beta}{8}e^{\frac{\beta}{4}u}\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}{\mathcal{H}}_{\beta}du,\quad u\leq 0,$ (6)

subject to the initial condition
$\lim_{u\to-\infty}\sup_{|z|<1}|{\mathcal{H}}_{\beta}(u,z)-\binom{1}{0}|=0$.
Then $\bm{\zeta}_{\beta}$ has the same distribution as the random function
${\mathcal{H}}_{\beta}(0,z)^{{\dagger}}\binom{1}{-q}$.

Our main result gives the edge scaling limit of the truncated circular beta
ensemble (5) together with the scaling limit of its normalized characteristic
polynomial. This result also provides a connection to the limit objects of the
original circular beta ensemble.

######  Theorem 2.

Under the edge-scaling (3) the truncated circular beta ensemble converges to a
point process $\mathcal{X}_{\beta}$, which has the same distribution as the
zero set of the random entire function
${\mathcal{E}}_{\beta}={\mathcal{H}}_{\beta}(0,\cdot)^{{\dagger}}\binom{1}{-i}$
defined via (6). Moreover, ${\mathcal{E}}_{\beta}$ is the scaling limit of the
normalized characteristic polynomials of the truncated circular beta ensemble.

Theorem 2 is proved in Section 6.1. In fact, we will show that there is a
coupling of the finite ensembles and the limiting object so that the stated
limits hold with probability one with effective (random) error bounds (see
Proposition 48). The proof of the theorem uses the random operator framework
to analyze the scaling limit of the normalized characteristic polynomial of
truncated CMV matrices.

Theorem 2 provides a connection between the scaling limits of the full and the
truncated circular beta ensemble that is new even in the classical $\beta=2$
case. It shows that the scaling limit of the characteristic polynomials of the
circular beta ensemble can be obtained from the corresponding limit of the
truncated model and an independent Cauchy random variable. Both
$\bm{\zeta}_{\beta}$ and ${\mathcal{E}}_{\beta}$ are random entire functions,
and hence they are determined by their restriction to ${\mathbb{R}}$. By
Theorems 1 and 2 we have the following equality in distribution:

$\\{\bm{\zeta}_{\beta}(s):s\in{\mathbb{R}}\\}\stackrel{{\scriptstyle
d}}{{=}}\\{\Re{\mathcal{E}}_{\beta}(s)+q\,\Im{\mathcal{E}}_{\beta}(s):s\in{\mathbb{R}}\\},$

where $q$ is a Cauchy random variable independent of ${\mathcal{E}}_{\beta}$.

If $M$ is a square matrix then the spectrum of $M^{\textstyle{\ulcorner}}$ can
also be studied by considering the spectrum of the rank one multiplicative
perturbation

$M\cdot\operatorname{diag}(0,1,1,\dots,1)$

instead. (Of course, this also adds an additional zero eigenvalue.) This
motivates the study of general rank one multiplicative perturbations. For
$r\in{\mathbb{R}}$ define

$\displaystyle M^{[r]}:=M\cdot\operatorname{diag}(r,1,1,\dots,1).$ (7)

If $M$ is a Haar unitary matrix then the distribution of $M^{[r]}$ has been
studied in [10]. (See also [11] and [13] for related results on rank-one
additive anti-Hermitian perturbations for the Gaussian unitary ensemble.) Note
that because of the various symmetries of the model, we may assume
$r\in[0,1]$.

Following the definition of the truncated circular beta ensemble, it is
natural to define the appropriate rank-one multiplicative perturbation of the
circular beta ensemble as the spectrum of $\mathsf{Circ}_{n,\beta}^{[r]}$. In
[19] Killip and Kozhan derived the joint eigenvalue distribution of
$\mathsf{Circ}_{n,\beta}^{[r]}$. Moreover, they showed that if ${\mathcal{C}}$
is a unitary CMV matrix then the spectrum of ${\mathcal{C}}^{[r]}$ is the same
as a certain explicitly determined CMV matrix. Using their results we are able
to extend the results of Theorem 2.

######  Theorem 3.

Fix $r\in[0,1]$. Consider the random function ${\mathcal{H}}_{\beta}$ defined
via (6), and let $q$ be a standard Cauchy random variable independent of
$b_{1},b_{2}$ appearing in (6). Under the edge-scaling (3) the eigenvalues of
$\mathsf{Circ}_{n,\beta}^{[r]}$ converge to a point process
$\mathcal{X}_{r,\beta}$, which has the same distribution as the zero set of
the random entire function
${\mathcal{E}}_{r,\beta}={\mathcal{H}}_{\beta}(0,\cdot)^{{\dagger}}\binom{1}{-c_{r}}$
with

$\displaystyle c_{r}=\frac{q+i\frac{1-r}{1+r}}{1-iq\frac{1-r}{1+r}}.$ (8)

Moreover, ${\mathcal{E}}_{r,\beta}$ is the limit of the normalized
characteristic polynomials of $\mathsf{Circ}_{n,\beta}^{[r]}$ under the same
scaling.

Note that $c_{r}=q$ for $r=1$ and $c_{r}=i$ for $r=0$, so this result gives an
interpolation between the scaling limits of the unperturbed and the truncated
circular beta ensemble.

Our approach extends to other matrix models as well. We provide versions of
Theorems 2 and 3 for the _real orthogonal beta ensemble_ and the _circular
Jacobi beta ensemble_.

The real orthogonal beta ensemble was introduced in [20] (see also [19]) as a
generalization of the joint eigenvalue distributions of a certain class of the
classical compact random matrix models. (See Section 4.1 for more detail.) The
operator level limit of the real orthogonal beta ensemble in the hard-edge
limit was derived in [24]. The real orthogonal beta ensemble can be naturally
transformed into another classical model, the (real) Jacobi beta ensemble,
whose edge scaling limits were studied in [15]. The truncated version of the
real orthogonal beta ensemble was introduced in [19], where the authors
constructed a sparse matrix model and derived the joint eigenvalue
distribution. In Theorem 52 and Corollary 53 of Section 6.2, we will establish
the scaling limit of the truncated (and perturbed) real orthogonal beta
ensemble, together with the scaling limit of its normalized characteristic
polynomial.

The circular Jacobi beta ensemble is a one-parameter extension of the circular
beta ensemble. For a complex parameter $\delta$ with $\Re\delta>-1/2$ it is
given by the joint density function

$\displaystyle\frac{1}{Z_{n,\beta,\delta}^{\operatorname{CJ}}}\prod_{j<k\leq
n}\left|e^{i\theta_{j}}-e^{i\theta_{k}}\right|^{\beta}\prod_{k=1}^{n}(1-e^{-i\theta_{k}})^{\delta}(1-e^{i\theta_{k}})^{\bar{\delta}}$
(9)

with respect to the uniform measure on the unit circle. For $\delta=0$ this is
just the circular beta ensemble. When $\delta=\beta\tfrac{k}{2}$ with a
positive integer $k$ then this model can be viewed as the circular beta
ensemble conditioned to have $k$ particles at $e^{i\theta}=1$. (See Section
4.1 for additional details.) In [4] the authors constructed a family of
unitary matrix models whose eigenvalues are distributed according to (9).
Following the Killip-Nenciu approach they studied a random probability measure
where the support is given by the circular Jacobi beta ensemble, and the
weights are given by an independently chosen beta-dependent Dirichlet
distribution. [4] showed that although the Verblunsky coefficients for this
measure are usually not independent, a modified version of these coefficients
are in fact independent, and their distributions can be described explicitly.
[24] used this representation to derive the point process and operator level
scaling limit of the circular Jacobi beta ensemble. We build on these results
together with the ideas of [19] to define the truncated (and the perturbed)
version of the circular Jacobi beta ensemble (see Section 7.1). We show that
the joint density of the truncated circular beta ensemble is given by a
constant multiple of

$\prod_{j,k=1}^{n}(1-z_{j}\bar{z}_{k})^{\frac{\beta}{2}-1}\prod_{j<k}|z_{k}-z_{j}|^{2}\prod_{j=1}^{n}\left((1-z_{j})^{\bar{\delta}}(1-\bar{z}_{j})^{\delta}\right),$

generalizing (5). We then derive the point process scaling limit of these new
models, together with the scaling limit of their normalized characteristic
polynomial. See Theorem 66 and Corollary 67 of Section 7.2 for the precise
statements.

###  1.4 Outline of the paper

In Section 2 we review the required background for the considered Dirac-type
differential operators. In Section 3 we give an overview of how finitely
supported probability measures on the unit circle can be represented with CMV
matrices and Dirac-type differential operators. Section 4 describes the
considered finite beta ensembles and their operator level limits, and
summarizes the known results on scaling limits of characteristic polynomials
of these models. In Section 5 we provide general results regarding the
convergence of the eigenvalues truncated and perturbed CMV matrices. Section 6
provides the proofs for Theorems 2 and 3 on the scaling limits of the
truncated and the perturbed circular beta ensemble, and proves the
corresponding results for the real orthogonal beta ensemble. Section 7 proves
our results on the truncated and the perturbed circular Jacobi beta ensemble.
Section 8 is an appendix that contains a more detailed discussion of the
determinantal point processes mentioned in the Introduction, together with a
few open problems.

Acknowledgments. We thank Bálint Virág for helpful comments and references.
B.V was partially supported by the University of Wisconsin – Madison Office of
the Vice Chancellor for Research and Graduate Education with funding from the
Wisconsin Alumni Research Foundation and by the National Science Foundation
award DMS-2246435. Y.L. is partially supported by the Shuimu Tsinghua Scholar
Program and China International Postdoctoral Exchange Fellowship Program
No.YJ20220279.

##  2 Dirac-type differential operators

This section provides a brief overview of Dirac-type operators. A more
detailed discussion can be found in [36], [38], and [40].

###  2.1 Basics of Dirac operators

Let ${\mathcal{I}}$ be $[0,1)$ or $(0,1]$. Suppose
$x+iy:{\mathcal{I}}\mapsto{\mathbb{H}}=\\{z\in{\mathbb{C}}:\Im z>0\\}$ is a
locally bounded measurable function, we define

$R=\frac{X^{{\dagger}}X}{2\det X},\qquad X=\begin{pmatrix}1&-x\\\
0&y\end{pmatrix}$ (10)

to be the positive definite matrix-valued function that encodes $x+iy$. We
consider differential operators of the form

$\tau:f\to R^{-1}(t)Jf^{\prime},\qquad
f:{\mathcal{I}}\mapsto{\mathbb{R}}^{2},\qquad J=\begin{pmatrix}0&-1\\\
1&0\end{pmatrix}.$ (11)

We call $\tau$ a _Dirac-type operator_ , $x+iy$ the _generating path_ of
$\tau$, and $R(t)$ the _weight function_ of $\tau$.

The boundary conditions for $\tau$ at $t=0,1$ are given by two non-zero, non-
parallel vectors $\mathfrak{u}_{0},\mathfrak{u}_{1}\in{\mathbb{R}}^{2}$. We
assume that these vectors are normalized and satisfy certain integrability
conditions with respect to the weight function.

######  Assumption 4.

We assume that

$\mathfrak{u}_{0}^{{\dagger}}J\mathfrak{u}_{1}=1,$ (12)
$\int_{0}^{1}\int_{0}^{t}\mathfrak{u}_{0}^{{\dagger}}R(s)\mathfrak{u}_{0}\,\mathfrak{u}_{1}^{{\dagger}}R(t)\mathfrak{u}_{1}dsdt<\infty,$
(13)

and

$\displaystyle\int_{0}^{1}\|R(s)\mathfrak{u}_{1}\|ds<\infty\text{ if
${\mathcal{I}}=[0,1)$, \quad
and\quad}\int_{0}^{1}\|\mathfrak{u}_{0}^{{\dagger}}R(s)\|ds<\infty\text{ if
${\mathcal{I}}=(0,1]$.}$

Let $L^{2}_{R}$ denote the $L^{2}$ space of functions
$f:{\mathcal{I}}\mapsto{\mathbb{R}}^{2}$ with the $L^{2}$-norm

$\|f\|_{R}^{2}:=\int_{\mathcal{I}}f(s)^{{\dagger}}R(s)f(s)ds.$

Under Assumption 4, the operator $\tau$ defined according to (10) and (11) is
self-adjoint on the following domain:

$\mathrm{dom}(\tau)=\\{v\in L^{2}_{R}\cap\text{AC}:\tau v\in
L^{2}_{R},\lim_{s\to 0}v(s)^{{\dagger}}J\mathfrak{u}_{0}=0,\lim_{s\to
1}v(s)^{{\dagger}}J\mathfrak{u}_{1}=0\\}.$ (14)

Here $\text{AC}({\mathcal{I}})$ is the set of absolutely continuous real
functions on ${\mathcal{I}}$.

Throughout the paper, we use the notations
$\mathtt{Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$ or
$\mathtt{Dir}(x+iy,\mathfrak{u}_{0},\mathfrak{u}_{1})$ interchangeably for the
operator $\tau$ defined via (11) and (10) on the domain (14). We can identify
a nonzero vector in ${\mathbb{R}}^{2}$ with a boundary point in
$\partial{\mathbb{H}}={\mathbb{R}}\cup\\{\infty\\}$ using the projection
operator

$\mathcal{P}\binom{x_{1}}{x_{2}}=\begin{cases}x_{1}/x_{2}&\mbox{if $x_{2}\neq
0$,}\\\ \infty&\mbox{if $x_{2}=0$.}\end{cases}$ (15)

Using this identification the boundary conditions can be parametrized by two
points in $\partial{\mathbb{H}}$.

Under Assumption 4, the operator $\tau$ is invertible, and $\tau^{-1}$ is a
Hilbert-Schmidt integral operator.

######  Proposition 5.

Suppose that $\tau={\tt Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$ satisfies
Assumption 4. Then

$\|\tau^{-1}\|_{\textup{HS}}^{2}=2\int_{0}^{1}\int_{0}^{t}\mathfrak{u}_{0}^{{\dagger}}R(s)\mathfrak{u}_{0}\,\mathfrak{u}_{1}^{{\dagger}}R(t)\mathfrak{u}_{1}dsdt<\infty,$

and $\tau^{-1}$ is a Hilbert-Schmidt integral operator on $L^{2}_{R}$ given by

$\tau^{-1}f(s)=\int_{\mathcal{I}}K_{\tau^{-1}}(s,t)f(t)dt,\quad
K_{\tau^{-1}}(s,t)=\big{(}\mathfrak{u}_{0}\mathfrak{u}_{1}^{{\dagger}}1_{s<t}+\mathfrak{u}_{1}\mathfrak{u}_{0}^{{\dagger}}1_{s\geq
t}\big{)}R(t).$ (16)

Consider the conjugated operator $X\tau X^{-1}$, where $X$ is defined as in
(15). We denote the inverse of this operator by $\mathtt{r\,}\tau:=\left(X\tau
X^{-1}\right)^{-1}$, this is an integral operator on $L^{2}({\mathcal{I}})$
with kernel

$K_{\mathtt{r\,}\tau}(s,t)=\tfrac{1}{2}\big{(}\mathfrak{a}_{0}(s)\mathfrak{a}_{1}(t)^{{\dagger}}\mathbf{1}_{s<t}+\mathfrak{a}_{1}(s)\mathfrak{a}_{0}(t)^{{\dagger}}\mathbf{1}_{s\geq
t}\big{)},\qquad\mathfrak{a}_{j}(s)=\frac{X(s)\mathfrak{u}_{j}}{\sqrt{\det
X}},\quad j=0,1.$ (17)

The conjugated operator $X\tau X^{-1}$ has the same spectrum as the operator
$\tau$, and $\|\mathtt{r\,}\tau\|_{\textup{HS}}=\|\tau^{-1}\|_{\textup{HS}}$.

Proposition 5 and the fact that $\tau$ is self-adjoint implies that $\tau$ has
a discrete spectrum, with countably many eigenvalues that are all nonzero real
numbers, and can only accumulate near $\pm\infty$. We label the eigenvalues of
$\tau$ in increasing order by $\lambda_{k},k\in{\mathbb{Z}}$ so that
$\lambda_{-1}<0<\lambda_{0}$.

Let us remark that if $\tau={\tt Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$
satisfies Assumptions 4, then for any $\sigma\in(0,1)$, the operator $\tau$
restricted in ${\mathcal{I}}\cap[0,\sigma]$ is well-defined with boundary
conditions $\mathfrak{u}_{0},\mathfrak{u}_{1}$ at endpoints $t=0,\sigma$ and
the restricted weight function $R|_{t\in{\mathcal{I}}\cap[0,\sigma]}$. We
denote the restricted operator and its resolvent as
$\tau_{\sigma},\mathtt{r\,}\tau_{\sigma}$ respectively.

###  2.2 Canonical systems and the secular functions

In order to study scaling limits of characteristic polynomials, [38]
introduced the secular function and structure function of a Dirac-type
operator. We briefly review the required definitions and results.

Suppose that ${\mathcal{I}}=(0,1]$ and consider the eigenvalue equation of a
Dirac operator $\tau H=zH$ as a canonical system, i.e.,

$J\frac{d}{dt}H(t,z)=zR(t)H(t,z).$

[38] showed that this system has a unique solution under our assumptions if we
set the initial condition to be $\mathfrak{u}_{0}$.

######  Proposition 6.

Suppose that ${\mathcal{I}}=(0,1]$ and $\tau={\tt
Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$ satisfies Assumptions 4. Then there
exists a unique vector valued function
$H:{\mathcal{I}}\times{\mathbb{C}}\mapsto{\mathbb{C}}^{2}$ so that for every
$z\in{\mathbb{C}}$, the function $H(\cdot,z)$ solves the following ordinary
differential equation

$J\frac{d}{dt}H(t,z)=zR(t)H(t,z),\qquad t\in{\mathcal{I}},\qquad\lim_{t\to
0}H(t,z)=\mathfrak{u}_{0}.$ (18)

Write $H=\binom{A}{B}$. For any $t\in{\mathcal{I}}$, the function $H(t,z)$
satisfies $\|H(t,z)\|\geq 0$, and its components $A(t,\cdot),B(t,\cdot)$ are
analytic functions such that $A(t,x),B(t,x)\in{\mathbb{R}}$ for
$x\in{\mathbb{R}}$.

######  Definition 7.

Under the assumptions of Proposition 6, we define the secular function of
$\tau$ as

$\zeta_{\tau}(\cdot)=H(1,\cdot)^{{\dagger}}J\mathfrak{u}_{1},$ (19)

and the structure function of $\tau$ as

${\mathcal{E}}_{\tau}(\cdot)=H(1,\cdot)^{{\dagger}}\binom{1}{-i}=A(1,\cdot)-iB(1,\cdot).$
(20)

We define the integral trace of $\mathtt{r\,}\tau$ as the integral of the
trace of the kernel $K_{\mathtt{r\,}\tau}$, and denote it by
$\mathfrak{t}_{\tau}$:

$\displaystyle\mathfrak{t}_{\tau}=\int_{0}^{1}\operatorname{tr}K_{\mathfrak{t}_{\tau}}(s,s)ds=\tfrac{1}{2}\int_{0}^{1}\mathfrak{a}_{0}(s)^{{\dagger}}\mathfrak{a}_{1}(s)ds=\int_{0}^{1}\mathfrak{u}_{0}^{{\dagger}}R(s)\mathfrak{u}_{1}ds.$
(21)

It was proved in [38] that the secular function $\zeta_{\tau}$ can be
represented as

$\displaystyle\zeta_{\tau}(z)=e^{-z{\mathfrak{t}}_{\tau}}{\det}_{2}(I-z\,\mathtt{r\,}\tau)=e^{-\tfrac{z}{2}\int_{0}^{1}\mathfrak{a}_{0}(s)^{{\dagger}}\mathfrak{a}_{1}(s)ds}\prod_{k}(1-z/\,\lambda_{k})e^{z/\lambda_{k}},$
(22)

where ${\det}_{2}$ is the second regularized determinant, see [33]. Note that
the integral trace is finite under Assumption 4, and the secular function
$\zeta_{\tau}$ is an entire function with zero set given by
${\text{spec}}(\tau)$, the spectrum of $\tau$. We refer to [38] for additional
details. The secular function of $\tau$ can be viewed as a generalization of
the normalized characteristic polynomial of a matrix.

The next proposition provides comparisons for the solutions of two canonical
systems of the form (18). It also shows that $H(t,\cdot)$ is continuous on
compacts in $z\in{\mathbb{C}}$ at $t=0$.

######  Proposition 8 (Proposition 12 of [39]).

Suppose that ${\mathcal{I}}=(0,1]$ and
$\tau=\mathtt{Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$,
$\tilde{\tau}=\mathtt{Dir}(\tilde{R},\mathfrak{u}_{0},\tilde{\mathfrak{u}}_{1})$
satisfy Assumption 4. Let $H,\tilde{H}$ be the solutions of the corresponding
canonical systems (18), and define $\mathfrak{a}_{0},\tilde{\mathfrak{a}}_{0}$
according to (17). Recall that $\tau_{t}$ is the operator $\tau$ restricted to
$(0,t]$. Then there is an absolute constant $c>1$ depending only on
$\mathfrak{u}_{0}$ so that for all $t\in{\mathcal{I}}$, $z\in{\mathbb{C}}$ we
have

$\displaystyle|H(t,z)-\tilde{H}(t,z)|\leq\left(c^{|z|\left(|\mathfrak{t}_{\tau_{t}}-\tilde{\mathfrak{t}}_{\tau_{t}}|+\|\mathtt{r\,}\tau_{t}-\mathtt{r\,}\tilde{\tau}_{t}\|+\sqrt{\int_{0}^{t}|\mathfrak{a}_{0}(s)-\tilde{\mathfrak{a}}_{0}(s)|^{2}ds\int_{0}^{t}(|\mathfrak{a}_{0}(s)|^{2}+|\tilde{\mathfrak{a}}_{0}(s)|^{2})ds}\right)}-1\right)$
$\displaystyle\hskip 150.0pt\times
c^{\left(|z|(|\mathfrak{t}_{\tau_{t}}|+|\mathfrak{t}_{\tilde{\tau}_{t}}|+\|\mathtt{r\,}\tau_{t}\|+\|\mathtt{r\,}\tilde{\tau}_{t}\|+\int_{0}^{t}\left(|\mathfrak{a}_{0}(s)|^{2}+|\tilde{\mathfrak{a}}_{0}(s)|^{2}\right)ds)+1\right)^{2}}.$
(23)

Moreover, we have for all $t\in{\mathcal{I}}$, $z\in{\mathbb{C}}$

$\displaystyle|H(t,z)-\mathfrak{u}_{0}|\leq\left(c^{|z|(|\mathfrak{t}_{\tau_{t}}|+\|\mathtt{r\,}\tau_{t}\|+\int_{0}^{t}|\mathfrak{a}_{0}(s)|^{2}ds)}-1\right)c^{\left(|z|(|\mathfrak{t}_{\tau_{t}}|+\|\mathtt{r\,}\tau_{t}\|+\int_{0}^{t}|\mathfrak{a}_{0}(s)|^{2}ds)+1\right)^{2}}.$
(24)

Proposition 8 together with the Hoffman-Wielandt inequality in infinite
dimensions (see e.g. [3]) provide similar comparisons for the secular
functions and the spectra of two Dirac operators.

######  Remark 9.

Let $\tau_{1}$, $\tau_{2}$ be two Dirac operators on ${\mathcal{I}}$
satisfying Assumption 4. Denote by
$\lambda_{k,i},\zeta_{i},\mathtt{r\,}\tau_{i},\mathfrak{t}_{\tau_{i}}$ the
eigenvalues, secular function, resolvent and integral trace of $\tau_{i}$.
Then we have

$\displaystyle\sum_{k}\left|\lambda_{k,1}^{-1}-\lambda_{k,2}^{-1}\right|^{2}\leq$
$\displaystyle\|\mathtt{r\,}{\tau_{1}}-\mathtt{r\,}{\tau_{2}}\|_{\textup{HS}}^{2},$
(25)

and there is an absolute constant $c>1$ so that for all $z\in{\mathbb{C}}$

$\displaystyle|\zeta_{1}(z)-\zeta_{2}(z)|\leq$
$\displaystyle\Big{(}c^{|z||\mathfrak{t}_{\tau_{1}}-\mathfrak{t}_{\tau_{2}}|}-1+|z|\big{\|}\mathtt{r\,}\tau_{1}-\mathtt{r\,}\tau_{2}\big{\|}\Big{)}c^{|z|^{2}(\|\mathtt{r\,}{\tau_{1}}\|^{2}+\|\mathtt{r\,}{\tau_{2}}\|^{2})+|z|(|\mathfrak{t}_{\tau_{1}}|+|\mathfrak{t}_{\tau_{2}}|)+1}.$
(26)

#### Transformations of Dirac-type operators

We finish this section with a short discussion on simple transformations of
Dirac-type operators. First note that the two cases ${\mathcal{I}}=(0,1]$ and
${\mathcal{I}}=[0,1)$ can be connected by a time reversal transformation
$\rho$ on functions defined on $(0,1]$ or $[0,1)$ by $\rho f(t)=f(1-t)$. Let
$\mathfrak{r}:{\mathbb{H}}\to{\mathbb{H}}$ be the reflection with respect to
the imaginary axis defined by $\mathfrak{r}(x+iy)=-x+iy$. The next statement
provides a description of the effect of the composition of the time reversal
$\rho$ and the reflection $\mathfrak{r}$, see [38] or [39].

######  Lemma 10 ([38]).

Suppose that the Dirac operator
$\tau=\mathtt{Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$ satisfies Assumption
4. Set $S=\operatorname{diag}(1,-1)$, then the operator $\rho^{-1}S\tau S\rho$
also satisfies Assumption 4 with boundary conditions
$S\mathfrak{u}_{1},S\mathfrak{u}_{0}$, weight function $\rho SRS$, and
generating path $\mathfrak{r}\rho z$. The operators $\tau$ and $\rho^{-1}S\tau
S\rho$ are orthogonally equivalent in the respective $L^{2}$ spaces, and they
have the same integral traces and secular functions.

Consider the projection operator $\mathcal{P}$ defined in (15). It can be
naturally generalized to the projection operator on non-zero two-dimensional
complex vectors with $\mathcal{P}\binom{z_{1}}{z_{2}}=z_{1}/z_{2}$ given
$z_{2}\neq 0$. In this way, a $2\times 2$ non-singular matrix $A$ can be
identified with a linear fractional transformation via
$z\to\mathcal{P}A\binom{z}{1}$. When $A$ is real, the corresponding linear
fractional transformation is an isometry of ${\mathbb{H}}$. The next lemma
describes the effect of a hyperbolic isometry on a Dirac operator.

######  Lemma 11 ([38]).

Let $Q$ be a $2\times 2$ orthogonal matrix with determinant 1. Let
$\mathcal{Q}:\overline{\mathbb{H}}\to\overline{\mathbb{H}}$ be the
corresponding linear isometry of $\overline{\mathbb{H}}$ mapping
$z\in\overline{\mathbb{H}}$ to the ratio of the entries of $Q\binom{z}{1}$.
Suppose that the Dirac operator
$\tau=\mathtt{Dir}(R,\mathfrak{u}_{0},\mathfrak{u}_{1})$ satisfies Assumption
4. Then the operator $Q\tau Q^{-1}$ also satisfies Assumption 4, with boundary
conditions $\mathcal{Q}\mathfrak{u}_{0},\mathcal{Q}\mathfrak{u}_{1}$ and
generating path $\mathcal{Q}(x+iy)$. The two operators are orthogonally
equivalent, they have the same integral traces and secular functions.

##  3 Finitely supported measures on the unit circle

In this section we provide an overview of the CMV construction for finitely
supported probability measures on the unit circle, and review how Dirac-type
operators can be used to study them.

###  3.1 CMV matrices

We briefly review some of the needed facts from the theory of orthogonal
polynomials on the unit circle, together with some properties of CMV matrices.
See [32] for a comprehensive treatment of the subject, or [31] for a shorter
summary.

Let $\nu$ be a probability measure on $\partial{\mathbb{D}}$ with a support of
$n$ distinct points $e^{i\lambda_{j}},1\leq j\leq n$. Let
$\Phi_{k},k=0,\dots,n$ be the Gram-Schmidt orthogonalization of the
polynomials $1,z,\cdots,z^{n}$ with respect to $\nu$. Then
$\Phi_{k},k=0,\dots,n-1$ are the monic orthogonal polynomials with respect to
$\nu$, and $\Phi_{n}(z):=\prod_{j=1}^{n}(z-e^{i\lambda_{j}})$. Together with
the reversed polynomials

$\displaystyle\Phi_{k}^{*}(z):=z^{k}\overline{\Phi_{k}(1/\bar{z})},$

these polynomials satisfy the famous Szegő recursion (see e.g. Section 1.5,
vol. 1 of [32]):

$\displaystyle\binom{\Phi_{k+1}}{\Phi_{k+1}^{*}}=\left(\begin{array}[]{cc}1&-\bar{\alpha}_{k}\\\
-\alpha_{k}&1\\\ \end{array}\right)\left(\begin{array}[]{cc}z&0\\\ 0&1\\\
\end{array}\right)\binom{\Phi_{k}}{\Phi_{k}^{*}},\qquad\binom{\Phi_{0}}{\Phi_{0}^{*}}=\binom{1}{1},\qquad
0\leq k\leq n-1.$ (31)

The constants $\alpha_{k}$, $0\leq k\leq n-1$ are called the Verblunsky
coefficients, they satisfy $|\alpha_{k}|<1$ for $0\leq k\leq n-2$ and
$|\alpha_{n-1}|=1$. The map between the probability measures supported on $n$
points on $\partial{\mathbb{D}}$ and the possible Verblunsky coefficients
$(\alpha_{0},\dots,\alpha_{n-1})\in{\mathbb{D}}^{n-1}\times\partial{\mathbb{D}}$
is invertible, and both the map and its inverse are continuous. (See e.g.
Theorem 1.7.11 of [32].)

The next definition introduces the CMV matrix.

######  Definition 12.

For fixed $n\geq 1$, let $\\{\alpha_{k},0\leq k\leq n-1\\}$ be a sequence of
complex coefficients with $|\alpha_{k}|\leq 1$. Define

$\Xi_{k}=\begin{pmatrix}\bar{\alpha}_{k}&\rho_{k}\\\
\rho_{k}&-\alpha_{k}\end{pmatrix},\qquad\rho_{k}=\sqrt{1-|\alpha_{k}|^{2}},\qquad
0\leq k\leq n-2$

and set $\Xi_{-1}=(1)$ and $\Xi_{n-1}=(\bar{\alpha}_{n-1})$ to be $1\times 1$
matrices.

For $n\geq 2$ we define the _CMV matrix_ corresponding to $\\{\alpha_{k},0\leq
k\leq n-1\\}$ as

$\displaystyle{\mathcal{C}}(\alpha_{0},\cdots,\alpha_{n-1}):={\mathcal{L}}{\mathcal{M}},$
(32)

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

$\displaystyle{\mathcal{L}}=\operatorname{diag}\left(\Xi_{0},\Xi_{2}\cdots,\Xi_{2\lfloor\tfrac{n-1}{2}\rfloor}\right),\qquad{\mathcal{M}}=\operatorname{diag}\left(\Xi_{-1},\Xi_{1}\cdots,\Xi_{2\lfloor\tfrac{n}{2}\rfloor-1}\right).$

For $n=1$ we define the $1\times 1$ CMV matrix as
${\mathcal{C}}(\alpha_{0})=(\bar{\alpha}_{0})$.

Note that ${\mathcal{C}}(\alpha_{0},\dots,\alpha_{n-1})$ is unitary if and
only if $|\alpha_{n-1}|=1$. The following proposition provides a crucial link
between CMV matrices and orthogonal polynomials of a discrete probability
measure on $\partial{\mathbb{D}}$.

######  Proposition 13 ([19, 32]).

Suppose that $\nu$ is a probability measure with a support of $n$ distinct
points on $\partial{\mathbb{D}}$, and let $\alpha_{k},0\leq k\leq n-1$ be its
sequence of Verblunsky coefficients. Then for any $1\leq k\leq n$ we have

$\det(zI_{k}-\mathcal{C}(\alpha_{0},\dots,\alpha_{k-1}))=\Phi_{k}(z).$ (33)

Suppose that $U$ is an $n\times n$ unitary matrix for which
$\mathbf{e}_{1}=(1,0,\dots,0)^{{\dagger}}$ is cyclic. The spectral measure of
$U$ with respect to $\mathbf{e}_{1}$ is given by the following discrete
probability measure:

$\mu=\sum_{k=1}^{n}\delta_{\lambda_{k}}\cdot\left|\langle\mathbf{e}_{1},\mathbf{v}_{k}\rangle\right|^{2}.$
(34)

Here $\lambda_{k},1\leq k\leq n$ are the eigenvalues of $U$, and
$\mathbf{v}_{k},1\leq k\leq n$ are the corresponding unit length (right)
eigenvectors. The following proposition summarizes how a unitary matrix and
the CMV matrix of its spectral measure are connected.

######  Proposition 14 ([5]).

Suppose that $U$ is an $n\times n$ unitary matrix for which $\mathbf{e}_{1}$
is cyclic. Let the spectral measure of $U$ with respect to $\mathbf{e}_{1}$ be
$\nu$, and denote the Verblunsky coefficients of $\nu$ by $\alpha_{k},0\leq
k\leq n-1$. Then there is a unitary matrix $V$ with
$V\mathbf{e}_{1}=\mathbf{e}_{1}$ and
$U=V{\mathcal{C}}(\alpha_{0},\dots,\alpha_{n-1})V^{-1}$. In particular, $U$
and ${\mathcal{C}}$ are unitary equivalent, and they have the same spectral
measure with respect to $\mathbf{e}_{1}$.

The next definition introduces a simple transformation on a collection of
Verblunsky coefficients and on the corresponding CMV matrix.

######  Definition 15.

Suppose that
$(\alpha_{0},\dots,\alpha_{n-1})\in{\mathbb{D}}^{n-1}\times\partial{\mathbb{D}}$
is the sequence of Verblunsky coefficients of the discrete probability measure
$\nu$ on $\partial{\mathbb{D}}$. Let ${\mathcal{C}}$ denote the CMV matrix
corresponding to these Verblunsky coefficients. We define the ‘reversed’
version of the Verblunsky coefficients as

$\displaystyle(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{n-1}):=(-\alpha_{n-1}\bar{\alpha}_{n-2},-\alpha_{n-1}\bar{\alpha}_{n-3},\dots,-\alpha_{n-1}\bar{\alpha}_{0},\alpha_{n-1}).$
(35)

We denote by $\widetilde{\nu}$ and $\widetilde{{\mathcal{C}}}$ the probability
measure and CMV matrix corresponding to the sequence
$\widetilde{\alpha}_{k},0\leq k\leq n-1$, and call these the reversed version
of $\mu$ and ${\mathcal{C}}$, respectively.

Note that since $|\alpha_{n-1}|=1$, the reversal operation is an involution.
This operation appeared in [20] where it was shown that
$\widetilde{{\mathcal{C}}}$ is unitary equivalent to ${\mathcal{C}}$ (if $n$
is even), and to ${\mathcal{C}}^{{\dagger}}$ (if $n$ is odd). This implies
that ${\mathcal{C}}$ and $\widetilde{{\mathcal{C}}}$ have the same
eigenvalues, or equivalently, $\mu$ and $\widetilde{\mu}$ have the same
support. In fact, the arguments of Proposition B.2 of [20] also imply that
$\widetilde{\mu}$ is the spectral measure of ${\mathcal{C}}$ with respect to
the unit vector $\mathbf{e}_{n}=(0,0,\dots,0,1)^{{\dagger}}$.

The following results were proved in [19]. They provide key ingredients for
our study of truncated and perturbed unitary matrices.

######  Proposition 16 ([19]).

Consider the same setup as in Proposition 14. Then if $n\geq 2$ then the
truncated matrix $U^{\textstyle{\ulcorner}}$ is unitary equivalent to
${\mathcal{C}}^{\textstyle{\ulcorner}}$, which in turn is unitary equivalent
to

$\displaystyle{\mathcal{C}}(\widetilde{\alpha}_{0},\dots,\widetilde{\alpha}_{n-2}),\quad\text{if
$n$ is even,
and}\quad{\mathcal{C}}(\widetilde{\alpha}_{0},\dots,\widetilde{\alpha}_{n-2})^{{\dagger}},\quad\text{if
$n$ is odd.}$ (36)

For $r\in[0,1]$ the perturbed matrix $U^{[r]}$ (as defined in (7)) is unitary
equivalent to

$\displaystyle{\mathcal{C}}(\widetilde{\alpha}_{0},\dots,\widetilde{\alpha}_{n-2},r\widetilde{\alpha}_{n-1}),\quad\text{if
$n$ is even,
and}\quad{\mathcal{C}}(\widetilde{\alpha}_{0},\dots,\widetilde{\alpha}_{n-2},r\widetilde{\alpha}_{n-1})^{{\dagger}},\quad\text{if
$n$ is odd.}$ (37)

###  3.2 Connection to Dirac-type operators

In this section, we show how the Dirac operator framework introduced in [36,
37, 38] can be used to study finitely supported probability measures on
$\partial{\mathbb{D}}$.

The following definition constructs a Dirac-type differential operator
corresponding to a finitely supported probability measure on
$\partial{\mathbb{D}}$.

######  Definition 17.

Suppose that $\mu$ is a probability measure on $\partial{\mathbb{D}}$
supported on $n$ distinct points $\\{e^{i\lambda_{j}},1\leq j\leq n\\}$, and
let $\alpha_{k},0\leq k\leq n-1$ be the corresponding Verblunsky coefficients.
For $0\leq k\leq n$ we define

$\displaystyle b_{0}=0,\quad
b_{k}=\mathcal{P}\begin{pmatrix}1&\bar{\alpha}_{0}\\\
\alpha_{0}&1\end{pmatrix}\cdots\begin{pmatrix}1&\bar{\alpha}_{k-1}\\\
\alpha_{k-1}&1\end{pmatrix}\begin{pmatrix}0\\\ 1\end{pmatrix}\quad\text{for
$1\leq k\leq n$}.$ (38)

Set $z_{k}=\mathcal{U}^{-1}(b_{k})$, where $\mathcal{U}$ is the Cayley
transform mapping $\overline{\mathbb{H}}$ to $\overline{\mathbb{D}}$ defined
via

$\mathcal{U}(z)=\mathcal{P}U\begin{pmatrix}z\\\
1\end{pmatrix}=\frac{z-i}{z+i},\quad\quad U=\begin{pmatrix}1&-i\\\
1&i\end{pmatrix}.$ (39)

We call $b_{k},0\leq k\leq n$ and $z_{k},0\leq k\leq n$ the (discrete) _path
parameters_ of $\mu$ in ${\mathbb{D}}$ and in ${\mathbb{H}}$, respectively. We
say that $z:[0,1)\to{\mathbb{H}}$ defined via $z(t)=(x+iy)(t):=z_{\lfloor
nt\rfloor}$ is the _generating path associated to $\mu$_. We set
$\mathfrak{u}_{0}=\binom{1}{0}$, $\mathfrak{u}_{1}=\binom{-z_{n}}{-1}$, and we
call $\tau=\mathtt{Dir}(z(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$ the
Dirac-type operator corresponding to $\mu$.

We may call $\tau$ the Dirac-type operator corresponding to the Verblunsky
coefficients $\alpha_{k},0\leq k\leq n-1$, or the path $b_{0},\dots,b_{n}$. We
may use the notation $\tau=\mathtt{Dir}(b_{-1},b_{0},\dots,b_{n})$ with
$b_{-1}=\mathcal{P}U^{-1}\mathfrak{u}_{0}=1$ to emphasize the path dependence.
We call $b_{-1}$ and $b_{n}$ the _left_ and _right_ _boundary point_ of
$\tau$, respectively.

As the next proposition shows, the Dirac-type operator corresponding to $\mu$
encodes the support of $\mu$ in the spectrum.

######  Proposition 18 ([38, 39]).

Suppose that $\mu$ is a probability measure on $\partial{\mathbb{D}}$
supported on $n$ distinct points $e^{i\lambda_{j}},1\leq j\leq n$, with
$\mu(\\{1\\})=0$. Then the Dirac-type operator $\tau$ corresponding to $\mu$
satisfies Assumption 4 with ${\mathcal{I}}=[0,1)$, and its spectrum is given
by

${\text{spec}}(\tau)=n\Lambda_{n}+2\pi
n{\mathbb{Z}},\qquad\Lambda_{n}=\\{\lambda_{1},\dots,\lambda_{n}\\}.$

In [36, 39] it was observed that the Dirac operator representation for $\mu$
can be simplified using the modified Verblunsky coefficients. These are
defined in terms of the Verblunsky coefficients via the recursion

$\displaystyle\gamma_{k}=\bar{\alpha}_{k}\prod_{j=0}^{k-1}\frac{1-\bar{\gamma}_{j}}{1-\gamma_{j}},\qquad
0\leq k\leq n-1.$ (40)

Note that the modified Verblunsky coefficients satisfy
$|\gamma_{k}|=|\alpha_{k}|$. Denote by ${\mathcal{T}}$ the mapping from the
Verblunsky coefficients to the modified ones. This mapping is invertible, in
fact for any $k\geq 1$ it provides a one-to-one correspondence between the
first $k$ Verblunsky coefficients and the first $k$ modified Verblunsky
coefficients. We will use the notation ${\mathcal{T}}_{k}$ for this restricted
map.

The modified Verblunsky coefficients of a discrete probability measure $\mu$
are connected to its normalized orthogonal polynomials (with normalization at
$1$). Let $\Phi_{k}$ be the monic orthogonal polynomials of $\mu$, with
$\Phi^{*}_{k}$ being the reversed polynomials. We set

$\Psi_{k}(z)=\frac{\Phi_{k}(z)}{\Phi_{k}(1)},\qquad\Psi_{k}^{*}(z)=\frac{\Phi_{k}^{*}(z)}{\Phi_{k}^{*}(1)}.$
(41)

These are always well defined for $0\leq k\leq n-1$, and $\Psi_{n}$ is defined
as long as $\mu(\\{1\\})=0$. The polynomials $\Psi,\Psi^{*}$ satisfy the
following modified version of the Szegő recursion (31):

$\displaystyle\binom{\Psi_{k+1}}{\Psi_{k+1}^{*}}=\left(\begin{array}[]{cc}\frac{1}{1-\gamma_{k}}&-\frac{\gamma_{k}}{1-\gamma_{k}}\\\
-\frac{\bar{\gamma}_{k}}{1-\bar{\gamma}_{k}}&\frac{1}{1-\bar{\gamma}_{k}}\\\
\end{array}\right)\left(\begin{array}[]{cc}z&0\\\ 0&1\\\
\end{array}\right)\binom{\Psi_{k}}{\Psi_{k}^{*}},\quad\binom{\Psi_{0}}{\Psi_{0}^{*}}=\binom{1}{1},\quad
0\leq k\leq n-1.$ (46)

The matrices appearing in this recursion correspond to affine transformations.

Affine transformations of ${\mathbb{H}}$ can be parametrized by the elements
of ${\mathbb{H}}$ as follows. For $z=x+iy\in{\mathbb{H}}$ we define the matrix
$A_{z}$ and the corresponding linear fractional transformation
$\mathcal{A}_{z,{\mathbb{H}}}:\overline{\mathbb{H}}\mapsto\overline{\mathbb{H}}$
as follows:

$A_{x+iy,{\mathbb{H}}}=\left(\begin{array}[]{cc}1&-x\\\ 0&y\\\
\end{array}\right),\qquad\mathcal{A}_{x+iy,{\mathbb{H}}}(w)=\mathcal{P}A_{x+iy,{\mathbb{H}}}\binom{w}{1}.$
(47)

Note that $x+iy$ is the pre-image of $i$ under
$\mathcal{A}_{x+iy,{\mathbb{H}}}$.

The transformations ${\mathcal{A}}_{z,{\mathbb{H}}}$ are isometries of the
half-plane model of the hyperbolic plane. The corresponding transformations on
the unit-disk model of the hyperbolic plane can be obtained by conjugating
with the Cayley transform. For $\gamma\in{\mathbb{D}}$ we set

$\displaystyle
A_{\gamma,{\mathbb{D}}}=UA_{{\mathcal{U}}^{-1}(\gamma),{\mathbb{H}}}U^{-1},\qquad\mathcal{A}_{\gamma,{\mathbb{D}}}=\mathcal{U}\circ\mathcal{A}_{{\mathcal{U}}^{-1}(\gamma),{\mathbb{H}}}\circ\mathcal{U}^{-1},$
(48)

which leads to

$A_{\gamma,{\mathbb{D}}}=\left(\begin{array}[]{cc}\frac{1}{1-\gamma}&\frac{\gamma}{\gamma-1}\\\
\frac{\bar{\gamma}}{\bar{\gamma}-1}&\frac{1}{1-\bar{\gamma}}\\\
\end{array}\right),\qquad\mathcal{A}_{\gamma,{\mathbb{D}}}(z)=\mathcal{P}A_{\gamma,{\mathbb{D}}}\binom{z}{1}.$
(49)

Note that the matrix coefficient in (46) is exactly
$A_{\gamma_{k},{\mathbb{D}}}$, and ${\mathcal{A}}_{\gamma,{\mathbb{D}}}$ maps
$\gamma$ to $0$.

The following proposition shows that the generating path of a discrete
probability measure $\mu$ can be expressed via a simple (affine) recursion
using the modified Verblunsky coefficients. Define $w_{k},v_{k}\in\mathbb{R}$
from the modified Verblunsky coefficients as

$\displaystyle
v_{k}+iw_{k}:=\frac{2i\gamma_{k}}{1-\gamma_{k}}={\mathcal{U}}^{-1}(\gamma_{k})-i.$
(50)

######  Proposition 19 ([38, 39]).

Suppose that $\mu$ is a probability measure on $\partial{\mathbb{D}}$
supported on $n$ distinct points $e^{i\lambda_{j}},1\leq j\leq n$, and
$\mu(\\{1\\})=0$. Let $\gamma_{k},0\leq k\leq n-1$ be the modified Verblunsky
coefficients of $\mu$. Let $b_{k},z_{k},0\leq k\leq n$ be the path parameters
of $\mu$ defined as in Definition 17. Then the following identities hold for
$0\leq k\leq n-1$:

$\displaystyle z_{k+1}$ $\displaystyle=z_{k}+(v_{k}+iw_{k})\Im z_{k},$ (51)

with $v_{k},w_{k}$ defined in (50), and

$b_{k+1}=\frac{b_{k}+\gamma_{k}\frac{1-b_{k}}{1-\bar{b}_{k}}}{1+\bar{b}_{k}\gamma_{k}\frac{1-b_{k}}{1-\bar{b}_{k}}}.$
(52)

We also have

$b_{k}={\mathcal{A}}_{\gamma_{0},{\mathbb{D}}}^{-1}\circ\cdots\circ{\mathcal{A}}_{\gamma_{k-1},{\mathbb{D}}}^{-1}(0).$
(53)

The normalized orthogonal polynomials of $\mu$ can be expressed using the
canonical system (18) of the Dirac operator $\tau$ corresponding to $\mu$.

######  Proposition 20 ([38, 39]).

Under the same setup as in Proposition 18, consider the solution to the
canonical system (18)

$\tau H=\lambda H,\qquad
H(0,\lambda)=\binom{1}{0},\qquad(t,\lambda)\in[0,1]\times{\mathbb{C}}.$

Then the normalized orthogonal polynomials $\Psi_{k},\Psi_{k}^{*}$ of $\mu$
(defined via (41)) satisfy

$\binom{\Psi_{k}(e^{i\lambda/n})}{\Psi_{k}^{*}(e^{i\lambda/n})}=e^{i\lambda
k/(2n)}\left(\begin{array}[]{cc}1&-z_{k}\\\ 1&-\bar{z}_{k}\\\
\end{array}\right)H(k/n,\lambda),\qquad 0\leq k\leq n.$ (54)

Note that Proposition 20 applied with $k=n$ gives

$\displaystyle
e^{-i\lambda/2}\Psi_{n}(e^{i\lambda/n})=\binom{1}{-z_{n}}^{\dagger}H(1,\lambda)=H(1,\lambda)^{{\dagger}}J\binom{1}{z_{n}}.$
(55)

Recall that $\Psi_{n}(\cdot)$ is just the normalized characteristic polynomial
corresponding to the support of $\mu$:

$\Psi_{n}(z)=\frac{\Phi_{n}(z)}{\Phi_{n}(1)}=\prod_{j=1}^{n}\frac{z-e^{i\lambda_{j}}}{1-e^{i\lambda_{j}}}$

Hence (19) and (55) imply that the scaled and normalized characteristic
polynomial of $\mu$ is the same as the secular function of the Dirac-type
operator corresponding to $\mu$.

##  4 beta ensembles from classical unitary and orthogonal random matrices

This section collects the matrix models for various beta-generalizations of
unitary and orthogonal random matrices, along with their Dirac operator
representations and operator limits.

###  4.1 Finite ensembles and their Dirac operator representations

#### Circular beta ensemble

Recall the definition of the size $n$ circular beta ensemble with density
function (4) and the spectral measure with respect to $\mathbf{e}_{1}$ (34).
Recall also that for $a_{1},\dots,a_{n}>0$ the Dirichlet$(a_{1},\dots,a_{n})$
distribution is a probability measure on the set

$\\{(x_{1},x_{2},\dots,x_{n})\in[0,1]^{n}:\sum_{i=1}^{n}x_{i}=1\\}$

with joint probability density function

$\displaystyle\frac{\Gamma(\sum_{i=1}^{n}a_{i})}{\prod_{i=1}^{n}\Gamma(a_{i})}\prod_{i=1}^{n}x_{i}^{a_{i}-1}.$

We define the size $n$ Killip-Nenciu measure as a random probability measure
on $\partial{\mathbb{D}}$ as

$\mu^{\textup{KN}}_{n,\beta}=\sum_{j=1}^{n}\pi_{j}\delta_{e^{i\lambda_{j}}},$
(56)

where the support is distributed as the size $n$ circular beta ensemble, and
the weights $\pi_{j},1\leq j\leq n$ are chosen according to
Dirichlet$(\beta/2,\dots,\beta/2)$ distribution, independently of the support.
The joint distribution of the Verblunsky coefficients with respect to
$\mu_{n,\beta}^{\textup{KN}}$ was computed in [20].

######  Definition 21.

For $a>0$ we denote by $\Theta(a+1)$ the distribution on ${\mathbb{D}}$ that
has probability density function

$\displaystyle\tfrac{a}{2\pi}(1-|z|^{2})^{a/2-1}.$ (57)

The definition is extended for $a=0$ as follows: $\Theta(1)$ is the uniform
distribution on $\partial{\mathbb{D}}$.

######  Proposition 22 ([20]).

For a fixed $n$ and $\beta>0$, the sequence of Verblunsky coefficients
$\alpha_{k},0\leq k\leq n-1$ of $\mu_{n,\beta}^{\textup{KN}}$ are independent,
and $\alpha_{k}\sim\Theta(\beta(n-k-1)+1)$ for $0\leq k\leq n-1$.

######  Definition 23.

For fixed $n$ and $\beta>0$, we denote by $\mathsf{Circ}_{n,\beta}$ and
$\mathtt{Circ}_{n,\beta}$ the CMV matrix and Dirac-type operator corresponding
to $\mu_{n,\beta}^{\textup{KN}}$.

#### Real orthogonal beta ensemble

The _real orthogonal beta ensemble_ is a family of distributions supported on
conjugated pairs of points on the unit circle indexed by real parameters
$\beta>0,a>-1,b>-1$. If we parametrize the size $2n$ real orthogonal beta
ensemble as $\\{\pm e^{i\theta_{1}},\dots,\pm e^{i\theta_{n}}\\}$ with
$\theta_{j}\in(0,\pi)$ then the joint density for
$(\theta_{1},\dots,\theta_{n})$ is proportional to

$\displaystyle\prod_{j<k\leq
n}|\cos(\theta_{j})-\cos(\theta_{k})|^{\beta}\times\prod_{k=1}^{n}|1-\cos(\theta_{k})|^{\frac{\beta}{2}(a+1)-1/2}|1+\cos(\theta_{k})|^{\frac{\beta}{2}(b+1)-1/2}.$
(58)

The ensemble was introduced in [20], it can be viewed as a generalization of
joint eigenvalue distributions of some classical orthogonal ensembles. For
example when $\beta=2$, $a=b=\frac{1}{\beta}-1$, (58) describes the joint
eigenvalue distribution of a $2n\times 2n$ special orthogonal matrix chosen
uniformly with respect to the Haar measure on $\mathbb{SO}(2n)$.

Consider the probability measure

$\mu^{\operatorname{RO}}_{2n,\beta,a,b}=\sum_{j=1}^{n}\frac{1}{2}\pi_{j}(\delta_{e^{i\theta_{j}}}+\delta_{e^{-i\theta_{j}}})$
(59)

on the unit circle, where the support is distributed as a $2n$ real orthogonal
beta ensemble, and the weights $(\pi_{1},\dots,\pi_{n})$ is an independent
random vector that has a Dirichlet$(\beta/2,\dots,\beta/2)$ distribution. The
joint distribution of the Verblunsky coefficients of
$\mu_{2n,\beta,a,b}^{\operatorname{RO}}$ was derived in [19].

######  Definition 24.

For $s,t>0$ let $\tilde{\mathrm{B}}(s,t)$ denote the scaled (and flipped) beta
distribution on $(-1,1)$ that has probability density function

$\displaystyle\tfrac{2^{1-s-t}\Gamma(s+t)}{\Gamma(s)\Gamma(t)}(1-x)^{s-1}(1+x)^{t-1}.$

######  Proposition 25 (Theorem 2 of [20], Proposition 4.5 in [19]).

For given $\beta>0$, $a,b>-1$ and fixed $n$, the Verblunsky coefficients
$\alpha_{k},0\leq k\leq 2n-1$ of the random probability measure
$\mu^{\operatorname{RO}}_{2n,\beta,a,b}$ are independent with
$\alpha_{2n-1}=-1$, and for $0\leq k\leq 2n-2$

$\displaystyle\alpha_{k}\sim\begin{cases}\mathrm{\tilde{B}}\big{(}\tfrac{\beta}{4}(2n-k+2a),\tfrac{\beta}{4}(2n-k+2b)\big{)},\quad&\text{if
$k$ is even,}\\\
\mathrm{\tilde{B}}\big{(}\tfrac{\beta}{4}(2n-k+2a+2b+1),\frac{\beta}{4}(2n-k-1)\big{)},\quad&\text{if
$k$ is odd.}\end{cases}$

######  Definition 26.

For fixed $n\geq 1$, $a,b>-1$ and $\beta>0$, we define
$\mathsf{RO}_{2n,\beta,a,b}$ and $\mathtt{RO}_{2n,\beta,a,b}$ as the CMV
matrix model and the Dirac-type operator corresponding to
$\mu_{2n,\beta,a,b}^{\operatorname{RO}}$, respectively.

#### Circular Jacobi beta ensemble

The circular Jacobi beta ensemble can be viewed as a one-parameter
generalization of the circular beta ensemble. Let $\delta$ be a complex
parameter such that $\Re\delta>-1/2$. The size $n$ circular Jacobi beta
ensemble is defined as the probability measure on the $n$ distinct points
$\\{e^{i\theta_{1}},\dots,e^{i\theta_{n}}\\}$ with $\theta_{j}\in[-\pi,\pi)$,
where the joint density function of the angles $\theta_{j}$ is given by (9).
For $\beta=2$ the distribution was studied by Hua [17] and Pickrell [29], this
special case is sometimes called the Hua-Pickrell measure.

Consider the random probability measure

$\mu^{\operatorname{CJ}}_{n,\beta,\delta}=\sum_{j=1}^{n}\pi_{j}\delta_{e^{i\theta_{j}}}$
(60)

on the unit circle, where the support is distributed as the size $n$ circular
Jacobi beta ensemble, the weights are Dirichlet$(\beta/2,\dots,\beta/2)$
distributed and are independent of the support. In [4] the authors showed that
the modified Verblunsky coefficients $\gamma_{k},0\leq k\leq n-1$ of
$\mu_{n,\beta,\delta}^{\operatorname{CJ}}$ are independent and described
explicitly their joint distribution.

We first introduce a generalization of the $\Theta(a+1)$ distribution defined
in Definition 21.

######  Definition 27.

For $a>0$ and $\Re\delta>-1/2$ let $\Theta(a+1,\delta)$ be the distribution on
${\mathbb{D}}$ that has probability density function

$\displaystyle
c_{a,\delta}(1-|z|^{2})^{a/2-1}(1-z)^{\bar{\delta}}(1-\bar{z})^{\delta},\qquad
c_{a,\delta}=\tfrac{\Gamma(a/2+1+\delta)\Gamma(a/2+1+\bar{\delta})}{\pi\Gamma(a/2)\Gamma(a/2+1+\delta+\bar{\delta})}.$
(61)

For the $a=0$, $\Re\delta>-1/2$ case we define $\Theta(1,\delta)$ to be the
distribution on $\partial{\mathbb{D}}=\\{|z|=1\\}$ with probability density
function

$\displaystyle\tfrac{\Gamma(1+\delta)\Gamma(1+\bar{\delta})}{\Gamma(1+\delta+\bar{\delta})}(1-z)^{\bar{\delta}}(1-\bar{z})^{\delta}.$
(62)

######  Proposition 28 ([4]).

For given $n\geq 1,\beta>0,\Re\delta>-1/2$, the sequence of modified
Verblunsky coefficients $\gamma_{k},0\leq k\leq n-1$ of
$\mu_{n,\beta,\delta}^{\textup{CJ}}$ are independent with
$\gamma_{k}\sim\Theta(\beta(n-k-1)+1,\delta)$ for $0\leq k\leq n-1$.

Note that for $\delta\neq 0$ the Verblunsky coefficients $\alpha_{k},0\leq
k\leq n-1$ are not independent.

######  Definition 29.

For given $n\geq 1,\beta>0,\Re\delta>-1/2$, we define
$\mathsf{CJ}_{n,\beta,\delta}$ and $\mathtt{CJ}_{n,\beta,\delta}$ as the CMV
matrix model and the Dirac-type operator corresponding to
$\mu_{n,\beta,\delta}^{\textup{CJ}}$, respectively.

###  4.2 Random operator limits from beta ensembles

This section discusses the operator limits of the finite ensembles in Section
4.1. The main idea is that under the appropriate scaling, the piece-wise
constant generating paths of the
$\mathtt{Circ}_{n,\beta},\mathtt{RO}_{2n,\beta,a,b}$ and
$\mathtt{CJ}_{n,\beta,\delta}$ operators converge to certain diffusions in the
hyperbolic plane. Then one can construct random differential operators in
terms of these diffusions. These limiting operators will be denoted as
$\mathtt{Sine}_{\beta},\mathtt{Bess}_{\beta,a}$ and
$\mathtt{HP}_{\beta,\delta}$, respectively. For convenience, we will define
the limiting operators with generating paths that lie in ${\mathbb{H}}$. We
also set

$v(t)=v_{\beta}(t)=-\frac{4}{\beta}\log(1-t)$

be the logarithmic time change function.

The $\mathtt{Sine}_{\beta}$ operator was introduced in [36] as the
$n\to\infty$ limit of the $\mathtt{Circ}_{n,\beta}$ operator. The operator
level convergence was proved in [37] with an explicit rate of convergence, see
Proposition 47 below.

######  Definition 30.

Fix $\beta>0$. Let $B_{1},B_{2}$ be independent standard Brownian motion, and
let $\mathsf{x}_{v}+i\mathsf{y}_{v},v\geq 0$ be the strong solution of the SDE

$\displaystyle d\mathsf{y}=\mathsf{y}dB_{1},\quad
d\mathsf{x}=\mathsf{y}dB_{2},\quad\mathsf{y}(0)=1,\mathsf{x}(0)=0.$ (63)

For $t\in[0,1)$ define $z(t)=x(t)+iy(t)=\mathsf{x}_{v}+i\mathsf{y}_{v}$ where
$v=v(t)$. Let $\mathfrak{u}_{0}=\binom{1}{0}$,
$\mathfrak{u}_{1}=\binom{-q}{-1}$, where $q=\lim_{t\to\infty}\mathsf{x}(t)$.
Set
$\mathtt{Sine}_{\beta}=\mathtt{Dir}(z(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$.

Note that $\mathsf{x}_{v}+i\mathsf{y}_{v},v\geq 0$ is just a hyperbolic
Brownian motion in ${\mathbb{H}}$, started from $i$.

The hard-edge Dirac operator $\mathtt{Bess}_{\beta,a}$ and the Hua-Pickrell
operator $\mathtt{HP}_{\beta,\delta}$ were also introduced in [36]. It has
been shown in [24] that they are indeed the operator level limits of the
$\mathtt{RO}_{2n,\beta,a,b}$ and $\mathtt{CJ}_{n,\beta,\delta}$ operators,
respectively.

######  Definition 31.

Fix $\beta>0,a>-1$, and let $B$ be a standard Brownian motion. Set
$\mathsf{y}(t)=e^{-\frac{\beta}{4}(2a+1)t-B(2t)}$, $y(t)=\mathsf{y}(v(t))$,
$\mathfrak{u}_{0}=\binom{1}{0}$, and $\mathfrak{u}_{1}=\binom{0}{-1}$. Define
the hard-edge operator as
$\mathtt{Bess}_{\beta,a}:=\mathtt{Dir}(iy,\mathfrak{u}_{0},\mathfrak{u}_{1})$.

######  Definition 32.

Fix $\beta>0$ and $\delta\in{\mathbb{C}}$ with $\Re\delta>-1/2$. Let
$B_{1},B_{2}$ be independent standard Brownian motion, and let
$\mathsf{x}_{v}+i\mathsf{y}_{v},v\geq 0$ be the strong solution of the SDE

$\displaystyle d\mathsf{y}=\left(-\Re\delta dt+dB_{1}\right)\mathsf{y},\quad
d\mathsf{x}=\left(\Im\delta
dt+dB_{2}\right)\mathsf{y},\quad\mathsf{y}(0)=1,\mathsf{x}(0)=0.$ (64)

For $t\in[0,1)$ define $z(t)=x(t)+iy(t)=\mathsf{x}_{v}+i\mathsf{y}_{v}$ where
$v=v(t)$. Let $\mathfrak{u}_{0}=\binom{1}{0}$,
$\mathfrak{u}_{1}=\binom{-q}{-1}$, where $q=\lim_{v\to\infty}\mathsf{x}(v)$.
Set
$\mathtt{HP}_{\beta,\delta}=\mathtt{Dir}(z(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$.

Note that the SDE (64) can be solved explicitly and the solution is given by

$\mathsf{y}(v)=e^{B_{1}(v)-(\Re\delta+\frac{1}{2})v},\quad\mathsf{x}(v)=\int_{0}^{v}\mathsf{y}(s)dB_{2}(s)+\Im\delta\int_{0}^{v}\mathsf{y}(s)ds.$
(65)

In the case when $\delta=0$ the equation reduces to (63). In particular, the
$\mathtt{HP}_{\beta,\delta}$ operator can be viewed as the
$\delta$-generalization of $\mathtt{Sine}_{\beta}$, and we have
$\mathtt{HP}_{\beta,0}=\mathtt{Sine}_{\beta}$.

###  4.3 Random analytic functions from beta ensembles

As explained in Section 2.2, the Hilbert–Schmidt convergence of the resolvents
of Dirac-type operators and the convergence of the integral traces imply the
uniform on compacts convergence of the secular functions. In Section 3.2 we
saw that we can express the characteristic polynomials of finite ensembles on
the unit circle via the secular function of an associated Dirac-type operator.
The operator level limits discussed in the previous section lead then lead to
convergence statements regarding the scaled and normalized characteristic
polynomials of the respective finite ensembles.

In this section, we briefly review the secular functions and structure
functions arising from the limits of the considered beta ensembles. The
constructions rely on certain time-reversed and transformed versions of the
limiting operators introduced in Section 4.2. Recall the transformations
introduced in Section 2.2.

In [38] Valkó and Virág constructed the following Dirac-type operator on
$(0,1]$ and showed that it is orthogonally equivalent to the
$\mathtt{Sine}_{\beta}$ operator. Consider the time change

$u(t)=u_{\beta}(t)=\frac{4}{\beta}\log t,\quad\quad t\in(0,1].$ (66)

######  Definition 33.

Let $b_{1},b_{2}$ be independent two-sided standard Brownian motion, and set

$\displaystyle\mathsf{y}_{u}=e^{b_{2}(u)-u/2},\qquad\mathsf{x}_{u}=-\int_{u}^{0}e^{b_{2}(s)-\tfrac{s}{2}}db_{1},$
(67)

For $t\in(0,1]$ set
$\hat{z}(t)=(\hat{x}+i\hat{y})(t):=\mathsf{x}_{u}+i\mathsf{y}_{u},\mathfrak{u}_{0}=\binom{1}{0}$,
and $\mathfrak{u}_{1}=\binom{-q}{-1}$, where $q$ is a Cauchy distributed
random variable independent of $b_{1},b_{2}$. Define
$\tau_{\beta}=\tau_{\beta}^{\textup{Sine}}=\mathtt{Dir}(\hat{z}(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$,
and denote by $\zeta_{\beta}:=\zeta_{\tau_{\beta}}$ and
${\mathcal{E}}_{\beta}:={\mathcal{E}}_{\tau_{\beta}}$ the corresponding
secular and structure function according to Definition (7).

######  Proposition 34 ([38]).

Let $q$ and $\tau_{\beta}$ be defined as in Definition 33. Then the
$\tau_{\beta}$ operator satisfies Assumption 4, and the orthogonally
equivalent operator

$\rho^{-1}SQ\tau_{\beta}Q^{-1}S\rho,\qquad
Q=\frac{1}{\sqrt{1+q^{2}}}\left(\begin{array}[]{cc}q&1\\\ -1&q\\\
\end{array}\right),$

has the same distribution as $\mathtt{Sine}_{\beta}$. Let $H_{\beta}(t,z)$ be
the solution to the canonical system (18) of the $\tau_{\beta}$ operator, and
let $\mathsf{x}_{u}$ and $\mathsf{y}_{u}$ be defined according to (67). Set

${\mathcal{H}}_{\beta}(u,z)=\left(\begin{array}[]{cc}1&-\mathsf{x}_{u}\\\
0&\mathsf{y}_{u}\\\ \end{array}\right)H_{\beta}(t(u),z),\qquad t=t(u)=e^{\beta
u/4},$ (68)

then ${\mathcal{H}}$ satisfies the stochastic differential equation (6). Since
${\mathcal{H}}_{\beta}(0,z)=H_{\beta}(1,z)$, the structure-function
${\mathcal{E}}_{\beta}(z)$ can be represented as
${\mathcal{E}}_{\beta}(z)={\mathcal{H}}_{\beta}(0,z)^{{\dagger}}\binom{1}{-i}$.

Using the results of Proposition 34 [38] showed the convergence of the scaled
and normalized characteristic polynomials of the circular beta ensemble to the
stochastic zeta function.

######  Proposition 35 ([38]).

Consider the size $n$ (unperturbed) circular beta ensemble, i.e., the
eigenvalues of $\mathsf{Circ}_{n,\beta}$, and its normalized characteristic
polynomial
$p_{n,\beta}(z):=\frac{\det(zI-\mathsf{Circ}_{n,\beta})}{\det(I-\mathsf{Circ}_{n,\beta})}$.
There exists a coupling of $p_{n,\beta}(z),n\geq 1$, the secular function
$\zeta_{\beta}(z)={\mathcal{H}}_{\beta}(0,z)^{\dagger}\binom{1}{-q}$, and an
a.s. finite $C$ so that for all $z\in{\mathbb{C}}$ we have

$\displaystyle|p_{n,\beta}(e^{iz/n})e^{-iz/2}-\zeta_{\beta}(z)|\leq
C^{|z|^{2}+1}\left(e^{|z|\frac{\log^{3}n}{\sqrt{n}}}-1\right).$

The time-reversed and transformed version of the $\mathtt{Bess}_{\beta,a}$ and
$\mathtt{HP}_{\beta,\delta}$ operators were constructed in [24] in a similar
spirit.

######  Definition 36.

Let $B$ be a standard two-sided Brownian motion. Let
$\mathsf{y}_{u}=e^{-\frac{\beta}{4}(2a+1)u+B(2u)}$ and
$\hat{y}(t)=\mathsf{y}(u(t))$ for $t\in(0,1]$, where $u$ is defined in (66).
Set $\mathfrak{u}_{0}=\binom{1}{0},\mathfrak{u}_{1}=\binom{0}{-1}$ and define
$\tau_{\beta,a}=\tau_{\beta,a}^{\textup{Bess}}=\mathtt{Dir}(i\hat{y}(t),\mathfrak{u}_{0},\mathfrak{u}_{1})$.
We also define $\zeta_{\beta,a}=\zeta_{\tau_{\beta,,a}}$ and
${\mathcal{E}}_{\beta,a}={\mathcal{E}}_{\tau_{\beta,a}}$ as the secular and
structure function of the $\tau_{\beta,a}$ operator via Definition 7.

######  Proposition 37.

The operator $\tau_{\beta,a}$ satisfies Assumption 4, and we have

$\rho^{-1}J\tau_{\beta,a}J\rho\stackrel{{\scriptstyle
d}}{{=}}\mathtt{Bess}_{\beta,a},\qquad J=\left(\begin{array}[]{cc}0&-1\\\
1&0\\\ \end{array}\right).$

Let $H_{\beta,a}(t,z)$ be the solution to the canonical system (18) of the
$\tau_{\beta,a}$ operator and set
${\mathcal{H}}_{\beta,a}(u,z)=\operatorname{diag}(1,\mathsf{y}_{u})H_{\beta,a}(e^{\frac{\beta}{4}u},z)$
for $u\leq 0,z\in{\mathbb{C}}$, where $\mathsf{y}_{u}$ is defined as in
Definition 36. Then ${\mathcal{H}}_{\beta,a}$ is the unique strong solution to
the SDE

$d{\mathcal{H}}=\begin{pmatrix}0&0\\\
0&\sqrt{2}dB+(1-\frac{\beta}{4}(2a+1))du\end{pmatrix}{\mathcal{H}}-z\frac{\beta}{8}e^{\beta
u/4}\begin{pmatrix}0&-1\\\ 1&0\end{pmatrix}{\mathcal{H}}du,$ (69)

with boundary conditions
$\lim\limits_{u\to-\infty}\sup_{|z|<1}\left|{\mathcal{H}}(u,z)-\binom{1}{0}\right|=0$.
Since ${\mathcal{H}}_{\beta,a}(0,z)=H_{\beta,a}(1,z)$, we have
${\mathcal{E}}_{\beta,a}(z)={\mathcal{H}}_{\beta,a}(0,z)^{{\dagger}}\binom{1}{-i}$
and $\zeta_{\beta,a}={\mathcal{H}}_{\beta,a}(1,z)^{{\dagger}}\binom{1}{0}$.

[24] showed that the secular function $\zeta_{\beta,a}$ of the
$\mathtt{Bess}_{\beta,a}$ operator is the limit in distribution of the
normalized characteristic polynomial of the real orthogonal beta ensemble
under the edge scaling (3).

######  Definition 38.

Let $B_{1},B_{2}$ be independent two-sided standard Brownian motion, and for
$u\leq 0$ define

$\displaystyle\mathsf{y}_{u}=e^{B_{2}(u)-(\Re\delta+\frac{1}{2})u},\quad\mathsf{x}_{u}=-\int_{u}^{0}e^{B_{2}(s)-(\Re\delta+\frac{1}{2})s}dB_{1}-\Im\delta\int_{u}^{0}e^{B_{2}(s)-(\Re\delta+\frac{1}{2})s}ds,$
(70)

For $t\in(0,1]$ let
$\hat{z}(t)=(\hat{x}+i\hat{y})(t)=\mathsf{x}_{u}+i\mathsf{y}_{u}$ with
$u=u(t)$ defined in (66). Set $\mathfrak{u}_{0}=\binom{1}{0}$, and
$\mathfrak{u}_{1}=\binom{-q}{-1}$, where $q\sim\Theta(1,\delta)$ is
independent of $B_{1},B_{2}$. Define
$\tau_{\beta,\delta}=\tau_{\beta,\delta}^{\textup{HP}}=\mathtt{Dir}(\hat{z}(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$,
and denote by $\zeta_{\beta,\delta}=\zeta_{\tau_{\beta,\delta}}$ and
${\mathcal{E}}_{\beta,\delta}={\mathcal{E}}_{\tau_{\beta,\delta}}$ the secular
and structure function of $\tau_{\beta,\delta}$, respectively.222From the
context it will always be clear if $\zeta_{\beta,\cdot}$,
${\mathcal{E}}_{\beta,\cdot}$ refer to the objects related to the
$\mathtt{Bess}_{\beta,a}$ or the $\mathtt{HP}_{\beta,\delta}$ operators.

######  Proposition 39 ([24]).

Let $q$ and $\tau_{\beta,\delta}$ be defined as in Definition 38. Then the
$\tau_{\beta,\delta}$ operator satisfies Assumption 4, and the orthogonally
equivalent operator

$\rho^{-1}SQ\tau_{\beta,\delta}Q^{-1}S\rho,\qquad
Q=\frac{1}{\sqrt{1+q^{2}}}\left(\begin{array}[]{cc}q&1\\\ -1&q\\\
\end{array}\right),$

has the same distribution as $\mathtt{HP}_{\beta,\delta}$. Let
$H_{\beta,\delta}$ be the solution of the canonical system (18) of the
$\tau_{\beta,\delta}$ operator, and let $\mathsf{x}_{u},\mathsf{y}_{u}$ be
defined as in (70). For $u\leq 0,z\in{\mathbb{C}}$ set

${\mathcal{H}}_{\beta,\delta}(u,z)=\left(\begin{array}[]{cc}1&-\mathsf{x}_{u}\\\
0&\mathsf{y}_{u}\\\ \end{array}\right)H_{\beta,\delta}(e^{\beta u/4},z),$

then ${\mathcal{H}}_{\beta,\delta}$ is the unique solution of the SDE

$d\mathcal{H}=\begin{pmatrix}0&-dB_{1}\\\
0&dB_{2}\end{pmatrix}\mathcal{H}+\begin{pmatrix}0&-\Im\delta du\\\
0&-\Re\delta du\end{pmatrix}\mathcal{H}-z\frac{\beta}{8}e^{\beta
u/4}\left(\begin{array}[]{cc}0&-1\\\ 1&0\\\ \end{array}\right)\mathcal{H}du,$
(71)

with the boundary condition
$\lim\limits_{u\to-\infty}\sup_{|z|<1}\left|{\mathcal{H}}(u,z)-\binom{1}{0}\right|=0$,
and we have
${\mathcal{E}}_{\beta,\delta}(z)={\mathcal{H}}_{\beta,\delta}(0,z)^{{\dagger}}\binom{1}{-i}$.

[24] also showed that the secular function
$\zeta_{\beta,\delta}={\mathcal{H}}_{\beta,\delta}(0,z)^{{\dagger}}\binom{1}{-q}$
of the $\mathtt{HP}_{\beta,\delta}$ operator is the limit in distribution of
the normalized characteristic polynomials of the circular Jacobi beta ensemble
under the edge scaling (3).

##  5 Convergence of the truncated models

This section establishes general convergence results for the rank-one
truncations and multiplicative perturbations of the finite ensembles when the
generating paths satisfy certain path bounds. The statements in this section
hold in the deterministic setting.

Throughout the section, we assume $\mu_{n}$ is a probability measure on
$\partial{\mathbb{D}}$ supported on $n$ distinct points, with Verblunsky
coefficients $\alpha_{0},\dots,\alpha_{n-1}$. We denote by $\tau_{n}$ the
Dirac operator corresponding to $\mu_{n}$.

Recall the reversed (discrete) probability measures defined in Definition 15.
We first introduce the reversed Dirac operator and its pulled-back version,
which are closely connected to the truncated ensembles.

######  Definition 40.

Let $\widetilde{\mu}_{n}$ be the reversed version of $\mu_{n}$, i.e., the
probability measure corresponding to the reversed Verblunsky coefficients
$(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{n-1})$.
We define $\widetilde{\tau}_{n}$ as the Dirac operator corresponding to
$\widetilde{\mu}_{n}$ and call it the reversed version of $\tau_{n}$. We
denote by $\widetilde{b}_{k},0\leq k\leq n$ the path parameters of
$\widetilde{\tau}_{n}$ in ${\mathbb{D}}$.

Recall the affine transformation $\mathcal{A}_{\gamma,{\mathbb{D}}}$ defined
in (49).

######  Definition 41.

Set
$\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k}:={\mathcal{A}}_{\widetilde{b}_{n-1},{\mathbb{D}}}(\widetilde{b}_{k})$
for $0\leq k\leq n$ with the extension that
$\overset{\text{\tiny${\hookleftarrow}$}}{b}_{-1}={\mathcal{A}}_{\widetilde{b}_{n-1},{\mathbb{D}}}(\widetilde{b}_{-1})=1$.
We define
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}:={\mathcal{A}}_{\widetilde{b}_{n-1},{\mathbb{D}}}(\widetilde{\tau}_{n}):=\mathtt{Dir}(1,\overset{\text{\tiny${\hookleftarrow}$}}{b}_{0},\dots,\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n-1},\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n})$
as the Dirac operator with path parameters
$\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$ in
${\mathbb{D}}$. We call $\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$
the pulled-back version of $\widetilde{\tau}_{n}$.

To summarize, we are considering three sets of path parameters corresponding
to $\mu_{n}$, and each one has a Dirac-type operator associated to it.
$b_{k},0\leq k\leq n$ are the path parameters constructed from the Verblunsky
coefficients $\alpha_{k},0\leq k\leq n-1$ according to Definition 17. The
reversed path $\widetilde{b}_{k},0\leq k\leq n$ is constructed via the same
definition, but from the reversed Verblunsky coefficients (35). Finally, the
path $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$ can be
obtained from the reversed path by applying an affine transformation that maps
$\widetilde{b}_{n-1}$ to 0 in ${\mathbb{D}}$. See Table 1 for a summary of the
defined objects.

Note that by the definition of ${\mathcal{A}}_{\gamma,{\mathbb{D}}}$ and the
recursion (52), we have $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n-1}=0$
and
$\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n}=\widetilde{\gamma}_{n-1}$,
where $\widetilde{\gamma}_{n-1}$ is the last modified Verblunsky coefficient
corresponding to $\widetilde{\mu}_{n}$.

original objects |  | reversed objects |  | pulled-back objects
---|---|---|---|---
$\mu$ |  | $\widetilde{\mu}$ |  |
discrete prob. measure on $\partial{\mathbb{D}}$ |  | reversed measure |  |
$\updownarrow$ (31) | (35) |  $\uparrow$(31) |  |
$\alpha_{k},0\leq k\leq n-1$ | $\longleftrightarrow$ | $\widetilde{\alpha}_{k},0\leq k\leq n-1$ |  |
Verblunsky coefficients |  | reversed Verbl. coeffs |  |
$\updownarrow$ (40) |  |  |  |
$\gamma_{k},0\leq k\leq n-1$ |  | ${\left\downarrow\vbox to28.45274pt{}\right.\kern-1.2pt}$ Def. 17 |  |
modified Verblunsky coeffs |  |  |
$\updownarrow$ Def. 17 |  |  | Def. 41 |
$b_{k},0\leq k\leq n$ |  | $\widetilde{b}_{k},0\leq k\leq n$ | $\longrightarrow$ | $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$
$z_{k},0\leq k\leq n$ |  | $\widetilde{z}_{k},0\leq k\leq n$ |  | $\overset{\text{\tiny${\hookleftarrow}$}}{z}_{k},0\leq k\leq n$
path parameters |  | reversed path |  | pulled-back path
$\downarrow$ Def. 17 |  |  $\downarrow$Def. 17 |  |  $\downarrow$Def. 17, 41
$\tau=\mathtt{Dir}(b_{-1},\dots,b_{n})$ |  | $\widetilde{\tau}=\mathtt{Dir}(\widetilde{b}_{-1},\dots,\widetilde{b}_{n})$ |  | $\overset{\text{\tiny${\hookleftarrow}$}}{\tau}=\mathtt{Dir}(\overset{\text{\tiny${\hookleftarrow}$}}{b}_{-1},\dots,\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n})$
Dirac-type operator |  | reversed operator |  | pulled-back operator
Table 1: Table of objects associated to a probability measure $\mu$ supported
on $n$ points on $\partial{\mathbb{D}}$.

The next statement represents the orthogonal polynomial of degree $n-1$ of
$\widetilde{\mu}_{n}$ using the operator
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$.

######  Proposition 42.

Let $\widetilde{\mu}_{n}$ be the reversed version of $\mu_{n}$, and
$\widetilde{\tau}_{n},\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$ be
defined as in Definitions 40 and 41. For $0\leq k\leq n-1$, let
$\widetilde{\Phi}_{k}(\cdot)$ be the monic orthogonal polynomials of degree
$k$ associated to $\widetilde{\mu}_{n}$, and set
$\widetilde{\Psi}_{k}(\cdot)=\widetilde{\Phi}_{k}(\cdot)/\widetilde{\Phi}_{k}(1)$.
Then we have

$\widetilde{\Psi}_{n-1}(e^{iz/n})=e^{iz(n-1)/(2n)}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}\big{(}\tfrac{n-1}{n},z\big{)}^{{\dagger}}\binom{1}{-i},$

where $\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(t,z)$ is the solution
of the canonical system (18) corresponding to the operator
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$.

###### Proof.

Let $\widetilde{b}_{k},0\leq k\leq n$ be the path parameters of
$\widetilde{\tau}_{n}$ and set
$\widetilde{z}_{k}=\mathcal{U}^{-1}(\widetilde{b}_{k}),0\leq k\leq n$ to be
the corresponding path parameters in ${\mathbb{H}}$. Here ${\mathcal{U}}$ is
the Cayley transform defined in (39). Consider the linear fractional
transformation ${\mathcal{A}}_{\widetilde{z}_{n-1},{\mathbb{H}}}$ (the upper
half plane representation of
${\mathcal{A}}_{\widetilde{b}_{n-1},{\mathbb{D}}}$) such that

${\mathcal{A}}_{\widetilde{z}_{n-1},{\mathbb{H}}}(w)=\mathcal{U}^{-1}\circ{\mathcal{A}}_{\widetilde{b}_{n-1},{\mathbb{D}}}\circ\mathcal{U}(w)=\frac{w-\Re\widetilde{z}_{n-1}}{\Im\widetilde{z}_{n-1}},\qquad
w\in{\mathbb{H}}.$

Define
$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{k}=\overset{\text{\tiny${\hookleftarrow}$}}{x}_{k}+i\overset{\text{\tiny${\hookleftarrow}$}}{y}_{k}={\mathcal{A}}_{\widetilde{z}_{n-1},{\mathbb{H}}}(\widetilde{z}_{k})$
for $0\leq k\leq n$. Then by (48) we have
$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{k}={\mathcal{U}}^{-1}(\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k}),0\leq
k\leq n$ with the extension that
$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{-1}={\mathcal{U}}^{-1}(\widetilde{b}_{-1})=\infty$.

Introduce the temporary notation

$\widetilde{X}_{k}=A_{\widetilde{z}_{k},{\mathbb{H}}}=\left(\begin{array}[]{cc}1&-\widetilde{x}_{k}\\\
0&\widetilde{y}_{k}\\\
\end{array}\right),\quad\overset{\text{\tiny${\hookleftarrow}$}}{X}_{k}=A_{\overset{\text{\tiny${\hookleftarrow}$}}{z}_{k},{\mathbb{H}}}=\left(\begin{array}[]{cc}1&-\overset{\text{\tiny${\hookleftarrow}$}}{x}_{k}\\\
0&\overset{\text{\tiny${\hookleftarrow}$}}{y}_{k}\\\ \end{array}\right),\qquad
0\leq k\leq n,$

then we have

$\overset{\text{\tiny${\hookleftarrow}$}}{X}_{k}=\widetilde{X}_{k}(\widetilde{X}_{n-1})^{-1},\quad\quad
0\leq k\leq n.$ (72)

For a matrix of the form

$X=\left(\begin{array}[]{cc}1&-x\\\ 0&y\\\ \end{array}\right)$

with $y>0$ we have the identity

$y^{-1}X^{{\dagger}}J=JX^{-1}.$

Using this together with (10) and (11) we obtain

$\displaystyle\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}=\widetilde{X}_{n-1}\widetilde{\tau}_{n}(\widetilde{X}_{n-1})^{-1}.$
(73)

Let $\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(t,z)$ and
$\widetilde{H}_{n}(t,z)$ be the solutions to the canonical systems (18) of
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$ and
$\widetilde{\tau}_{n}$ respectively, then (73) leads to

$\displaystyle\widetilde{H}_{n}(t,z)=(\widetilde{X}_{n-1})^{-1}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(t,z).$
(74)

Let
$\widetilde{\Psi}_{k}^{*}(z)=z^{k}\overline{\widetilde{\Psi}_{k}(1/\bar{z})}$
be the reversed polynomial of $\widetilde{\Psi}_{k}(z)$. Using (39), (54),
(72), and (74) we get

$\displaystyle\binom{\widetilde{\Psi}_{k}(e^{iz/n})}{\widetilde{\Psi}_{k}^{*}(e^{iz/n})}$
$\displaystyle=e^{izk/(2n)}U\widetilde{X}_{k}\widetilde{H}_{n}(k/n,z)=e^{izk/(2n)}U\overset{\text{\tiny${\hookleftarrow}$}}{X}_{k}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(k/n,z).$

Since
$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{n-1}={\mathcal{A}}_{\widetilde{z}_{n-1},{\mathbb{H}}}(\widetilde{z}_{n-1})=i$,
and $\overset{\text{\tiny${\hookleftarrow}$}}{X}_{n-1}=I$, we have

$\widetilde{\Psi}_{n-1}(e^{iz/n})=e^{iz(n-1)/(2n)}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}((n-1)/n,z)^{{\dagger}}\binom{1}{-i},$

finishing the proof. ∎

Suppose now that the probability measure $\mu_{n}$ is the spectral measure of
an $n\times n$ unitary or orthogonal matrix $U$ with respect to
$\mathbf{e}_{1}$. By Propositions 13 and 16, the eigenvalues of the truncated
matrix $U^{\textstyle{\ulcorner}}$ are exactly the zeros of the normalized
orthogonal polynomial $\widetilde{\Psi}_{n-1}(\cdot)$, hence they can be
expressed from the vector-valued function
$\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}$.

Next we turn to the discussion of the scaling limit of the eigenvalues of the
truncated matrices. If we can prove uniform on compacts convergence of the
normalized orthogonal polynomials, then this leads to the convergence of the
eigenvalues of the truncated models. By Proposition 42 it is sufficient to
show the convergence the convergence of
$\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}$. Using Proposition 8, this
can be done if we have sufficient control of the integral traces, resolvent
norms, and the proper integrals of the kernels of
$\mathtt{r\,}\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$. We summarize
this idea in the following statement.

######  Proposition 43.

Suppose that $\mu_{n},n\geq 1$ is a sequence of probability measures on
$\partial{\mathbb{D}}$, with $\mu_{n}$ supported on $n$ distinct points.
Consider the setup of Proposition 42 and denote by $z_{n}(\cdot)$ the
generating path of $\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$ for
$n\geq 1$. Suppose that there exists a Dirac operator
$\tau_{\infty}=\mathtt{Dir}(z_{\infty}(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$
with $\mathfrak{u}_{0}=\binom{1}{0}$ and ${\mathcal{I}}=(0,1]$ so that

$\|\mathtt{r\,}\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}-\mathtt{r\,}\tau_{\infty}\|_{\textup{HS}}\to
0,\qquad\mathfrak{t}_{\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}}-\mathfrak{t}_{\tau_{\infty}}\to
0,\qquad\int_{0}^{1}|\mathfrak{a}_{0,n}(s)-\mathfrak{a}_{0,\infty}(s)|^{2}ds\to
0,$ (75)

where $\mathfrak{a}_{0,n}(\cdot)=\binom{\Im
z_{n}(\cdot)^{-\frac{1}{2}}}{0},n\in{\mathbb{Z}}_{+}\cup\\{\infty\\}$ is
defined according to (17). Let $\widetilde{\Psi}_{n-1,n}$ be the normalized
orthogonal polynomial of degree $n-1$ of $\widetilde{\mu}_{n}$, and let
${\mathcal{E}}(z):=H(1,z)^{{\dagger}}\binom{1}{-i}$ be the structure function
of $\tau_{\infty}$. Then we have

$|e^{-iz/2}\widetilde{\Psi}_{n-1,n}(e^{iz/n})-{\mathcal{E}}(z)|\to
0\quad\text{uniformly on compacts in ${\mathbb{C}}$ as $n\to\infty$.}$ (76)

Note that (76) implies that the zeros of $\widetilde{\Psi}_{n-1,n}$ converge
to the zeros of ${\mathcal{E}}$ under the edge scaling (3).

###### Proof of Proposition 43.

Let $\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n},n\geq 1$ be the solution
of the canonical system (18) of
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}$. By Proposition 42 and
the triangle inequality, we have

$\displaystyle|\widetilde{\Psi}_{n-1,n}(e^{iz/n})e^{-iz/2}-{\mathcal{E}}(z)|$
$\displaystyle\leq|e^{-iz/(2n)}|\left|\Big{(}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(\tfrac{n-1}{n},z)^{{\dagger}}-H(\tfrac{n-1}{n},z)^{{\dagger}}\Big{)}\cdot\binom{1}{-i}\right|$
(77)
$\displaystyle+|e^{-iz/(2n)}|\left|\Big{(}H(\tfrac{n-1}{n},z)^{{\dagger}}-H(1,z)^{{\dagger}}\Big{)}\cdot\binom{1}{-i}\right|$
$\displaystyle+\left|e^{-iz/(2n)}-1\right|\Big{|}H(1,z)^{{\dagger}}\cdot\binom{1}{-i}\Big{|}.$

For $z$ in a compact set of ${\mathbb{C}}$ and large enough $n$, the sum of
the second and the third term of (77) is upper bounded by an error term of the
order $O(n^{-1})$. Hence it remains to provide an upper bound for the
difference
$|\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n}(\tfrac{n-1}{n},z)-H(\tfrac{n-1}{n},z)|$.
By Proposition 8, it suffices to control the terms

$|\mathfrak{t}_{\tau_{\infty}}-\mathfrak{t}_{\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}}|,\quad\quad\|\mathtt{r\,}\tau_{\infty}-\mathtt{r\,}\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n}\|_{\textup{HS}},\quad\quad\int_{0}^{1}|\mathfrak{a}_{0,\infty}(s)-\mathfrak{a}_{0,n}(s)|^{2}ds.$

Applying the assumptions (75) finishes the proof. ∎

Next, we turn to the discussion of the rank-one multiplicative perturbation of
the matrix models. Suppose $U$ is an $n\times n$ unitary or orthogonal matrix
with CMV matrix form ${\mathcal{C}}(\alpha_{0},\dots,\alpha_{n-1})$. Let
$r\in[0,1]$ and $U^{[r]}=U\cdot\operatorname{diag}(r,1,1,\dots,1)$ be its
rank-one multiplicative perturbation. Recall from Proposition 16 that the
perturbed matrix $U^{[r]}$ has the same eigenvalues as the CMV matrix

${\mathcal{C}}_{n}^{[r]}:={\mathcal{C}}(\widetilde{\alpha}_{0},\dots,\widetilde{\alpha}_{n-2},r\widetilde{\alpha}_{n-1}).$
(78)

Note that this matrix is a function of the spectral measure of $U$. In
particular, for any probability measure $\mu_{n}$ on $\partial{\mathbb{D}}$
(supported on $n$ points) we can define the matrix ${\mathcal{C}}_{n}^{[r]}$
from its Verblunsky coefficients.

Suppose now that we have a sequence of probability measures $\mu_{n},n\geq 1$,
just as in Proposition 43. Let $\widetilde{\gamma}_{n-1}$ be the last modified
Verblunsky coefficient corresponding to the Verblunsky coefficients
$\widetilde{\alpha}_{0},\dots,\widetilde{\alpha}_{n-1}$ of
$\widetilde{\mu}_{n}$. (Note the abuse of notation: we should write
$\widetilde{\alpha}_{k,n},0\leq k\leq n-1$ here, but we drop the extra $n$
from the notation.) The next result shows that if $\widetilde{\gamma}_{n-1}$
converges and the assumptions of Proposition 43 hold, then the eigenvalues of
${\mathcal{C}}_{n}^{[r]}$ converge as well.

######  Proposition 44.

Consider the same setup as in Proposition 43. Fix $r\in[0,1]$, let
$\Psi_{n}^{[r]}(z)=\prod_{i=1}^{n}\frac{z-\lambda_{i}}{1-\lambda_{i}}$ be the
normalized characteristic polynomial of ${\mathcal{C}}_{n}^{[r]}$. Assume that
the convergence (76) holds, and assume further that
$\lim_{n\to\infty}\widetilde{\gamma}_{n-1}=\gamma\neq 1$. Then we have

$|\Psi_{n}^{[r]}(e^{iz/n})e^{-iz/2}-{\mathcal{E}}^{[r]}(z)|\to
0,\qquad{\mathcal{E}}^{[r]}(z)=H(1,z)^{{\dagger}}\binom{1}{-i\frac{1-r\gamma}{1+r\gamma}}$
(79)

uniformly on compacts in ${\mathbb{C}}$ as $n\to\infty$.

Note that (79) implies the convergence of the eigenvalues of $\Psi_{n}^{[r]}$
to the zeros of ${\mathcal{E}}^{[r]}$ under the edge scaling (3).

###### Proof of Proposition 44.

By the modified Szegő recursion (46), we have

$\Psi_{n}^{[r]}(z)=\tfrac{1}{1-r\widetilde{\gamma}_{n-1}}z\Psi_{n-1}^{[r]}(z)-\tfrac{r\widetilde{\gamma}_{n-1}}{1-r\widetilde{\gamma}_{n-1}}\Psi_{n-1}^{[r],*}(z),$
(80)

where $\Psi_{n-1}^{[r],*}(\cdot)$ is the reversed polynomial of
$\Psi_{n-1}^{[r]}(\cdot)$. Note that the polynomials $\Psi_{n-1}^{[r]}$ and
$\Psi_{n-1}^{[r],*}$ do not depend on $\widetilde{\gamma}_{n-1}$.

Introduce ${\mathcal{E}}^{*}(z)=\overline{{\mathcal{E}}(\bar{z})}$, by the
definition of the reversed polynomials, (54) and (76), we have

$\left|\Psi_{n-1}^{[r],*}(e^{iz/n})e^{-iz/2}-{\mathcal{E}}^{*}(z)\right|\to
0\quad\text{uniformly on compacts as $n\to\infty$}.$ (81)

Together with (80) and the convergence of $\widetilde{\gamma}_{n-1}\to\gamma$
as $n\to\infty$, we get

$\displaystyle\lim_{n\to\infty}\Psi_{n}^{[r]}(e^{iz/n})e^{-iz/2}=\tfrac{1}{1-r\gamma}{\mathcal{E}}(z)-\tfrac{r\gamma}{1-r\gamma}{\mathcal{E}}^{*}(z)=H(1,z)^{{\dagger}}\binom{1}{-i\frac{1-r\gamma}{1+r\gamma}},$

uniformly on compacts in $z$. ∎

##  6 Edge limits of the truncated circular and real orthogonal beta ensembles

Our goal is to prove Theorems 2 and 3, the edge scaling limits of the rank-one
truncation and multiplicative perturbation of the circular beta ensemble using
the framework developed in Section 5. We will also prove the analogue results
for the real orthogonal beta ensemble.

In both cases we will check that the random measures associated to the
respective beta ensembles (introduced in Section 4.1) satisfy the conditions
in Propositions 43 and 44.

###  6.1 The truncated circular beta ensemble

Recall the definition of the random probability measure
$\mu_{n,\beta}^{\textup{KN}}$ introduced in Section 4.1 and the corresponding
CMV matrix $\mathsf{Circ}_{n,\beta}$. Proposition 22 gives the distribution of
the Verblunsky coefficients $\alpha_{k},0\leq k\leq n-1$ corresponding to
$\mu_{n,\beta}^{\textup{KN}}$. The truncated circular beta ensemble is the
joint distribution of the eigenvalues of the truncated CMV matrix
$\mathsf{Circ}^{\textstyle{\ulcorner}}_{n,\beta}$. The rank-one multiplicative
perturbation of the circular beta ensemble (corresponding to a parameter
$r\in[0,1]$) is defined as the distribution of the eigenvalues of
$\mathsf{Circ}_{n,\beta}^{[r]}$.

Note that by Proposition 22 the Verblunsky coefficients $\alpha_{k},0\leq
k\leq n-1$ are independent and rotationally invariant. Hence from (35) we get
the following distributional identity:

$(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{n-2},\widetilde{\alpha}_{n-1})\stackrel{{\scriptstyle
d}}{{=}}(\alpha_{n-2},\alpha_{n-3},\dots,\alpha_{0},\alpha_{n-1}).$ (82)

Using this together with Proposition 16, we recover the following result of
Killip and Kozhan [19].

######  Proposition 45 ([19]).

For fixed $n$ and $\beta>0$, let $\alpha_{k},0\leq k\leq n-1$ be distributed
according to Proposition 22. Then the joint eigenvalue distribution of the CMV
matrix ${\mathcal{C}}(\alpha_{n-2},\alpha_{n-3},\dots,\alpha_{0})$ is the same
as that of $\mathsf{Circ}^{\textstyle{\ulcorner}}_{n,\beta}$, which is given
by the truncated circular beta ensemble. For fixed $r\in[0,1]$, the matrix
${\mathcal{C}}(\alpha_{n-2},\alpha_{n-3},\dots,\alpha_{0},r\alpha_{n-1})$ has
the same distribution as $\mathsf{Circ}_{n,\beta}^{[r]}$, in particular its
joint eigenvalue distribution is given by the rank-one multiplicative
perturbation of the circular beta ensemble.

Note that [19] also proved that the joint density of the truncated circular
beta ensemble is given by (5), and described explicitly the joint eigenvalue
distribution of the perturbed matrix $\mathsf{Circ}_{n,\beta}^{[r]}$ (see
Proposition 7.2 of [19]).

Recall the Dirac operator $\mathtt{Circ}_{n,\beta}$ in Definition 23. We
denote by $\widetilde{\tau}_{n,\beta}$ the reversed version of
$\mathtt{Circ}_{n,\beta}$, and by
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}$ the pulled-back
version of $\widetilde{\tau}_{n,\beta}$ (see Definitions 40 and 41). By
studying the generating paths of the operators
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}$ and
$\mathtt{Circ}_{n,\beta}$, we claim that these operators are orthogonally
equivalent. Indeed, this observation follows from Proposition 52 of [39] using
the rotational invariance of the models and a conditioning argument. We will
provide a different proof that holds for the circular Jacobi beta ensemble
($\delta=0$ corresponding to the circular beta ensemble). To avoid repetition,
we will state the result without proof here.

######  Proposition 46.

Recall the definitions of the operators $\rho$ and $S$ from Lemma 10. Denote
by $\overset{\text{\tiny${\hookleftarrow}$}}{b}_{k},0\leq k\leq n$ the path
parameters of $\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}$, and
let

$Q=\frac{1}{\sqrt{1+q^{2}}}\begin{pmatrix}q&1\\\ -1&q\end{pmatrix},\qquad
q={\mathcal{U}}^{-1}(\overset{\text{\tiny${\hookleftarrow}$}}{b}_{n}).$

Then the operator
$\rho^{-1}(SQ)\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}(SQ)^{-1}\rho$
has the same distribution as $\mathtt{Circ}_{n,\beta}$.

Recall the definition of the $\mathtt{Sine}_{\beta}$ operator from Definition
30, and its reversed and transformed version $\tau_{\beta}$ defined in
Definition 33. Proposition 46 shows that one can obtain the operators
$\tau_{n,\beta}$ and $\tau_{\beta}$ from $\mathtt{Circ}_{n,\beta},n\geq 1$ and
$\mathtt{Sine}_{\beta}$ under the same orthogonal transformations. The final
ingredient of proving Theorems 2 and 3 is the following strong operator level
convergence of $\mathtt{Circ}_{n,\beta}$ to the $\mathtt{Sine}_{\beta}$
operator. Let
$d_{\mathbb{H}}(z_{1},z_{2})=\operatorname{arccosh}(1+\frac{|z_{1}-z_{2}|^{2}}{2\Im
z_{1}\Im z_{2}})$ denote the hyperbolic distance in ${\mathbb{H}}$.

######  Proposition 47 ([37, 38]).

There exists an explicit coupling of the operators
$\mathtt{Circ}_{n,\beta}=\mathtt{Dir}(z_{n}(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1}^{(n)}),n\geq
1$ and
$\mathtt{Sine}_{\beta}=\mathtt{Dir}(z(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$
such that $\mathfrak{u}_{1}^{(n)}=\mathfrak{u}_{1}$, and for large enough $n$,

$\displaystyle d_{{\mathbb{H}}}(z_{n}(t),z(t))$
$\displaystyle\leq\frac{\log^{3-1/8}n}{\sqrt{(1-t)n}},\quad\quad 0\leq t\leq
t_{n}:=1-\frac{\log^{6}n}{n},$ (83) $\displaystyle
d_{{\mathbb{H}}}(z_{n}(t_{n}),z(t))$ $\displaystyle\leq(\log\log
n)^{4},\quad\quad t_{n}\leq t<1.$

Under this coupling, we have

$\|\mathtt{r\,}\mathtt{Circ}_{n,\beta}-\mathtt{r\,}\mathtt{Sine}_{\beta}\|_{\text{HS}}^{2}\leq\frac{\log^{6}n}{n},\quad\quad|\mathfrak{t}_{\mathtt{Circ}_{n,\beta}}-\mathfrak{t}_{\mathtt{Sine}_{\beta}}|\leq\frac{\log^{3}n}{\sqrt{n}}.$
(84)

Here the first inequality of (84) was proved in [37] and the second one
follows from the estimates used in the proof of Theorem 49 of [38] (in
particular equations (107)-(109)). ‘Large enough $n$’ means that there is a
finite random variable $N_{0}$ so that the statements hold for $n\geq N_{0}$.

Proposition 47 provides us with the necessary ingredients so that we can apply
Proposition 43 to prove the convergence of the normalized characteristic
polynomials of the truncated circular beta ensemble.

######  Proposition 48.

Let $\lambda_{i},1\leq i\leq n-1$ be the size $n-1$ truncated circular beta
ensemble and set
$p_{n-1,\beta}(z)=\prod_{i=1}^{n-1}\frac{z-\lambda_{i}}{1-\lambda_{i}}$ be the
normalized characteristic polynomial. Let ${\mathcal{E}}_{\beta}$ be the
structure function of $\tau_{\beta}$ defined via (20). Then there is a
coupling of $p_{n-1,\beta},n\geq 2$ and ${\mathcal{E}}_{\beta}$ such that

$|p_{n-1,\beta}(e^{iz/n})e^{-iz/2}-{\mathcal{E}}_{\beta}(z)|\leq\left(e^{|z|\frac{\log^{3}n}{\sqrt{n}}}-1\right)C^{1+|z|^{2}}$
(85)

for all $z\in{\mathbb{C}}$ and $n\geq 1$, where $C$ is an a.s. finite
constant.

###### Proof.

Let $\widetilde{\mu}_{n}$ be the reversed version of the Killip-Nenciu
probablity measures $\mu_{n}:=\mu_{n,\beta}^{\textup{KN}}$, and let
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}$ be the pulled-back
operator. By Propositions 13 and 45, the normalized characteristic polynomial
$p_{n-1,\beta}$ has the same distribution as $\widetilde{\Psi}_{n-1,\beta}$,
the monic orthogonal polynomial of degree $n-1$ associated to
$\widetilde{\mu}_{n}$. Hence it is enough to provide a coupling of $\mu_{n}$
and ${\mathcal{E}}_{\beta}$ where (85) holds with
$\widetilde{\Psi}_{n-1,\beta}$ in place of $p_{n-1,\beta}$.

By Proposition 42, we have

$\widetilde{\Psi}_{n-1,\beta}(e^{iz/n})=e^{iz(n-1)/(2n)}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n,\beta}((n-1)/n,z)^{{\dagger}}\binom{1}{-i},$

where $\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n,\beta}$ solves the ODE
(18) of $\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}$. Recall
that ${\mathcal{E}}_{\beta}(z)=H_{\beta}(1,z)^{{\dagger}}\binom{1}{-i}$.

Consider the coupling of Proposition 47. Using the triangle inequality as in
(77) within the proof of Proposition 43, we have

$\displaystyle|\widetilde{\Psi}_{n-1,\beta}(e^{iz/n})e^{-iz/2}-{\mathcal{E}}_{\beta}(z)|$
$\displaystyle\leq|e^{-iz/(2n)}|\left|\Big{(}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n,\beta}(\tfrac{n-1}{n},z)^{{\dagger}}-H_{\beta}(\tfrac{n-1}{n},z)^{{\dagger}}\Big{)}\cdot\binom{1}{-i}\right|$
(86)
$\displaystyle+|e^{-iz/(2n)}|\left|\Big{(}H_{\beta}(\tfrac{n-1}{n},z)^{{\dagger}}-H_{\beta}(1,z)^{{\dagger}}\Big{)}\cdot\binom{1}{-i}\right|$
$\displaystyle+\left|e^{-iz/(2n)}-1\right|\Big{|}H_{\beta}(1,z)^{{\dagger}}\cdot\binom{1}{-i}\Big{|}.$

Note that for any compact subset of $(0,1]$ the operator norm of the weight
function $R(s)$ of $\tau_{\beta}$ is bounded by a finite random constant.
Hence by the standard theory of ordinary differential equations, for $z$ in a
compact set of ${\mathbb{C}}$, the second term on the right hand side of (86)
can be upper bounded by $Cn^{-1}$ where $C$ is an a.s. finite constant. The
third term on the right side of (86) can also be bounded similarly, since
$z\mapsto H_{\beta}(1,z)^{{\dagger}}\cdot\binom{1}{-i}$ is a random entire
function.

It remains to estimate the first term on the right hand side of (86). By
Propositions 34 and 46, the operators
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta},n\geq 1$ and
$\tau_{\beta}$ can be obtained from $\mathtt{Circ}_{n,\beta},n\geq 1$ and
$\mathtt{Sine}_{\beta}$ under the same orthogonal transformations. Hence in
the coupling of Proposition 47, for large enough $n$ we have

$\displaystyle\|\mathtt{r\,}\tau_{\beta}-\mathtt{r\,}\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}\|\leq\frac{\log^{3}n}{\sqrt{n}},\quad\quad|\mathfrak{t}_{\tau_{\beta}}-\mathfrak{t}_{\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{n,\beta}}|\leq\frac{\log^{3}n}{\sqrt{n}},$
(87)

It has also been shown in the proof of Proposition 3 of [39] (see equations
(69)-(72) therein) that

$\displaystyle\int_{0}^{1}|\mathfrak{a}_{0,n}(s)-\mathfrak{a}_{0}(s)|^{2}ds\leq\frac{\log^{6}n}{n}.$
(88)

The estimates (87) and (88) allow us to use Proposition 8 to bound
$\|\overset{\text{\tiny${\hookleftarrow}$}}{H}_{n,\beta}(\tfrac{n-1}{n},z)-H_{\beta}(\tfrac{n-1}{n},z)\|$
for $n$ large enough. From this it follows that (86) can be bounded from above
according to (85), which proves the proposition. ∎

We are now ready to prove Theorems 2 and 3.

###### Proof of Theorem 2.

First note that by (68) we have
${\mathcal{H}}_{\beta}(0,\cdot)=H_{\beta}(1,\cdot)$, hence
${\mathcal{E}}_{\beta}(\cdot)={\mathcal{H}}_{\beta}(0,\cdot)^{{\dagger}}\binom{1}{-q}$.
Then by Proposition 48, we obtain the convergence of the normalized
characteristic polynomials of the truncated circular beta ensemble to the
random analytic function ${\mathcal{E}}_{\beta}$ with an explicit error bound.
Under the edge scaling (3), the size $n$ truncated circular beta ensemble has
the same distribution as the zeros of the function $p_{n,\beta}(e^{iz/n})$
defined in Proposition 48. The weak convergence of the truncated circular beta
ensemble to
$\mathcal{X}_{\beta}:=\\{z\in{\mathbb{H}}:{\mathcal{E}}_{\beta}(z)=0\\}$
follows directly from Proposition 48 and Hurwitz’s theorem. ∎

###### Proof of Theorem 3.

Consider the coupling of $\tau_{n,\beta}$ and $\tau_{\beta}$ described in
Proposition 48 again, the right boundary condition of these operators are
coupled together. By Propositions 44 and 48 we get the uniform-on-compacts
convergence of the normalized characteristic polynomials of
$\mathsf{Circ}_{n,\beta}^{[r]}$ to ${\mathcal{E}}_{r,\beta}$ with a similar
error bound. This in turn gives the weak convergence of the eigenvalues of
$\mathsf{Circ}_{n,\beta}^{[r]}$ under the edge scaling and completes the
proof. ∎

###  6.2 Edge limits of the truncated real orthogonal beta ensemble

This section discusses the edge limits of the rank-one truncation and
multiplicative perturbation of the real orthogonal beta ensemble. We will
follow the same approach as in Section 6.1.

Recall the size $2n$ real orthogonal ensemble with joint density (58) and the
random probability measure $\mu_{2n,\beta,a,b}^{\textup{RO}}$ in (59).
Proposition 25 describes the distribution of the corresponding Verblunsky
coefficients $\alpha_{k},0\leq k\leq 2n-1$. Since $\alpha_{k}$’s are all real
and $\alpha_{2n-1}=-1$, from (35) we get that the reversed Verblunsky
coefficients $\widetilde{\alpha}_{k},0\leq k\leq 2n-1$ satisfy

$(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\cdots,\widetilde{\alpha}_{2n-2},\widetilde{\alpha}_{2n-1})=(\alpha_{2n-2},\alpha_{2n-3},\cdots,\alpha_{0},\alpha_{2n-1}).$
(89)

The rank-one truncation and multiplicative perturbation of the real orthogonal
beta ensemble are defined as the joint eigenvalue distributions of the
truncated CMV matrix $\mathsf{RO}^{\textstyle{\ulcorner}}_{2n,\beta,a,b}$ and
the perturbed CMV matrix $\mathsf{RO}_{2n,\beta,a,b}^{[r]}$ (indexed by
$r\in[0,1])$, respectively. Using (89) with Proposition 16 yields the
following result of Killip and Kozhan [19].

######  Proposition 49 ([19]).

For given $\beta>0$, $a,b>-1$ and $n\geq 1$, let $\alpha_{k},0\leq k\leq 2n-1$
be distributed according to Proposition 25. Then the CMV matrix
${\mathcal{C}}(\alpha_{2n-2},\alpha_{2n-3}\dots,\alpha_{0})$ has the same
joint eigenvalue distribution as
$\mathsf{RO}^{\textstyle{\ulcorner}}_{2n,\beta,a,b}$. For fixed $r\in[0,1]$,
the CMV matrix ${\mathcal{C}}(\alpha_{2n-2},\dots,\alpha_{0},-r)$ has the same
joint eigenvalue distribution as $\mathsf{RO}_{2n,\beta,a,b}^{[r]}$.

Note that the joint distributions of the rank-one truncation and
multiplicative perturbation of the real orthogonal beta ensemble were
described explicitly in Theorem 6.4 and Proposition 7.2 (b) of [19].

Recall the Dirac operator $\mathtt{RO}_{2n,\beta,a,b}$ in Definition 26.
Denote by $\widetilde{\tau}_{2n,\beta,a,b}$ the reversed version of
$\mathtt{RO}_{2n,\beta,a,b}$, and by
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}$ the pulled-
back version of $\widetilde{\tau}_{2n,\beta,a,b}$. Recall also the limiting
$\mathtt{Bess}_{\beta,a}$ operator in Definition 31 and its reversed version
$\tau_{\beta,a}$ in Definition 36. By Proposition 37 we get that
$\rho^{-1}J\tau_{\beta,a}J\rho\stackrel{{\scriptstyle
d}}{{=}}\mathtt{Bess}_{\beta,a}$. The next result shows that under the same
transformations, the operators
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}$ and
$\mathtt{RO}_{2n,\beta,a,b}$ are orthogonally equivalent.

######  Proposition 50.

$\rho^{-1}J\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}J\rho\stackrel{{\scriptstyle
d}}{{=}}\mathtt{RO}_{2n,\beta,a,b}.$

###### Proof.

Let $\alpha_{k},0\leq k\leq 2n-1$ be distributed as in Proposition 25, and let
$\gamma_{k},0\leq k\leq 2n-1$ be the corresponding modified Verblunsky
coefficients. Since $\alpha_{k}$’s are all real, from (40) we have
$\gamma_{k}=\alpha_{k}$ for all $0\leq k\leq 2n-1$.

Let $z_{k},0\leq k\leq 2n$ and $\widetilde{z}_{k},0\leq k\leq 2n$ be the path
parameters of $\mathtt{RO}_{2n,\beta,a,b}$ and
$\widetilde{\tau}_{2n,\beta,a,b}$ in ${\mathbb{H}}$, respectively. Using (51)
and (89) we get for

$z_{k}=i\prod_{j=0}^{k-1}\frac{1+\gamma_{j}}{1-\gamma_{j}},\qquad\widetilde{z}_{k}=i\prod_{j=0}^{k-1}\frac{1+\gamma_{2n-2-j}}{1-\gamma_{2n-2-j}},\qquad
0\leq k\leq 2n.$

Recall the affine transformation ${\mathcal{A}}_{z,{\mathbb{H}}}$ in (47) and
the corresponding transformation on the uni-disk model
${\mathcal{A}}_{\gamma,{\mathbb{D}}}$ in (49). By Definition 41 and (48), the
path parameters of
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}$ in
${\mathbb{H}}$, denoted by
$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{k},0\leq k\leq 2n$, are obtained
from $\widetilde{z}_{k},0\leq k\leq 2n$ via the affine transformation
${\mathcal{A}}_{\widetilde{z}_{2n-1},{\mathbb{H}}}$. More precisely, we have

$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{k}={\mathcal{P}}\left(\begin{array}[]{cc}1&0\\\
0&\Im\widetilde{z}_{2n-1}\\\
\end{array}\right)\binom{\widetilde{z}_{k}}{1}=i\prod_{j=0}^{2n-2-k}\frac{1-\gamma_{j}}{1+\gamma_{j}},\qquad
0\leq k\leq 2n-1,$

with
$\overset{\text{\tiny${\hookleftarrow}$}}{z}_{2n}={\mathcal{A}}_{\widetilde{z}_{2n-1},{\mathbb{H}}}(\widetilde{z}_{2n})=0$.

Note that conjugating with the permutation matrix $J$ maps $z\mapsto-1/z$ for
$z\in i{\mathbb{R}}$. Together with the time-reversal, we see that the path
parameters of the operator $\rho
J\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}J\rho$ are the
same as the path parameters of $\mathtt{RO}_{2n,\beta,a,b}$. By Lemmas 10 and
11, the left and right boundary points of
$\rho^{-1}J\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}J\rho$
are given by $1$ and $-1$, which corresponds to the vectors
$\mathfrak{u}_{0}=\binom{1}{0}$ and $\mathfrak{u}_{1}=\binom{0}{-1}$ as
desired. This completes the proof. ∎

In the rest of the section, we aim to prove the convergence of the normalized
characteristic polynomials of the truncated real orthogonal beta ensemble
using Proposition 43. The main ingredient will be the operator level
convergence of $\mathtt{RO}_{2n,\beta,a,b}$ to its limit
$\mathtt{Bess}_{\beta,a}$ proved in [24].

######  Proposition 51 ([24]).

Let $\mathfrak{u}_{0}=\binom{1}{0},\mathfrak{u}_{1}=\binom{0}{-1}$. There
exists a coupling of the operators
$\mathtt{RO}_{2n,\beta,a,b}=\mathtt{Dir}(iy_{n}(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1}),n\geq
1$ and
$\mathtt{Bess}_{\beta,a}=\mathtt{Dir}(iy(\cdot),\mathfrak{u}_{0},\mathfrak{u}_{1})$
such that as $n\to\infty$ we have almost surely $|y_{n}-y|\to 0$ point-wise on
$[0,1)$, and

$\|\mathtt{r\,}\mathtt{RO}_{2n,\beta,a,b}-\mathtt{r\,}\mathtt{Bess}_{\beta,a}\|\to
0.$

Moreover, for $\varepsilon>0$ small there exists a sequence of tight random
variables $\kappa_{n}$ and an a.s. finite random variable $\kappa>0$ such that
for $0\leq t<1$

$\kappa_{n}^{-1}(1-\lfloor nt\rfloor/n)^{2a+1+\varepsilon}\leq
y_{n}(t)\leq\kappa_{n}(1-\lfloor nt\rfloor/n)^{2a+1-\varepsilon},$ (90)

and similarly,

$\kappa^{-1}(1-t)^{2a+1+\varepsilon}\leq
y(t)\leq\kappa(1-t)^{2a+1-\varepsilon}.$ (91)

Now we are ready to prove the main result of this section.

######  Theorem 52.

For fixed $\beta>0,a,b>-1$ and $n\geq 1$, let $\lambda_{i},1\leq i\leq 2n-1$
be the size $(2n-1)$ truncated real orthogonal beta ensemble and set
$p_{2n-1,\beta,a,b}(z)=\prod_{i=1}^{2n-1}\frac{z-\lambda_{i}}{1-\lambda_{i}}$
be the normalized characteristic polynomial. Let ${\mathcal{E}}_{\beta,a}$ be
the structure function of $\tau_{\beta,a}$ defined via (20). Then there is a
coupling of $p_{2n-1,\beta,a,b},n\geq 1$ and ${\mathcal{E}}_{\beta,a}$ such
that almost surely

$\left|p_{2n-1,\beta,a,b}(e^{iz/(2n)})e^{-iz/2}-{\mathcal{E}}_{\beta,a}(z)\right|\to
0\quad\text{ uniformly on compacts as $n\to\infty$.}$ (92)

Consequently, the truncated real orthogonal beta ensembles converge weakly to
the zeros of ${\mathcal{E}}_{\beta,a}(\cdot)$ under the edge scaling (3) as
$n\to\infty$.

###### Proof.

Let $\widetilde{\mu}_{2n}$ be the reversed version of the random probability
measures $\mu_{2n}:=\mu_{2n,\beta,a,b}^{\textup{RO}}$, and let
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}$ be the pulled-
back operator. By Propositions 13 and 49, the normalized characteristic
polynomial $p_{2n-1,\beta,a,b}$ has the same distribution as the monic
orthogonal polynomial of degree $2n-1$ associated to $\widetilde{\mu}_{2n}$.
Hence it is enough to show (92) holds with $\widetilde{\Psi}_{2n-1,\beta,a,b}$
in place of $p_{2n-1,\beta,a,b}$.

By Proposition 42, we have

$\widetilde{\Psi}_{2n-1,\beta,a,b}(e^{iz/(2n)})=e^{iz(2n-1)/(4n)}\overset{\text{\tiny${\hookleftarrow}$}}{H}_{2n,\beta,a,b}((2n-1)/(2n),z)^{{\dagger}}\binom{1}{-i},$

where $\overset{\text{\tiny${\hookleftarrow}$}}{H}_{2n,\beta,a,b}$ solves the
ODE (18) of $\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}$.
Recall that
${\mathcal{E}}_{\beta,a}=H_{\beta,a}(1,\cdot)^{{\dagger}}\binom{1}{-i}$. It
suffices to provide a coupling of $\mu_{2n}$ and ${\mathcal{E}}_{\beta,a}$
under which the uniform-on-compacts convergence of
$\overset{\text{\tiny${\hookleftarrow}$}}{H}_{2n,\beta,a,b}(1,z)$ to
$H_{\beta,a}(1,z)$ holds.

By Propositions 37 and 50, the operators
$\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b},n\geq 1$ and
$\tau_{\beta,a}$ can be obtained from $\mathtt{RO}_{2n,\beta,a,b},n\geq 1$ and
$\mathtt{Bess}_{\beta,a}$ under the same orthogonal transformations.
Therefore, in the coupling of Proposition 51, we have

$\|\mathtt{r\,}\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}-\mathtt{r\,}\tau_{\beta,a}\|\to
0,\quad\quad\mathfrak{t}_{\overset{\text{\tiny${\hookleftarrow}$}}{\tau}_{2n,\beta,a,b}}=\mathfrak{t}_{\tau_{\beta,a}}=0.$
(93)

By the triangle inequality, the path bounds (90) and (91), and a standard
subsequence argument, we have
$\lim_{n\to\infty}\int_{0}^{1}|\hat{y}(s)^{-1/2}-\hat{y}_{n}(s)^{-1/2}|^{2}ds=0$.
Together with (93), this verifies the conditions in (75). Using Proposition 43
completes the proof of the proposition. ∎

Note that by Proposition 39 the limit random analytic function
${\mathcal{E}}_{\beta,a}$ can also be characterized as
${\mathcal{E}}_{\beta,a}={\mathcal{H}}_{\beta,a}(0,z)^{{\dagger}}\binom{1}{-i}$,
where ${\mathcal{H}}_{\beta,a}$ solves the SDE (71).

Using Proposition 44 and Theorem 52, one obtains the following result on the
convergence of normalized characteristic polynomial of the perturbed matrix
$\mathsf{RO}^{[r]}_{2n,\beta,a,b}$.

######  Corollary 53.

For fixed $r\in[0,1]$ let $\Lambda_{2n}=\\{\lambda_{1},\dots,\lambda_{2n}\\}$
be the size $2n$ rank-one multiplicative perturbed real orthogonal ensemble
and set
$p_{2n,\beta,a,b}^{[r]}(z)=\prod_{i=1}^{2n}\frac{z-\lambda_{i}}{1-\lambda_{i}}$
to be the normalized characteristic polynomial. Then under the coupling of
Proposition 51, we have almost surely as $n\to\infty$

$\displaystyle\left|\Psi_{2n,\beta,a,b}^{[r]}(e^{iz/n})e^{-iz/(2n)}-{\mathcal{E}}_{\beta,a}^{[r]}(z)\right|\to
0,\quad\text{ uniformly on compacts in ${\mathbb{C}}$,}$

where
${\mathcal{E}}_{\beta,a}^{[r]}(z)=H_{\beta,a}(1,z)^{{\dagger}}\binom{1}{-i\frac{1-r}{1+r}}$.
Consequently, the rank-one multiplicative perturbed real orthogonal beta
ensemble $\Lambda_{2n}$ converges weakly to the zero set of the random
analytic function ${\mathcal{E}}_{\beta,a}^{[r]}$ under the edge scaling (3)
as $n\to\infty$.

Recall that the secular function $\zeta_{\beta,a}$ of the
$\mathtt{Bess}_{\beta,a}$ operator is given by
$\zeta_{\beta,a}=H_{\beta,a}(1,z)^{{\dagger}}\binom{1}{0}$. The above result
gives an interpolation between the scaling limits of the normalized
characteristic polynomials of the unperturbed and the truncated real
orthogonal beta ensemble.

##  7 The truncated circular Jacobi beta ensemble

In Section 7.1 we construct the truncated and the multiplicative perturbed
circular Jacobi beta ensemble, and we compute the joint eigenvalue density of
the truncated model. In Section 7.2, we derive the edge scaling limits of the
truncated and perturbed models using Propositions 43 and 44.

###  7.1 Matrix model and joint eigenvalue density for truncated circular
Jacobi beta ensemble

Consider the random probability measure
$\mu_{n,\beta,\delta}^{\operatorname{CJ}}$ in (60), with support given by the
circular Jacobi beta ensemble. Recall also the matrix model
$\mathsf{CJ}_{n,\beta,\delta}$ defined in Proposition 28 via the sequence of
regular Verblunsky coefficients $\alpha_{k},0\leq k\leq n-1$ of
$\mu_{n,\beta,\delta}^{\operatorname{CJ}}$.

######  Definition 54.

For fixed $n\geq 1$, $\beta>0$, $\Re\delta>-1/2$ we define the _truncated
circular Jacobi beta ensemble_ as the joint eigenvalue distribution of the
truncated matrix $\mathsf{CJ}^{\textstyle{\ulcorner}}_{n,\beta,\delta}$. For a
fixed $r\in[0,1]$ we define the perturbed circular Jacobi beta ensemble as the
joint eigenvalue distribution of the perturbed matrix
$\mathsf{CJ}_{n,\beta,\delta}^{[r]}:=\mathsf{CJ}_{n,\beta,\delta}\cdot\operatorname{diag}(r,1,1,\dots,1)$.

The main challenge to study these ensembles is that the Verblunsky
coefficients of $\mu_{n,\beta,\delta}^{\operatorname{CJ}}$ are not
independent, hence one cannot expect a nice description of the CMV matrices
appearing in Proposition 16. However, as the next proposition shows, we can
still preserve the independence by expressing the appearing CMV matrices in
terms of the _modified_ Verblunsky coefficients.

Recall the definition of modified Verblunsky coefficients given by the
recursion (40). The recursion provides a one-to-one map between the first
$k\leq n-1$ Verblunsky coefficients and the first $k$ modified Verblunsky
coefficients, we denoted this map by ${\mathcal{T}}_{k}$.

######  Proposition 55.

Fix $\beta>0,\delta\in{\mathbb{C}}$ with $\Re\delta>-1/2$, and let
$\gamma_{k},0\leq k\leq n-1$ be the sequence of modified Verblunsky
coefficients of $\mu_{n,\beta,\delta}^{\operatorname{CJ}}$. Then the sub-
unitary CMV matrix
${\mathcal{C}}({\mathcal{T}}_{n-1}^{-1}(\gamma_{n-2},\gamma_{n-3}\cdots,\gamma_{0}))$
has the same joint eigenvalue distribution as the truncated model
$\mathsf{CJ}^{\textstyle{\ulcorner}}_{n,\beta,\delta}$. For fixed $r\in[0,1]$,
the matrix
${\mathcal{C}}({\mathcal{T}}_{n}^{-1}(\gamma_{n-2},\gamma_{n-3}\dots,\gamma_{0},r\gamma_{n-1}))$
has the same joint eigenvalue distribution as the perturbed model
$\mathsf{CJ}_{n,\beta,\delta}^{[r]}$.

Before proving the proposition, we introduce a simple mapping on
${\mathbb{D}}$. For $\gamma\in{\mathbb{D}}$ recall the linear fractional
transformation ${\mathcal{A}}_{\gamma,{\mathbb{D}}}$ from (49). This is an
isometry of the Poincaré disk that corresponds to an affine transformation in
the Poincaré half-plane ${\mathbb{H}}$. The inverse of
${\mathcal{A}}_{\gamma,{\mathbb{D}}}$ is also an isometry, and it also
corresponds to an affine transformation in ${\mathbb{H}}$, we denote the
corresponding element in ${\mathbb{D}}$ by $\gamma^{\iota}$:

$\displaystyle{\mathcal{A}}_{\gamma,{\mathbb{D}}}^{-1}={\mathcal{A}}_{\gamma^{\iota},{\mathbb{D}}},\qquad\gamma^{\iota}=-\gamma\frac{1-\bar{\gamma}}{1-\gamma}.$
(94)

Note that the $\iota:\gamma\mapsto\gamma^{\iota}$ mapping is an involution
such that $\mathcal{A}_{\gamma,{\mathbb{D}}}(0)=\gamma^{\iota}$.

We also need the following distributional identity for $\Theta(a+1,\delta)$
random variables (see Definition 27).

######  Claim 56.

Fix $\delta\in{\mathbb{C}}$ with $\Re\delta>-1/2$. Let
$\zeta_{0},\zeta_{1},\dots,\zeta_{n-1}$ be a sequence of independent random
variables such that $\zeta_{i}\sim\Theta(a_{i}+1,\delta)$, with $a_{0}=0$ and
$a_{i}\geq 0$ for $i\geq 1$. Then

$\left(\frac{\zeta_{1}^{\iota}}{\zeta_{0}},\frac{\zeta_{2}^{\iota}}{\zeta_{1}},\dots\frac{\zeta_{n-1}^{\iota}}{\zeta_{n-2}},\frac{1}{\zeta_{n-1}}\right)\stackrel{{\scriptstyle
d}}{{=}}\left(\bar{\zeta}_{1},\frac{\bar{\zeta}_{2}}{\bar{\zeta}_{1}^{\iota}},\dots,\frac{\bar{\zeta}_{n-1}}{\bar{\zeta}_{n-2}^{\iota}},\frac{\bar{\zeta}_{0}}{\bar{\zeta}_{n-1}^{\iota}}\right).$
(95)

###### Proof.

Our statement will follow from the following (simpler) distributional
identity. Let $\zeta\sim\Theta(a+1,\delta)$ and $\eta\sim\Theta(1,\delta)$ be
independent. Then

$(\eta\bar{\zeta}^{\iota},\zeta)\stackrel{{\scriptstyle
d}}{{=}}(\zeta,\eta\bar{\zeta}^{\iota}).$ (96)

Since $|\zeta^{\iota}|=|\zeta|$, it is sufficient to prove that the unit
length random variables $\eta\bar{\zeta}^{\iota}/|\zeta|$ and $\zeta/|\zeta|$
are conditionally exchangeable given $|\zeta|=r$. Let $z=\zeta/|\zeta|$, then
under the condition $|\zeta|=r$ we have

$\eta\bar{\zeta}^{\iota}/|\zeta|=-\eta\frac{1-rz}{z-r}.$

By the independence of $\eta$ and $\zeta$, the conditional joint density of
$(\eta,z)$ (given $|\zeta|=r$) is proportional

$(1-\eta)^{\bar{\delta}}(1-\eta^{-1})^{\delta}(1-rz)^{\bar{\delta}}(1-rz^{-1})^{\delta}.$

Since $z\mapsto\frac{1-rz}{z-r}$ is an isometry of the unit circle, the
Jacobian of the mapping from
$\left(-\eta\frac{1-rz}{z-r},z\right)\mapsto(\eta,z)$ to is equal to $1$.
Therefore, the conditional joint density of
$(\xi_{1},\xi_{2}):=\left(-\eta\frac{1-rz}{z-r},z\right)$ (given $|\zeta|=r)$
is proportional to

$(1-r(\xi_{1}+\xi_{2})+\xi_{1}\xi_{2})^{\bar{\delta}}(1-r(\xi_{1}^{-1}+\xi_{2}^{-1})+\xi_{1}^{-1}\xi_{2}^{-1})^{\delta}.$

This shows the (conditional) exchangeability
$(\xi_{1},\xi_{2})=(\eta\bar{\zeta}^{\iota}/|\zeta|,\zeta/|\zeta|)$ and proves
(96).

Now we turn to the proof of the statement. Note that (96) implies that

$(\zeta_{1}^{\iota}/\zeta_{0},\bar{\zeta}_{1})\stackrel{{\scriptstyle
d}}{{=}}(\bar{\zeta}_{1},\zeta_{1}^{\iota}/\zeta_{0}).$

Starting from the random vector on the left-hand side of (95), we apply (96)
repeatedly and get

$\displaystyle\left(\frac{\zeta_{1}^{\iota}}{\zeta_{0}},\frac{\zeta_{2}^{\iota}}{\zeta_{1}},\dots\frac{\zeta_{n-1}^{\iota}}{\zeta_{n-2}},\frac{1}{\zeta_{n-1}}\right)$
$\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\left(\bar{\zeta}_{1},\frac{\bar{\zeta}_{0}\zeta_{2}^{\iota}}{\bar{\zeta}_{1}^{\iota}},\dots\frac{\zeta_{n-1}^{\iota}}{\zeta_{n-2}},\frac{1}{\zeta_{n-1}}\right)\stackrel{{\scriptstyle
d}}{{=}}\dots$ $\displaystyle\stackrel{{\scriptstyle
d}}{{=}}\left(\bar{\zeta}_{1},\frac{\bar{\zeta}_{2}}{\bar{\zeta}_{1}^{\iota}},\dots,\frac{\bar{\zeta}_{0}\zeta_{n-1}^{\iota}}{\bar{\zeta}_{n-2}^{\iota}},\frac{1}{\zeta_{n-1}}\right)\stackrel{{\scriptstyle
d}}{{=}}\left(\bar{\zeta}_{1},\frac{\bar{\zeta}_{2}}{\bar{\zeta}_{1}^{\iota}},\dots,\frac{\bar{\zeta}_{n-1}}{\bar{\zeta}_{n-2}^{\iota}},\frac{\bar{\zeta}_{0}}{\bar{\zeta}_{n-1}^{\iota}}\right),$

proving (95). ∎

Now we return to the proof of Proposition 55.

###### Proof of Proposition 55.

Let $\alpha_{k},0\leq k\leq n-1$ be the (regular) Verblunsky coefficients of
$\mu_{n,\beta,\delta}^{\operatorname{CJ}}$ and set
$\widetilde{\alpha}_{k},0\leq k\leq n-1$ to be the reversed version of
$\alpha_{k},0\leq k\leq n-1$ defined via (35). By Proposition 16, the
truncated and perturbed models
$\mathsf{CJ}^{\textstyle{\ulcorner}}_{n,\beta,\delta}$ and
$\mathsf{CJ}_{n,\beta,\delta}^{[r]}$ have the same eigenvalues as the CMV
matrices
${\mathcal{C}}(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\cdots,\widetilde{\alpha}_{n-2})$
and
${\mathcal{C}}(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\cdots,\widetilde{\alpha}_{n-2},r\widetilde{\alpha}_{n-1})$,
respectively. Hence, the statement of the proposition follows if we can show

$(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{n-2},\widetilde{\alpha}_{n-1})\stackrel{{\scriptstyle
d}}{{=}}{\mathcal{T}}_{n}^{-1}(\gamma_{n-2},\gamma_{n-3},\cdots,\gamma_{0},\gamma_{n-1}).$
(97)

Introduce the temporary notation

$(\hat{\alpha}_{0},\hat{\alpha}_{1},\dots,\hat{\alpha}_{n-1})={\mathcal{T}}_{n}^{-1}(\gamma_{n-2},\gamma_{n-3},\dots,\gamma_{0},\gamma_{n-1}).$

We need to show
$(\widetilde{\alpha}_{0},\widetilde{\alpha}_{1},\dots,\widetilde{\alpha}_{n-2},\widetilde{\alpha}_{n-1})\stackrel{{\scriptstyle
d}}{{=}}(\hat{\alpha}_{0},\hat{\alpha}_{1},\dots,\hat{\alpha}_{n-1})$, which
will follow from

$\left(\widetilde{\alpha}_{0},-\frac{\widetilde{\alpha}_{1}}{\widetilde{\alpha}_{0}}\dots,-\frac{\widetilde{\alpha}_{n-1}}{\widetilde{\alpha}_{n-2}}\right)\stackrel{{\scriptstyle
d}}{{=}}\left(\hat{\alpha}_{0},-\frac{\hat{\alpha}_{1}}{\hat{\alpha}_{0}}\dots,-\frac{\hat{\alpha}_{n-1}}{\hat{\alpha}_{n-2}}\right).$

From 40 and (94) it follows that

$\displaystyle-\frac{\alpha_{k}}{\alpha_{k-1}}=\frac{\bar{\gamma}_{k}}{\bar{\gamma}_{k-1}^{\iota}},\quad
k\geq 1,\qquad\alpha_{0}=\bar{\gamma}_{0}.$ (98)

Hence

$\left(\hat{\alpha}_{0},-\frac{\hat{\alpha}_{1}}{\hat{\alpha}_{0}}\dots,-\frac{\hat{\alpha}_{n-1}}{\hat{\alpha}_{n-2}}\right)=\left(\bar{\gamma}_{n-2},\frac{\bar{\gamma}_{n-3}}{\bar{\gamma}_{n-2}^{\iota}},\dots,\frac{\bar{\gamma}_{0}}{\bar{\gamma}_{1}^{\iota}},\frac{\bar{\gamma}_{n-1}}{\bar{\gamma}_{0}^{\iota}}\right).$

On the other hand, from 35 and (98) we have

$\left(\widetilde{\alpha}_{0},-\frac{\widetilde{\alpha}_{1}}{\widetilde{\alpha}_{0}}\dots,-\frac{\widetilde{\alpha}_{n-1}}{\widetilde{\alpha}_{n-2}}\right)=\left(-\frac{\bar{\alpha}_{n-2}}{\bar{\alpha}_{n-1}},-\frac{\bar{\alpha}_{n-3}}{\bar{\alpha}_{n-2}},\dots,-\frac{\bar{\alpha}_{0}}{\bar{\alpha}_{1}},\frac{1}{\bar{\alpha}_{0}}\right)=\left(\frac{\gamma_{n-2}^{\iota}}{\gamma_{n-1}},\frac{\gamma_{n-3}^{\iota}}{\gamma_{n-2}},\dots,\frac{\gamma_{0}^{\iota}}{\gamma_{1}},\frac{1}{\gamma_{0}}\right)$

From Proposition 28 we know that $\gamma_{k},0\leq k\leq n-1$ are independent
with $\gamma_{k}\sim\Theta(\beta(n-k-1)+1),\delta)$. Using Claim 56 with
$\zeta_{j}=\gamma_{n-1-j},0\leq j\leq n-1$ we get

$\left(\frac{\gamma_{n-2}^{\iota}}{\gamma_{n-1}},\frac{\gamma_{n-3}^{\iota}}{\gamma_{n-2}},\dots,\frac{\gamma_{0}^{\iota}}{\gamma_{1}},\frac{1}{\gamma_{0}}\right)\stackrel{{\scriptstyle
d}}{{=}}\left(\bar{\gamma}_{n-2},\frac{\bar{\gamma}_{n-3}}{\bar{\gamma}_{n-2}^{\iota}},\dots,\frac{\bar{\gamma}_{0}}{\bar{\gamma}_{1}^{\iota}},\frac{\bar{\gamma}_{n-1}}{\bar{\gamma}_{0}^{\iota}}\right),$

and the statement of the proposition follows. ∎

Proposition 55 allows us to derive the joint eigenvalue distribution of the
finite truncated circular Jacobi beta ensemble.

######  Theorem 57.

Fix $\beta>0,\delta\in{\mathbb{C}}$ with $\Re\delta>-1/2$. Then the
eigenvalues of $\mathsf{CJ}^{\textstyle{\ulcorner}}_{n+1,\beta,\delta}$ are
distributed in ${\mathbb{D}}^{n}$ according to the density

$c_{n,\beta,\delta}\prod_{j,k=1}^{n}(1-z_{j}\bar{z}_{k})^{\frac{\beta}{2}-1}\prod_{j<k}|z_{k}-z_{j}|^{2}\prod_{j=1}^{n}\left((1-z_{j})^{\bar{\delta}}(1-\bar{z}_{j})^{\delta}\right)$
(99)

with respect to the Lebesgue measure on $\mathbb{D}^{n}$. Here
$c_{n,\beta,\delta}=\frac{1}{\pi^{n}n!}\prod_{j=1}^{n}\frac{\Gamma(\frac{\beta}{2}j+1+\delta)\Gamma(\frac{\beta}{2}j+1+\bar{\delta})}{\Gamma(\frac{\beta}{2}j)\Gamma(\frac{\beta}{2}j+1+\delta+\bar{\delta})}$
is the normalizing constant.

###### Proof.

The proof of the statement relies on the following computations of Jacobian
determinants. We refer to Section 6 and Appendix B of [19] for more details.

Let $\gamma_{k},0\leq k\leq n-1$ be distributed as in Proposition 55, and let
$(\alpha_{0},\alpha_{1},\cdots,\alpha_{n-1})={\mathcal{T}}_{n}^{-1}(\gamma_{0},\gamma_{1},\cdots,\gamma_{n-1})$.
From (40) we have

$\left|\frac{\partial(\alpha_{0},\dots,\alpha_{n-1})}{\partial(\gamma_{0},\dots,\gamma_{n-1})}\right|=1.$
(100)

Denote by $z_{k},1\leq k\leq n$ the eigenvalues of
$\mathsf{CJ}^{\textstyle{\ulcorner}}_{n+1,\beta,\delta}$. It has been shown in
[19] that

$\left|\frac{\partial(\alpha_{0},\dots,\alpha_{n-1})}{\partial(z_{1},\dots,z_{n})}\right|=|\Delta(z_{1},\dots,z_{n})|^{2}\prod_{j=0}^{n-1}(1-|\alpha_{j}|^{2})^{-j},$
(101)

where $\Delta(z_{1},\dots,z_{n})=\prod_{1\leq j<k\leq n}(z_{j}-z_{k})$ denotes
the Vandermonde determinant of $z_{k},1\leq k\leq n$. Using
$|\alpha_{k}|=|\gamma_{k}|$, (100), and (101), we obtain

$\left|\frac{\partial(\gamma_{0},\dots,\gamma_{n-1})}{\partial(z_{1},\dots,z_{n})}\right|=|\Delta(z_{1},\dots,z_{n})|^{2}\prod_{j=0}^{n-1}(1-|\gamma_{j}|^{2})^{-j}.$
(102)

Proposition 2.5 of [4] shows that we have

$\prod_{j=0}^{n-1}(1-\gamma_{j})=\Phi_{n}(1)=\prod_{j=1}^{n}(1-z_{j}),\quad\prod_{j=0}^{n-1}(1-\bar{\gamma}_{j})=\bar{\Phi}_{n}(1)=\prod_{j=1}^{n}(1-\bar{z}_{j}).$
(103)

Since $|\alpha_{k}|=|\gamma_{k}|$, by Lemma B.1 (vi) of [19], we have

$\prod_{j=0}^{n-1}(1-|\gamma_{j}|^{2})^{(\frac{\beta}{2}-1)(j+1)}=\prod_{j=0}^{n-1}(1-|\alpha_{j}|^{2})^{(\frac{\beta}{2}-1)(j+1)}=\prod_{j,k=1}^{n}(1-z_{j}\bar{z}_{k})^{\frac{\beta}{2}-1}.$
(104)

By (102)-(104) and the explicit joint distribution of $\gamma_{k},0\leq k\leq
n-1$, the joint density of $z_{k},1\leq k\leq n$ is given by

$\displaystyle c_{n,\beta,\delta}\prod_{j=0}^{n-1}$
$\displaystyle(1-|\gamma_{j}|^{2})^{(\frac{\beta}{2}-1)(j+1)}(1-\gamma_{j})^{\bar{\delta}}(1-\bar{\gamma}_{j})^{\delta}|\Delta(z_{1},\dots,z_{n})|^{2}$
$\displaystyle=c_{n,\beta,\delta}\prod_{j,k=1}^{n}(1-z_{j}\bar{z}_{k})^{\frac{\beta}{2}-1}\prod_{j=1}^{n}\left((1-z_{j})^{\bar{\delta}}(1-\bar{z}_{j})^{\delta}\right)|\Delta(z_{1},\dots,z_{n})|^{2}.$

Collecting the normalizing constants of the joint distribution of
$\gamma_{k},0\leq k\leq n-1$, we get