# Recurrence of multiples of composition operators on weighted Dirichlet
spaces

Noureddine Karim  ,  Otmane Benchiheb  and  Mohamed Amouch  Chouaib Doukkali
University. Department of Mathematics, Faculty of science El Jadida, Morocco
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A bounded linear operator $T$ acting on a Hilbert space $\mathcal{H}$ is said
to be recurrent if for every non-empty open subset $U\subset\mathcal{H}$ there
is an integer $n$ such that $T^{n}(U)\cap U\neq\emptyset$. In this paper, we
completely characterize the recurrence of scalar multiples of composition
operators, induced by linear fractional self maps of the unit disk, acting on
weighted Dirichlet spaces $S_{\nu}$; in particular on the Bergman space, the
Hardy space, and the Dirichlet space. Consequently, we complete a previous
work of Costakis et al. [19] on recurrence of linear fractional composition
operators on Hardy space. In this manner, we determine the triples
$(\lambda,\nu,\phi)\in\mathbb{C}\times\mathbb{R}\times LFM(\mathbb{D})$ for
which the scalar multiple of composition operator $\lambda C_{\phi}$ acting on
$S_{\nu}$ fails to be recurrent.

###### Key words and phrases:

Hypercyclicity, Recurrence, Composition operator, Dirichlet spaces.

###### 2010 Mathematics Subject Classification:

Primarily 47A16, 37B20, Secondarily 46E50, 46T25

##  1\. Introduction and preliminaries

Throughout this paper, $\mathbb{C}$ will represent the complex plane,
$\mathbb{C}^{*}$ the punctured plane $\mathbb{C}\backslash\\{0\\}$, and
$\hat{\mathbb{C}}=\mathbb{C}\cup\\{\infty\\}$ will be the one-point
compactification of $\mathbb{C}$. Moreover, $\mathbb{D}$ will stand for the
open unit disk of $\mathbb{C}$, while $\mathbb{T}$ will represent the unit
circle of $\mathbb{C}$.

A bounded linear operator $T$ acting on a Hilbert space $\mathcal{H}$ is said
to be hypercyclic if there is a vector $f\in\mathcal{H}$ whose $T$-orbit;

$\mathcal{O}(f,T)=\\{T^{n}f;\ n\in\mathbb{N}\\},$

is dense in $\mathcal{H}$. In such a case the vector $f$ is called a
hypercyclic vector. An operator $T$ is said to be cyclic if there is a vector
$f\in\mathcal{H}$ such that the space generated by $\mathcal{O}(f,T)$;

$\textrm{span}(\mathcal{O}(f,T))=\\{p(T)f;\ p\ \textrm{a polynomial}\\},$

is dense in $\mathcal{H}$. In this case the vector $f$ is called a cyclic
vector. A strong form of cyclicity and weaker than hypercyclicity is
supercyclicity. An operator $T$ is said to be supercyclic if there is a vector
$f\in\mathcal{H}$ whose projective orbit;

$\mathbb{C}.\mathcal{O}(f,T)=\\{\lambda T^{n}f;\ n\in\mathbb{N}\ \textrm{and}\
\lambda\in\mathbb{C}\\},$

is dense in $\mathcal{H}$. The vector $f$ is called a supercyclic vector.

Recall [18], that an operator $T$ on a Hilbert space $\mathcal{H}$ is
hypercyclic if and only if it is topologically transitive; that is, for any
pair of non-empty open subsets $U$, $V$ of $\mathcal{H}$ there exists some
$n\in\mathbb{N}$ such that

$T^{n}(U)\cap V\neq\emptyset.$

Recently, the notions of hypercyclic and transitivity has been generalized and
studied see [4, 3, 1, 2].
Another important concept in topological dynamics is that of recurrence. This
notion has been initiated by Poincaré and Birkhoff, while a systematic study
was given by Costakis et al. in [19]. A bounded linear operator on a Hilbert
space $\mathcal{H}$ is called recurrent if for every non-empty open subset
$U\subset\mathcal{H}$ there is a positive integer $n$ such that

$T^{n}(U)\cap U\neq\emptyset.$

A vector $f\in\mathcal{H}$ is said to be recurrent for $T$ if there exists a
strictly increasing sequence of positive integers $(n_{k})_{k\in\mathbb{N}}$
such that

$T^{n_{k}}f\rightarrow f,\\\ \mbox{ as }\ \ k\rightarrow\infty.$

In that sense, every hypercyclic operator is recurrent, and every hypercyclic
vector is recurrent. For more information on linear dynamics we refer to [6]
and [5].

The study of linear dynamics has become a very active area of research. This
work will be devoted to studying the recurrence of composition operators.
Recall that if $\mathcal{H}$ is a Hilbert space of analytic functions in the
unit disk $\mathbb{D}$, and if $\phi$ is a nonconstant self map of
$\mathbb{D}$, then the composition operator $C_{\phi}$ associated to $\phi$ on
$\mathcal{H}$ is defined by

$C_{\phi}f=f\circ\phi\ \ \mbox{ for all }\ \ f\in\mathcal{H}.$

In which case, the function $\phi$ called a symbol of $C_{\phi}$. For general
references on the theory of composition operators, see, e.g., Cowen and
MacCluer’s book [20], Shapiro’s book [8] and K. H. Zhu’s book [7]. The special
about the composition operator is that the properties of $C_{\phi}$ depend
significantly on the behaviour of the symbol $\phi$. In this paper, we show
that the recurrence of the composition operator is influenced by the location
of the fixed points of its symbol.

For each real number $\nu$ the weighted Dirichlet space $\mathcal{S}_{\nu}$ is
the space of functions $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}$ analytic on
$\mathbb{D}$ such that the following norm

(1.1) $\|f\|_{\nu}^{2}=\sum_{n=0}^{\infty}|a_{n}|^{2}(n+1)^{2\nu}$

is finite. Endowed with the inner product

$\left\langle\sum_{n=0}^{\infty}a_{n}z^{n},\sum_{n=0}^{\infty}b_{n}z^{n}\right\rangle=\sum_{n=0}^{\infty}a_{n}\bar{b}_{n}(n+1)^{2\nu},$

the spaces $\mathcal{S}_{\nu}$ are Hilbert spaces, see [20, p. 16] or [13, p.
1]. For some values of $\nu$ the spaces $\mathcal{S}_{\nu}$ are very well
known classical analytic function spaces: for instance if $\nu=1/2$, $S_{\nu}$
is the Dirichlet space $\mathcal{D}$; for $\nu=0$ it is the Hardy space
$H^{2}$ and for $\nu=-1/2$ it is the Bergman space $\mathcal{A}^{2}$. Observe
that if $\nu_{1}>\nu_{2}$, then the space $\mathcal{S}_{\nu_{1}}$ is strictly
contained in $\mathcal{S}_{\nu_{2}}$, and that
$\|f\|_{\nu_{2}}\leq\|f\|_{\nu_{1}}$, for every $f\in\mathcal{S}_{\nu_{1}}$.

Also, we can define the Dirichlet space as the collection of functions
analytic on $\mathbb{D}$ whose first derivatives have square integrable
modulus over $\mathbb{D}$. For $f\in\mathcal{D}$ the norm in $\mathcal{D}$ has
the representation

$\|f\|_{\mathcal{D}}^{2}=|f(0)|^{2}+\int_{\mathbb{D}}|f^{\prime}(z)|^{2}dA(z),$

where here $dA(z)$ is the Lebesgue area measure on $\mathbb{D}$ normalized to
have unit mass. In the Hardy space $H^{2}$ there is also an integral
representation of the norm. This representation is the following

$\|f\|_{H^{2}}^{2}=\frac{1}{2\pi}\sup_{0<r<1}\int_{-\pi}^{\pi}|f(re^{i\theta})|^{2}d\theta.$

The Dirichlet space and the Hardy space will play a central role in our study.

Let $\nu\in\mathbb{R}$, we define the function on $\mathbb{D}$ as:

$k(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+1)^{2\nu}},$

which is analytic on $\mathbb{D}$. Then for each $w\in\mathbb{D}$, the
reproducing kernel is defined by

$K_{w}(z)=k(\bar{w}z).$

This is easily seen that $\|K_{w}\|^{2}=k(|w|^{2})$ and for every
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in S_{\nu}$ we have that

$\langle f,K_{w}\rangle=\sum_{n=0}^{\infty}a_{n}w^{n}=f(w).$

It is known according to a result of P.R. Hurst [9] that the composition
operator $C_{\phi}$ is always bounded on $S_{\nu}$ when $\phi$ is a linear
fractional map on $\mathbb{D}$.
Recall (see, e.g., [22, Chapter 3], [8, Chapter 0]) that linear fractional
maps are those maps of the form

$\phi(z)=\frac{az+b}{cz+d},$

where $a,b,c$ and $d$ are complex numbers satisfying $ad-bc\neq 0$. They
extend to the extended complex plane $\mathbb{\hat{C}}$ by defining
$\phi(\infty)=a/c$, and $\phi(-d/c)=\infty$ if $c\neq 0$, while
$\phi(\infty)=\infty$ if $c=0$. The linear fractional maps can be classified
according to their fixed point, which are at most two fixed points if $\phi$
is not the identity. Two linear fractional maps $\phi$ and $\psi$ are called
conjugate if there is another linear fractional map $T$ such that

$\phi=T^{-1}\psi T.$

It is called parabolic if $\phi$ has only one fixed point $\alpha$ or,
equivalently, it is conjugate to a translation $\psi(z)=z+\tau$, were
$\tau\neq 0$. If $\phi$ has two distinct fixed points $\alpha$ and $\beta$,
then $\phi$ is conjugate to a dilation $\psi(z)=\mu z$. In this case, the
linear fractional map $\phi$ is called elliptic if $|\mu|=1$; hyperbolic if
$\mu>0$ and loxodromic otherwise, see [22] for more details. The chain role
proves that the value of the derivative at the fixed point is preserved under
conjugation. Therefore, the derivative of a parabolic linear fractional map at
its fixed point is $1$, while the derivative of a hyperbolic one is strictly
less than $1$ at its attractive fixed point and greater than $1$ at its
repulsive point. We will note $LFM(\mathbb{D})$ to refer to the set of all
such maps, which are additionally self-maps of the unit disk $\mathbb{D}$. It
is known that if $\phi\in LFM(\mathbb{D})$, then the derivative
$\phi^{\prime}(\eta)$ exists and is finite for every $\eta$ in the unit circle
$\mathbb{T}$. The condition $\phi(\mathbb{D})\subset\mathbb{D}$ make some
exigences on the location of the fixed points of $\phi$. We have that (see [8,
p. 5])

  1. (1)

If $\phi$ is parabolic, then its fixed point $\eta$ is in $\mathbb{T}$ and it
satisfies $\phi^{\prime}(\eta)=1$.

  2. (2)

If $\phi$ is hyperbolic, the attractive fixed point is in $\bar{\mathbb{D}}$
and the other fixed point outside of $\mathbb{D}$ and both fixed points are on
$\mathbb{T}$ if and only if $\phi$ is an automorphism of $\mathbb{D}$.

  3. (3)

If $\phi$ is loxodromic or elliptic, one fixed point is in $\mathbb{D}$ and
the other fixed point outside of $\bar{\mathbb{D}}$. The elliptic ones are
always automorphisms of $\mathbb{D}$. The fixed point in $\mathbb{D}$ for the
loxodromic ones is attractive.

Concerning the dynamics of composition operator on Dirichlet spaces, many
works have been realized:

In her dissertation [14], Nina Zorboska studied composition operators induced
by non-elliptic disk automorphism and proved that they are all cyclic on the
Hardy space. In addition, in [15] she has obtained some results about
cyclicity and hypercyclicity on the so-called smooth weighted Hardy spaces,
which are the weighted Hardy spaces whose functions have continuous first
derivatives on the boundary of the unit disk (basically weighted Hardy spaces
strictly smaller than $\mathcal{S}_{3/2}$).

In [16, 17] Bourdon and Shapiro thoroughly characterized the cyclicity, the
supercyclicity, and the hypercyclicity of linear fractional composition
operators on the Hardy space $H^{2}(\mathbb{D})$ (see Table I). Moreover, in
[17] they gave a program of transferring the cyclic and hypercyclic properties
of linear fractional composition operators to general composition operators on
the Hardy space.

In [13] answering some open questions posed by Zorboska [15], Gallardo-
Gutiérrez and Montes-Rodríguez obtained a complete characterization of the
cyclic and hypercyclic composition operators induced by linear fractional maps
on weighted Dirichlet spaces $\mathcal{S}_{\nu}$ (see Table II).

In the present work, we give a complete characterization of recurrence of
scalar multiples of composition operator whose symbols are linear fractional
maps, acting on weighted Dirichlet spaces $\mathcal{S}_{\nu}$; in particular,
on the Bergman space $\mathcal{S}_{-1/2}$, the Hardy space $\mathcal{S}_{0}$
and the classical Dirichlet space $\mathcal{S}_{1/2}$. To do that, spectral
technics will play a significant role.

The paper is organized as follows$:$ In the third section, we characterize the
recurrence of linear fractional composition operator $C_{\phi}$ on
$\mathcal{S}_{\nu}$ by providing a necessary and sufficient condition on
$\phi$ and the real number $\nu$.

The fourth section is devoted to studying the recurrence of scalar multiples
of linear fractional composition operators $\lambda C_{\phi}$ on the weighted
Dirichlet spaces by providing a necessary and sufficient condition on $\phi$,
the real $\nu$, and the complex scalar $\lambda$.

In tables I and II, we expose the classification of Gallardo-Gutiérrez and
Montes-Rodríguez of the dynamics of composition operators and their scalar
multiples on $\mathcal{S}_{\nu}$. In table III, we summarize our results about
the recurrence of linear fractional composition operator on
$\mathcal{S}_{\nu}$. Finally, in table IV, we give a summarize of the
recurrence of scalar multiples of linear fractional composition operator on
$\mathcal{S}_{\nu}$.

Type of $\phi$ | Cyclic | Supercyclic | Hypercyclic
---|---|---|---
Hyperbolic Automorphism | $\nu<1/2$ | $\nu<1/2$ | $\nu<1/2$
Parabolic Automorphism | $\nu<1/2$ | $\nu<1/2$ | $\nu<1/2$
Hyperbolic Non-Automorphism | Always | $\nu\leq 1/2$ | $\nu<1/2$
Parabolic Non-Automorphism | $\nu\leq 3/2$ | Never | Never
Interior and Exterior | Always | Never | Never
Interior and Boundary | Never | Never | Never
Elliptic Irrational rotation | Always | Never | Never
Elliptic Rational rotation | Never | Never | Never
Table I. Cyclic properties of $C_{\phi}$ on $\mathcal{S}_{\nu}$, $\phi$ linear fractional map. Type of $\phi$ | Hypercyclicity
---|---
Hyperbolic Automorphism |  $\nu<1/2$ and $\phi^{\prime}(\eta)^{(1-2\nu)/2}<|\lambda|<\phi^{\prime}(\eta)^{(2\nu-1)/2}$
Parabolic automorphism |  $\nu<1/2$ and $|\lambda|=1$
Hyperbolic Non-Automorphism |  $\nu\leq 1/2$ and $|\lambda|>\phi^{\prime}(\eta)^{(1-2\nu)/2}$
Table II. Cyclic properties of $\lambda C_{\phi}$ on $\mathcal{S}_{\nu}$ with
$\phi$ linear fractional map and $\eta$ denotes the attractive fixed point of
$\phi$

Our goal in the next two sections will be completing the previous tables by
adding the recurrence of $C_{\phi}$ and also the recurrence of $\lambda
C_{\phi}$ on $\mathcal{S}_{\nu}$. All the results about the recurrence of
composition operators and its scalar multiples on $\mathcal{S}_{\nu}$ spaces
are summarized in Table III and Table IV below.

Type of $\phi$ | Recurrence | Examples
---|---|---
Hyperbolic Automorphism | $\nu<1/2$ | $\frac{3z+1}{z+3}$
Parabolic Automorphism | $\nu<1/2$ | $\frac{(1+i)z-1}{z+i-1}$
Hyperbolic Non-Automorphism | $\nu<1/2$ | $\frac{1+z}{2}$
Parabolic Non-Automorphism | Never | $\frac{1}{2-z}$
Interior and Exterior | Never | $\frac{-z}{2+z}$
Interior and Boundary | Never | $\frac{z}{2-z}$
Elliptic Irrational rotation | Always | $e^{i\sqrt{2}}z$
Elliptic Rational rotation | Always | $e^{2i\pi/3}z$
Table III. Recurrence of $C_{\phi}$ on $\mathcal{S}_{\nu}$, $\phi$ linear fractional map. Type of $\phi$ | Recurrence
---|---
Elliptic Automorphism |  $\nu\in\mathbb{R}$ and $|\lambda|=1$
Hyperbolic Automorphism |  $\nu<1/2$ and $\phi^{\prime}(\eta)^{(1-2\nu)/2}<|\lambda|<\phi^{\prime}(\eta)^{(2\nu-1)/2}$
Parabolic automorphism |  $\nu<1/2$ and $|\lambda|=1$
Hyperbolic Non-Automorphism |  $\nu\leq 1/2$ and $|\lambda|>\phi^{\prime}(\eta)^{(1-2\nu)/2}$
Table IV. Recurrence of $\lambda C_{\phi}$ on $\mathcal{S}_{\nu}$, $\phi$
linear fractional map and $\eta$ the attractive fixed point of $\phi$.

For the cases not appearing in the Table IV, the operator $\lambda C_{\phi}$
cannot be recurrent, for any $\lambda\in\mathbb{C}$.

##  2\. Recurrence of $C_{\phi}$ on $\mathcal{S}_{\nu}$

A useful tool in the study of any of the dynamic properties is the following
proposition known as Comparison principle (see [8, p. 111] and [21]).

######  Proposition 2.1.

Let $\mathcal{E}$ be a metric space and $\mathcal{F}$ be a subspace that it
self a linear metric space with a stronger topology. Suppose that $T$ is a
continuous linear transformation on $\mathcal{E}$ that maps the smaller space
$\mathcal{F}$ into itself and also continuous in the topology of this space.
If $T$ is cyclic on $\mathcal{F}$, then it is also cyclic on $\mathcal{E}$ and
has an $\mathcal{E}$-cyclic vector that belongs to $\mathcal{F}$. Furthermore,
the same is true for supercyclic and hypercyclic operators.

######  Remark 2.2.

Observe that the comparison principle is also true for recurrent operators. If
$\nu_{1}>\nu_{2}$, we have that $\mathcal{S}_{\nu_{1}}$ is a dense subspace of
the space $\mathcal{S}_{\nu_{2}}$. Thus we can apply the comparison principle
in the sense that, if an operator $T$ acting on $\mathcal{S}_{\nu_{1}}$ is
recurrent, then it is recurrent on $\mathcal{S}_{\nu_{2}}$.

######  Theorem 2.3.

Let $\phi$ be a linear fractional map on $\mathbb{D}$ with an interior fixed
point $p\in\mathbb{D}$. Then $C_{\phi}$ is recurrent on any of the
$\mathcal{S}_{\nu}$ spaces if and only if $\phi$ is an elliptic automorphism
of $\mathbb{D}$.

###### Proof.

If $\phi$ is an elliptic automorphism then $\phi$ is conjugate to a rotation
(see [8, Chapter 0]). Thus, there exists a linear fractional map composition
operator $S$ and a complex number $\lambda\in\mathbb{T}$ such that
$C_{\phi}=S^{-1}C_{\phi_{\lambda}}S$, where $\phi_{\lambda}(z)=\lambda z$,
$z\in\mathbb{D}$. In order to prove that $C_{\phi}$ is recurrent, we need to
prove that $C_{\phi_{\lambda}}$ is recurrent. Since $\lambda\in\mathbb{T}$,
there exists $(n_{k})_{k\in\mathbb{N}}$ such that $\lambda^{n_{k}}\rightarrow
1$. For any $f(z)=\sum_{m\geq 0}a_{m}z^{m}\in\mathcal{S}_{\nu}$ we have that

$\|C_{\phi_{\lambda}}^{n_{k}}(f)-f\|_{\mathcal{S}_{\nu}}=\sum_{m\geq
0}|a_{m}|^{2}|\lambda^{mn_{k}}-1|^{2}(m+1)^{2\nu}\rightarrow 0,\ \textrm{as}\
k\rightarrow\infty.$

Thus, every $f\in\mathcal{S}_{\nu}$ is a recurrent vector for
$C_{\phi_{\lambda}}$, and so $C_{\phi_{\lambda}}$ is recurrent on
$\mathcal{S}_{\nu}$. Now, if $\phi$ is not an elliptic automorphism then the
Denjoy-Wolff Iteration Theorem, [8, Proposition 1, Chapter 5], implies that
$\phi_{n}(z)$ converges to $p$, for every $z\in\mathbb{D}$. Hence, if $f$ is a
recurrent function of $C_{\phi}$ on $\mathcal{S}_{\nu}$, then there exists a
sequence of positive integers $(n_{k})_{k\in\mathbb{N}}$, such that
$f\circ\phi_{n_{k}}\rightarrow f$ in $\mathcal{S}_{\nu}$. Thus, by the
continuity of point evaluation functional on $S_{\nu}$, we have that for each
$z\in\mathbb{D}$:

$f(z)=\lim_{k\rightarrow\infty}f(\phi_{n_{k}}(z))=f(p).$

We conclude that the only recurrent vectors of $C_{\phi}$ are the constant
functions. ∎

######  Theorem 2.4.

Let $\phi$ be a parabolic automorphism of the unit disk. Then $C_{\phi}$ is
recurrent on $\mathcal{S}_{\nu}$ if and only if $\nu<1/2$.

###### Proof.

First, if $\nu<1/2$, then $C_{\phi}$ is hypercyclic, hence it is recurrent.
Now we shall prove that this condition is necessary. Let us start by the case
$\nu=1/2$. Since $\phi$ is parabolic, it has a fixed point on the unit circle.
Without loss of generality we may assume this fixed point is $1$. Suppose that
$C_{\phi}$ is recurrent, and let $f$ be a recurrent vector for $C_{\phi}$.
Then there exists a sequence $(n_{k})_{k\in\mathbb{N}}$ of positive integers
such that

$C_{\phi_{n_{k}}}(f)\rightarrow f,$

in $\mathcal{D}$. Since $\phi$ has no fixed point in $\mathbb{D}$, by Denjoy-
Wolff Iteration Theorem, the fixed point $1$ is actually attractive. Thus,
$\phi_{n_{k}}$ converge to $1$ uniformly on compact subsets of $\mathbb{D}$;
in particular, $\phi_{n_{k}}(z)$ converge to $1$, for every $z\in\mathbb{D}$.
Therefore we have that

$\|C_{\phi_{n_{k}}}(f)-f(1)\|_{\mathcal{D}}^{2}=\|f\circ\phi_{n_{k}}-f(1)\|_{\mathcal{D}}^{2}=|f\circ\phi_{n_{k}}(0)-f(1)|^{2}+\int_{\mathbb{D}}|f^{\prime}(\phi_{n_{k}}(z))|^{2}|\phi_{n_{k}}^{\prime}(z)|^{2}dA(z)\rightarrow
0,$

when $k\rightarrow\infty$. Thus, all recurrent vectors for $C_{\phi}$ are
constants, and so $C_{\phi}$ cannot be recurrent in this case. Therefore,
$C_{\phi}$ cannot be recurrent on $\mathcal{D}$. Hence, neither on any
$S_{\nu}$ with $\nu>1/2$, by Comparison principle. ∎

######  Theorem 2.5.

Let $\phi$ be a hyperbolic automorphism of the unit disk. Then $C_{\phi}$ is
recurrent on $\mathcal{S}_{\nu}$ if and only if $\nu<1/2$.

###### Proof.

If $\nu<1/2$ then the operator $C_{\phi}$ is recurrent. For $\nu\geq 1/2$, the
non-recurrence of $C_{\phi}$ follows exactly as in the case of the parabolic
automorphism. ∎

Now in the sequel let $\mathcal{S}_{\nu}^{0}$ be the space of functions
obtained from the space $\mathcal{S_{\nu}}$ by identifying functions that
differ by a constant; that is $f\in\mathcal{S}_{\nu}^{0}$ if and only if there
is $g\in\mathcal{S}_{\nu}$ such that $f-g$ is constant. Then the space
$\mathcal{S}_{\nu}^{0}$ is invariant under the operator $C_{\phi}$, and so we
can consider $\tilde{C}_{\phi}$ the restriction of $C_{\phi}$ on
$\mathcal{S}_{\nu}^{0}$.

######  Lemma 2.6.

Let $\phi$ be an analytic self-map of the unit disk. If $C_{\phi}$ is
recurrent on $\mathcal{S}_{\nu}$, then so is $\tilde{C}_{\phi}$ on
$\mathcal{S}_{\nu}^{0}$.

###### Proof.

We have that for every $f\in\mathcal{S}_{\nu}$ and $k\in\mathbb{N}$,

$\displaystyle\big{\|}C_{\phi_{k}}f-f\big{\|}_{S_{\nu}}$
$\displaystyle\geq\big{\|}C_{\phi_{k}}f-f+f(0)-f(\phi_{k}(0))\big{\|}_{S_{\nu}}$
$\displaystyle=\big{\|}\tilde{C}_{\phi_{k}}f-f\big{\|}_{\mathcal{S}_{\nu}^{0}}.$

Thus, $Rec(\tilde{C}_{\phi})=Rec(C_{\phi})\cap\mathcal{S}_{\nu}^{0}$. Assume
that $C_{\phi}$ is recurrent, then $Rec(C_{\phi})$ is dense in
$\mathcal{S}_{\nu}$, which implies that $Rec(\tilde{C}_{\phi})$ is dense in
$\mathcal{S}_{\nu}^{0}$. ∎

######  Theorem 2.7.

Let $\phi$ be a linear fractional map on $\mathbb{D}$ without interior fixed
point. If $\phi$ is hyperbolic non-automorphism, then $C_{\phi}$ is recurrent
on $\mathcal{S}_{\nu}$ if and if $\nu<1/2$.

###### Proof.

If $\nu<1/2$ then $C_{\phi}$ is hypercyclic, thus it is recurrent. Now,
suppose that $\nu>1/2$. Since $\phi$ has no interior fixed point, $\phi$ must
have an attractive fixed point $p\in\mathbb{T}$. Thus, $\phi$ is hyperbolic
non-automorphism with attractive fixed point $p\in\mathbb{T}$. Hence, the
spectrum of $C_{\phi}$ on $S_{\nu}$ is given by:

$\sigma(C_{\phi})=\\{\alpha\in\mathbb{C};\
|\alpha|\leq\phi^{\prime}(p)^{-\gamma}\\}\cup\\{\phi^{\prime}(p)^{k};\
k=0,1,...\\},$

where $\phi^{\prime}(p)$ is the angular derivation of $\phi$ on $p$ and
$\gamma=(1-2\nu)/2$, see [9]. If $\nu>1/2$, then $\gamma<0$. In this case the
singleton $\\{\phi^{\prime}(p)\\}$ is a component of the spectrum which isn’t
intersect $\mathbb{T}$, and so by [19, Proposition 2.11], $C_{\phi}$ cannot be
recurrent. Now, suppose that $\nu=1/2$, that is, $\mathcal{S}_{\nu}$ is the
Dirichlet space $\mathcal{D}$. If $C_{\phi}$ is recurrent on the Dirichlet
space, then, by Lemma 2.6, so is $\tilde{C}_{\phi}$ on
$\mathcal{D}_{0}:=\mathcal{S}_{1/2}^{0}$. Since $\phi$ is not an automorphism,
$\phi(\mathbb{D})\varsubsetneq\mathbb{D}$, and the recurrence of $C_{\phi}$
implies that $\phi$ is injective, then there exists a non-empty open
$U\varsubsetneq\mathbb{D}$, such that $\phi(\mathbb{D})\cap U=\emptyset$.
Hence, on $\mathcal{D}_{0}$ we have

$\int_{\mathbb{D}}|f^{\prime}(\phi(z))|^{2}|\phi^{\prime}(z)|^{2}dA(z)=\int_{\phi(\mathbb{D})}|f^{\prime}(z)|^{2}dA(z)\leq\int_{\phi(\mathbb{D})}|f^{\prime}(z)|^{2}dA(z)+\int_{U}|f^{\prime}(z)|^{2}dA(z)<\int_{\mathbb{D}}|f^{\prime}(z)|^{2}dA(z).$

Thus, $\|C_{\phi}\|_{\mathcal{D}_{0}}<1$, and therefore, $\tilde{C}_{\phi}$
cannot be recurrent on $\mathcal{D}_{0}$. ∎

######  Theorem 2.8.

Let $\phi$ be a parabolic non-automorphism of the unit disk. Then $C_{\phi}$
is never recurrent in any of the weighted Dirichlet spaces $S_{\nu}$.

###### Proof.

Since $\phi$ is parabolic non-automorphism, the spectrum of $C_{\phi}$ on
$\mathcal{S}_{\nu}$ is given by (see [23])

$\sigma(C_{\phi})=\\{e^{-\alpha t};\ t\geq 0\\}\cup\\{0\\},$

thus $\\{0\\}$ is a component of the spectrum which does not intersect the
unit circle. Hence, by [19, Proposition 2.11], the operator $C_{\phi}$ cannot
be recurrent. ∎

##  3\. Recurrence of $\lambda C_{\phi}$ on $\mathcal{S}_{\nu}$

######  Theorem 3.1.

Let $\phi$ be a holomorphic self map of $\mathbb{D}$ with an interior fixed
point. Then the operator $\lambda C_{\phi}$ is recurrent on
$\mathcal{S}_{\nu}$ if and only if $\phi$ is an elliptic automorphism and
$\lambda\in\mathbb{T}$.

###### Proof.

Assume that $\phi$ is an elliptic automorphism and $|\lambda|=1$. Then
$C_{\phi}$ is recurrent by Theorem 2.3, and since $|\lambda|=1$, then $\lambda
C_{\phi}$ is recurrent. If $\phi$ is not an elliptic automorphism, then
$C_{\phi}$ is not recurrent, which implies that $\lambda C_{\phi}$ is not
recurrent as well. Now, if $|\lambda|\neq 1$ and $\phi$ is not an elliptic
automorphism. Let $p\in\mathbb{D}$ be the fixed point of $\phi$. Suppose that
$\lambda C_{\phi}$ is recurrent and let $f$ be a recurrent vector. Then there
exists a sequence $(n_{k})_{k\in\mathbb{N}}$ of positive integers such that

$\lambda^{n_{k}}f\circ\phi_{n_{k}}\rightarrow f,$

in $\mathcal{S}_{\nu}$. Thus,

$f(p)=\lim_{k\rightarrow\infty}\lambda^{n_{k}}(C_{\phi_{n_{k}}}f)(p)=\lim_{k}\lambda^{n_{k}}f(\phi_{n_{k}}(p)).$

Now, observe that $f(\phi_{n_{k}}(p))\rightarrow f(p)$ as
$k\rightarrow\infty$. Therefore, if $|\lambda|<1$, then $f(p)=0$ for every
$z\in\mathbb{D}$. If $|\lambda|>1$, then $f(p)$ is not even defined, unless
$f(p)=0$. But $f(p)$ cannot be zero for every recurrent vector, because the
set of recurrent vectors is a dense subset. ∎

######  Theorem 3.2.

Let $\phi$ be a hyperbolic non-automorphism and $\eta$ its boundary fixed
point. Then $\lambda C_{\phi}$ is recurrent on $\mathcal{S}_{\nu}$ if and only
if $|\lambda|>\phi^{\prime}(\eta)^{(1-2\nu)/2}$ and $\nu\leq 1/2$. In
particular, we have: For the Dirichlet space ($\nu=1/2$) the operator $\lambda
C_{\phi}$ is recurrent if and only if $|\lambda|>1$.

To prove Theorem 3.2, we need the following lemma and proposition:

######  Lemma 3.3.

Let $T$ be a bounded operator on a separable Hilbert space $\mathcal{H}$.
Suppose that for each non zero vector $g\in\mathcal{H}$ such that the set of
complex numbers

$A:=\\{\langle Tf,g\rangle;\ f\in\mathcal{H}\\}$

is not dense in $\mathbb{C}$. Then $T$ is not recurrent.

###### Proof.

Every non zero vector $g\in\mathcal{H}$ defines a linear functional
$\psi:f\rightarrow\langle f,g\rangle$ whose range is $\mathbb{C}$. Suppose
that $T$ is recurrent, then its range is dense in $\mathcal{H}$. Thus,

$\mathbb{C}=\psi(\mathcal{H})=\psi(\overline{\textrm{Im}(T)})\subset\overline{\psi(\textrm{Im}(T))}=\overline{A},$

which implies that the set $A$ is dense in $\mathbb{C}$, a contradiction. ∎

######  Proposition 3.4.

[13, Proposition 2.16] If $\nu<1/2$, then for each
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}\in\mathcal{S}_{\nu}$ there is a constant
$C>0$ such that

$|f^{\prime}(z)|\leq\frac{C}{(1-|z|^{2})^{(3-2\nu)/2}}\ (z\in\mathbb{D}).$

Now we prove Theorem 3.2.

###### Proof of Theorem 3.2.

First, if $\nu\leq 1/2$ and $|\lambda|>\phi^{\prime}(\eta)^{(1-2\nu)/2}$, then
$C_{\phi}$ is hypercyclic on $\mathcal{S}_{\nu}$, thus it is recurrent on
$\mathcal{S}_{\nu}$. Now we prove that the conditions are necessary. Since the
recurrence is invariant under similarity, we may suppose that the boundary
fixed point is $1$.

Suppose that $\nu>1/2$. Then the reproducing kernel at $1$

$K_{1}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{(n+1)^{2\nu}}$

is in $\mathcal{S}_{\nu}$. Furthermore, for any $f\in\mathcal{S}_{\nu}$ we
have

$\langle\overline{\lambda}C_{\phi}^{*}K_{1},f\rangle=\overline{\lambda}\langle
K_{1},C_{\phi}f\rangle=\overline{\lambda}\langle
K_{1},f\circ\phi\rangle=\overline{\lambda}f(\phi(1))=\overline{\lambda}f(1)=\langle\overline{\lambda}K_{1},f\rangle.$

Then $\overline{\lambda}$ is an eigenvalue of
$\overline{\lambda}C_{\phi}^{*}$. Thus, if $|\lambda|\neq 1$, the operator
$\lambda C_{\phi}$ is not recurrent, since the point spectrum of its adjoint
$\overline{\lambda}C_{\phi}^{*}$, $\sigma_{p}(\overline{\lambda}C_{\phi}^{*})$
must be in the unit circle by [19, Proposition 2.15]. If $|\lambda|=1$, also
in this case $\lambda C_{\phi}$ is not recurrent, since $C_{\phi}$ is not
recurrent on $\mathcal{S}_{\nu}$.

Now, if $\nu=1/2$, in this case $\mathcal{S}_{\nu}$ will be the Dirichlet
space $\mathcal{D}$. Assume that $\lambda C_{\phi}$ is recurrent on
$\mathcal{D}$, for some $|\lambda|\leq 1$, then so is
$\lambda\tilde{C}_{\phi}$ on $\mathcal{D}_{0}$. But on $\mathcal{D}_{0}$ we
have

$\int_{\mathbb{D}}|\lambda
f^{\prime}(\phi(z))|^{2}|\phi^{\prime}(z)|^{2}dA(z)=|\lambda|^{2}\int_{\phi(\mathbb{D})}|f^{\prime}(z)|^{2}dA(z)<\int_{\mathbb{D}}|f^{\prime}(z)|^{2}dA(z).$

Thus $\|\lambda C_{\phi}\|_{D_{0}}<1$, and therefore,
$\lambda\tilde{C}_{\phi}$ cannot be recurrent on $\mathcal{D}_{0}$, a
contradiction. It remains to prove that if $|\lambda|\leq\mu^{(1-2\nu)/2}$ and
$\nu<1/2$, then the $\lambda C_{\phi}$ cannot be recurrent on
$\mathcal{S}_{\nu}$. The growth estimate for the derivative in Proposition 3.4
provides a constant $C$ such that the first inequality below holds. Thus we
have that

$\displaystyle|\langle\lambda C_{\phi}f,z\rangle|$
$\displaystyle=2^{2\nu}|\lambda||(f\circ\phi)^{\prime}(0)|$
$\displaystyle=2^{2\nu}|\lambda||f^{\prime}(\phi(0))||\phi^{\prime}(0)|$
$\displaystyle=2^{2\nu}|\lambda||f^{\prime}(1-\mu)|\mu$
$\displaystyle\leq\frac{2^{2\nu}C|\lambda|\mu}{(1-|1-\mu|^{2})^{(3-2\nu)/2}}$
$\displaystyle\leq\frac{2^{2\nu}C\mu^{(3-2\nu)/2}}{(2\mu-\mu^{2})^{(3-2\nu)/2}}$
$\displaystyle=\frac{2^{2\nu}C}{2-\mu}.$

By applying Lemma 3.3 we see that $\lambda C_{\phi}$ is not recurrent. ∎

######  Theorem 3.5.

Let $\phi$ be a parabolic automorphism of the unit disk. Then the operator
$\lambda C_{\phi}$ is recurrent on $\mathcal{S}_{\nu}$ if and only if
$\nu<1/2$ and $|\lambda|=1$.

###### Proof.

First if $\nu<1/2$ and $|\lambda|=1$, then $\lambda C_{\phi}$ is hypercyclic
on $\mathcal{S}_{\nu}$, which implies that it is recurrent. Now we prove that
these conditions are necessary. Let us first examine the case $\nu=1/2$.
Suppose that $|\lambda|\leq 1$. Let $f$ be a recurrent vector for $\lambda
C_{\phi}$, then there is a sequence $(n_{k})_{k\in\mathbb{N}}$ of positive
integers such that $\lambda^{n_{k}}C_{\phi}^{n_{k}}f\rightarrow f$ in
$\mathcal{D}$. Since $\phi$ is parabolic automorphism with fixed point $1$. If
$|\lambda|>1$, then, by what we have just proved, $\lambda^{-1}C_{\phi_{-1}}$
is not recurrent. Now, an invertible operator is recurrent if and only if its
inverse is. Consequently, $\lambda C_{\phi}$ is not recurrent on
$\mathcal{D}$.
Now suppose that $\nu>1/2$, we distingue two cases:
Case 1: If $|\lambda|\neq 1$, the reproducing kernel at $1$ in
$\mathcal{S}_{\nu}$ is an eigenvalue for the adjoint
$\overline{\lambda}C_{\phi}^{*}$, a contradiction.
Case 2: If $|\lambda|=1$, we know that $C_{\phi}$ is not recurrent on
$\mathcal{S}_{\nu}$, then $\lambda C_{\phi}$ is not recurrent on
$\mathcal{S}_{\nu}$ as well.
Now suppose that $\nu<1/2$. By Proposition the fixed point of $\phi$ must be
on the boundary of the unit disk. We may suppose that the fixed point is $1$.
Let

$\sigma(w)=\frac{i(1+w)}{1-w},\ \textrm{and}\
\psi=\sigma\circ\phi\circ\sigma^{-1}.$

then $\sigma$ is a linear fractional map of the unit disk onto the upper half
plane that takes $1$ to $\infty$, which implies $\infty$ is the only fixed
point of $\psi$, and so $\psi(w)=w+a$, where $a\neq 0$ and $\Re a=0$. The fact
that $\Re a=0$ comes form that $\phi$ corresponds to an automorphism of the
upper half plane. Thus $\phi$ satisfies the following formula

(3.1) $\phi(z)=\frac{(2-a)z+a}{-az+2+a}\ \textrm{with}\ a\neq 0\ \textrm{and}\
\Re a=0.$

If $|\lambda|<1$. By the growth estimate for the derivative’s proposition,
there is a constant $C$ such that the inequality below valid

$\displaystyle|\langle\lambda C_{\phi}f,z\rangle|$
$\displaystyle=2^{2\nu}|\lambda||f^{\prime}(\phi(0))||\phi^{\prime}(0)|$
$\displaystyle=2^{2\nu}|\lambda|\big{|}f^{\prime}\big{(}\frac{a}{2+a}\big{)}\big{|}\frac{4}{4+|a|^{2}}$
$\displaystyle\leq
2^{2\nu+2}C|\lambda|\big{(}1-\frac{|a|^{2}}{4+|a|^{2}}\big{)}^{(2\nu-3)/2}\frac{1}{4+|a|^{2}}$
$\displaystyle=\frac{2^{4\nu-1}C|\lambda|}{(4+|a|^{2})^{(2\nu-1)/2}}$
$\displaystyle<\frac{2^{4\nu-1}C}{(4+|a|^{2})^{(2\nu-1)/2}}.$

Thus, by Lemma 3.3, the operator $\lambda C_{\phi}$ is not recurrent. If
$|\lambda|>1$, then $\lambda^{-1}C_{\phi_{-1}}$ is not recurrent and,
therefore, neither is $\lambda C_{\phi}$. ∎

######  Theorem 3.6.

Let $\phi$ be a hyperbolic automorphism of the unit disk and $\eta$ its
attractive fixed point. Then $\lambda C_{\phi}$ is recurrent if and only if
$\nu<1/2$ and
$\phi^{\prime}(\eta)^{(1-2\nu)/2}<|\lambda|<\phi^{\prime}(\eta)^{(2\nu-1)/2}$.

###### Proof.

For $\nu\geq 1/2$, the non-recurrence of $\lambda C_{\phi}$ follows exactly as
in the case of the parabolic automorphism.
Now, suppose that $\nu<1/2$. We need an expression of $\phi$. Without loss of
generality, we may suppose that $\phi$ has $-1$ and $1$ as its fixed points.
Moreover, we may assume that $1$ is the attractive fixed point. In order to
compute $\phi$ explicitly, we use the change of variables

$\sigma(z)=\frac{i(1-z)}{1+z}$

that sends the unit disk onto the upper half-plane, the fixed points $1$ and
$-1$ to 0 and $\infty$, respectively, and $\phi$ to the contraction map
$\phi(w)=\mu w$, where $0<\mu<1$. Coming back to the unit disk we have

$\phi(z)=\frac{(1+\mu)z+1-\mu}{(1-\mu)z+1+\mu}\ \textrm{with}\ 0<\mu<1.$

Observe that the derivative at the attractive fixed point is
$\phi^{\prime}(1)=\mu$.
Now, for any $f\in\mathcal{S}_{\nu}$ we have the following estimate for some
constant $C$,

$\displaystyle|\langle\lambda C_{\phi}f,z\rangle|$
$\displaystyle=2^{2\nu}|\lambda||f^{\prime}(\phi(0))||\phi^{\prime}(0)|$
$\displaystyle=2^{2\nu}|\lambda|\big{|}f^{\prime}\big{(}\frac{1-\mu}{1+\mu}\big{)}\big{|}\frac{4\mu}{(1+\mu)^{2}}$
$\displaystyle\frac{\leq
2^{2\nu}C|\lambda|\mu^{(2\nu-1)/2}}{(1+\mu)^{2\nu-1}},$

that remain bounded for $|\lambda|\mu^{(2\nu-1)/2}\leq 1$. Therefor, if
$\lambda C_{\phi}$ is recurrent, then $|\lambda|>\mu^{(1-2\nu)/2}$. In
addition, the inverse operator $\lambda^{-1}C_{\phi_{-1}}$ must also be
recurrent. The attractive fixed point of $\phi_{-1}$ is $-1$ and
$\phi^{\prime}_{-1}(-1)=\mu$. Therefore, we must also have
$|\lambda^{-1}|>\mu^{(1-2\nu)/2}$. Thus the conditions on $\lambda$ are
necessary for $\lambda C_{\phi}$ to be recurrent. ∎

######  Theorem 3.7.

Let $\phi$ be a parabolic non-automorphism of the unit disk. Then the operator
$\lambda C_{\phi}$ is never recurrent on any $\mathcal{S}_{\nu}$.

###### Proof.

Since the singleton $\\{0\\}$ is a component of the spectrum of $\lambda
C_{\phi}$, but every component of the spectrum of recurrent operator must
intersect the unit circle, it implies that $\lambda C_{\phi}$ is not
recurrent. ∎

## References

  * [1] Amouch, M., Karim, N. Strong transitivity of composition operators. Acta Math. Hungar. 164, 458–469 (2021).
  * [2] S. Shkarin, On the spectrum of frequently hypercyclic operators, Proc. Amer. Math. Soc., 137 (2009), 123-134.
  * [3] A. Bonilla and K.-G. Grosse-Erdmann, Upper frequent hypercyclicity and related notions, Rev. Mat. Complut., 31 (2018), 673-711.
  * [4] J. Bes, Q. Menet, A. Peris and Y. Puig, Recurrence properties of hypercyclic operators, Math. Ann., 366 (2016), 545-5722.
  * [5] F. Bayart and E. Matheron, Dynamics of linear operators, Cambridge University Press, Cambridge 2009\.
  * [6] K.-G. Grosse-Erdmann and A. Peris Manguillot, Linear chaos, Springer, London 2011.
  * [7] K. H. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Texts in Mathematics 226, Springer, New York, 2005.
  * [8] Shapiro J. H.: Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993.
  * [9] Hurst P. R.: Relating composition operators on different weighted Hardy spaces. Arch. Math. (Basel), 68(6)1997, 503-513.
  * [10] Nordgren E. A., Radjavi H. and Rosenthal P.: Composition operators and the invariant subspace problem, C. R. Mat. Rep. Acad. Sci. Canada 6 (1984), 279-282.
  * [11] Nordgren E. A., Rosenthal P. and Wintrobe F. S.: Invertible composition operator on $H^{p}$, J. Func. Anal. 73 (1987), 324-344.
  * [12] Shapiro J. H.: The essential norm of a composition operator, Annals Math. 125 (1987), 375-404.
  * [13] Gallardo-Gutiérrez E. A. and Montes-Rodríguez A.: The role of the spectrum in the cyclic behavior of composition operators. Mem. Amer. Math. Soc., 167(791):x+81,2004.
  * [14] Zorboska N.: Composition operators on weighted Hardy spaces, Thesis, Univ. Toronto, 1988.
  * [15] Zorboska N., Cyclic composition operators on smooth weighted Hardy spaces, Rocky Mountain J. Math. 29 (1999), 725-740.
  * [16] P. S. Bourdon and J. H. Shapiro, Cyclic composition operators on $H^{2}$, Proc. Symp. Pure Math. 51 Part 2 (1990), 43-53.
  * [17] P. S. Bourdon and J. H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 596 (1997).
  * [18] Grosse-Erdmann K.-G.: Universal families and hypercyclic operators. Bull. Amer. Math. Soc. (N.S.) 36 (1999), 345-381.
  * [19] Costakis G., Manoussos A., Parissis I.: Recurrent Linear Operators.Complex Anal. Oper. Theory 8 (2014), 1601-1643.
  * [20] C. Cowen and B. MacCluer, Composition Operators on Spaces of Analytic Functions, CRC Press, 1995.
  * [21] Salas H. N.: Supercyclicity and weighted shifts, Studia Math. 135 (1999), 55-74.
  * [22] Ahlfors L. V.: Complex Analysis, McGraw-Hill, New York, 1979.
  * [23] Gallardo-Gutiérrez E. A. and Schroderus R.: The spectra of linear fractional composition operators on the weighted Dirichlet spaces. Journal of Functional Analysis, Volume 271, Issue 3, 720-745, (2016).


1},\dots,\mathbf{f}_{L-1}(1-z)^{\pm 1}$ are analytic in an annulus
$\mathfrak{A}_{\eta}$ where $\eta<1-2\mathfrak{c}$. Finally, assume we have
the factorisations
$\mathbf{f}_{0,L}(1-z)=\mathbf{S}_{+}(z)\mathbf{S}_{-}(z)=\tilde{\mathbf{S}}_{-}(z)\tilde{\mathbf{S}}_{+}(z)$,
where $\mathbf{S}_{\pm}$, $\tilde{\mathbf{S}}_{\pm}$ are
$\mathfrak{p}\times\mathfrak{p}$ matrix-valued functions satisfying

  * •

$\mathbf{S}_{+}^{\pm 1}(z),\tilde{\mathbf{S}}_{+}^{\pm 1}(z)$ are analytic for
$|z|<1$ and continuous for $|z|\leq 1$,

  * •

$\mathbf{S}_{-}^{\pm 1}(z),\tilde{\mathbf{S}}_{-}^{\pm 1}(z)$ are analytic for
$|z|>1$ and continuous for $|z|\geq 1$,

  * •

$\mathbf{S}_{-}(z)\sim z^{M}\mathbf{I}_{\mathfrak{p}}$ and
$\tilde{\mathbf{S}}_{-}(z)\sim z^{M}\mathbf{I}_{\mathfrak{p}}$ as
$z\to\infty$.

Consider the $2\mathfrak{p}\times 2\mathfrak{p}$ matrix-valued function
$\tilde{\mathbf{Y}}:\mathbb{C}\setminus\\{|z|=1\\}\to\mathbb{C}^{2\mathfrak{p}\times
2\mathfrak{p}}$ given as the unique solution to the $2\mathfrak{p}\times
2\mathfrak{p}$ Riemann-Hilbert problem:

  * •

$\tilde{\mathbf{Y}}$ is analytic,

  * •

$\tilde{\mathbf{Y}}_{+}(z)=\tilde{\mathbf{Y}}_{-}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}&\mathbf{f}_{0,L}(1-z)p_{M+N}(1-z)^{-1}\\\
\mathbf{0}_{\mathfrak{p}}&\mathbf{I}_{\mathfrak{p}}\end{pmatrix}$ on $|z|=1$,

  * •

$\tilde{\mathbf{Y}}(z)=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}(z^{-1})\right)\begin{pmatrix}z^{N}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&z^{-N}\mathbf{I}_{\mathfrak{p}}\end{pmatrix}$ as
$z\to\infty$.

Then, as $N\to\infty$ and for $|z|<1-2\mathfrak{c}$, we have

$\tilde{\mathbf{Y}}(z)=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right)\right)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{+}(z)\\\
-\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{+}^{-1}(z)&\mathbf{0}_{\mathfrak{p}}\end{pmatrix},$

for any $r\in(\mathfrak{c},1-2\mathfrak{c})$ such that $\max\\{|z|,\eta\\}<r$.
Similarly, as $N\to\infty$ and for $|z|>(1-2\mathfrak{c})^{-1}$,

$\tilde{\mathbf{Y}}(z)=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right)\right)\begin{pmatrix}\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{-}^{-1}(z)p_{M+N}(1-z)&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{-}(z)p_{M+N}(1-z)^{-1}\end{pmatrix},$

for any $r\in(\mathfrak{c},1-2\mathfrak{c})$ such that
$r^{-1}<\min\\{|z|,\eta^{-1}\\}$.

###### Proof.

We transform, in several steps, the Riemann-Hilbert problem for
$\tilde{\mathbf{Y}}(z)$ to a problem which can be solved explicitly, up to
asymptotically negligible terms, by virtue of the factorisation for
$\mathbf{f}_{0,L}(1-z)$. The strategy itself is standard and was followed in
[16] but there are a couple of interesting complications due to the non-
constant inhomogeneity sequence $\mathbf{a}$.

Step 1 We make the following transformation to normalise (in a slightly less
standard way) the Riemann-Hilbert problem for $\tilde{\mathbf{Y}}(z)$ at
infinity. Namely, define $\mathbf{X}(z)$ by

$\displaystyle\mathbf{X}(z)=\begin{cases}\tilde{\mathbf{Y}}(z)\begin{pmatrix}p_{M}(1-z)p_{M+N}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&p_{M+N}(1-z)p_{M}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix},&|z|>1,\\\
\tilde{\mathbf{Y}}(z),&|z|<1.\end{cases}$

Then, a simple computation shows that $\mathbf{X}(z)$ solves the Riemann-
Hilbert problem with conditions, since
$p_{m}(1-z)\sim\left(\prod_{k=0}^{m-1}a_{k}^{-1}\right)z^{m}$ as $z\to\infty$,

$\displaystyle\mathbf{X}_{+}(z)$
$\displaystyle=\mathbf{X}_{-}(z)\begin{pmatrix}p_{M+N}(1-z)p_{M}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}&\mathbf{f}_{0,L}(1-z)p_{M}(1-z)^{-1}\\\
\mathbf{0}_{\mathfrak{p}}&p_{M}(1-z)p_{M+N}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle|z|=1,$ $\displaystyle\mathbf{X}(z)$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}(z^{-1})\right)\begin{pmatrix}\prod_{k=M}^{M+N-1}a_{k}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\prod_{k=M}^{M+N-1}a_{k}^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle\textnormal{ as }z\to\infty.$

Step 2 We make another transformation. Consider $\mathbf{V}(z)$ given by,
where $r\in(\mathfrak{c},1-2\mathfrak{c})$ is picked such that the circles
$|z|=r$ and $|z|=r^{-1}$ are within the annulus of analyticity
$\mathfrak{A}_{\eta}$ from the statement, see Figure 28 for an illustration,

$\displaystyle\mathbf{V}(z)=\begin{cases}\mathbf{X}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
p_{M}^{2}(1-z)p_{M+N}(1-z)^{-1}\mathbf{f}_{0,L}(1-z)^{-1}&\mathbf{I}_{\mathfrak{p}}\end{pmatrix},&1<|z|<r^{-1},\\\
\mathbf{X}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
-p_{M+N}(1-z)\mathbf{f}_{0,L}(1-z)^{-1}&\mathbf{I}_{\mathfrak{p}}\end{pmatrix}&r<|z|<1,\\\
\mathbf{X}(z),&|z|<r\textnormal{ or }|z|>r^{-1}.\end{cases}$

Then, a computation shows that $\mathbf{V}(z)$ solves the following RHP

$\displaystyle\mathbf{V}_{+}(z)$
$\displaystyle=\mathbf{V}_{-}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
-\mathbf{f}_{0,L}(1-z)^{-1}p_{M+N}(1-z)&\mathbf{I}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle|z|=r,$ $\displaystyle\mathbf{V}_{+}(z)$
$\displaystyle=\mathbf{V}_{-}(z)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}&\mathbf{f}_{0,L}(1-z)p_{M}(1-z)^{-1}\\\
-\mathbf{f}_{0,L}(1-z)^{-1}p_{M}(1-z)&\mathbf{0}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle|z|=1,$ $\displaystyle\mathbf{V}_{+}(z)$
$\displaystyle=\mathbf{V}_{-}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
p_{M}^{2}(1-z)p_{M+N}(1-z)^{-1}\mathbf{f}_{0,L}(1-z)^{-1}&\mathbf{I}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle|z|=r^{-1},$ $\displaystyle\mathbf{V}(z)$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}(z^{-1})\right)\begin{pmatrix}\prod_{k=M}^{M+N-1}a_{k}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\prod_{k=M}^{M+N-1}a_{k}^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle\textnormal{ as }z\to\infty.$

$0$$\eta$$r$$1$$r^{-1}$$\eta^{-1}$ Figure 28: The contours for the Riemann-
Hilbert problem for $\mathbf{V}$. The area enclosed in the dotted circles is
the annulus $\mathfrak{A}_{\eta}$.

Step 3 Now, consider the following RHP problem for $\tilde{\mathbf{V}}(z)$,
which disregards the jumps on $|z|=r$ and $|z|=r^{-1}$ for $\mathbf{V}(z)$,
jumps that we will show next are exponentially small as $N\to\infty$,

$\displaystyle\tilde{\mathbf{V}}_{+}(z)$
$\displaystyle=\tilde{\mathbf{V}}_{-}(z)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}&\mathbf{f}_{0,L}(1-z)p_{M}(1-z)^{-1}\\\
-\mathbf{f}_{0,L}(1-z)^{-1}p_{M}(1-z)&\mathbf{0}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle|z|=1,$ $\displaystyle\tilde{\mathbf{V}}(z)$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}(z^{-1})\right)\begin{pmatrix}\prod_{k=M}^{M+N-1}a_{k}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\prod_{k=M}^{M+N-1}a_{k}^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix},$
$\displaystyle\textnormal{ as }z\to\infty.$

Then, by our factorisation assumption on $\mathbf{f}_{0,L}(1-z)$, we can
construct the explicit solution to this RHP for $\tilde{\mathbf{V}}(z)$ by,
since recall $p_{M}(1-z)\sim\left(\prod_{k=0}^{M-1}a_{k}^{-1}\right)z^{M}$ as
$z\to\infty$,

$\tilde{\mathbf{V}}(z)=\begin{cases}\begin{pmatrix}\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{-}^{-1}(z)p_{M}(1-z)&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{-}(z)p_{M}(1-z)^{-1}\end{pmatrix},\
&|z|>1,\\\
\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{+}(z)\\\
-\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{+}^{-1}(z)&\mathbf{0}_{\mathfrak{p}}\end{pmatrix},\
&|z|<1.\end{cases}$ (144)

Step 4 We now show that the jumps on the circles $|z|=r$ and $|z|=r^{-1}$ for
$\mathbf{V}(z)$ are exponentially small. Let $||\cdot||$ be any norm on
$\mathfrak{p}\times\mathfrak{p}$ matrices. Then, for
$r\in(\mathfrak{c},1-2\mathfrak{c})$ picked such that the circles $|z|=r$ and
$|z|=r^{-1}$ are within the annulus of analyticity $\mathfrak{A}_{\eta}$ of
$\mathbf{f}_{0,L}(1-z)^{-1}$, we have, with the implicit constant being
independent of $N$ and $r$,

$\displaystyle\sup_{|z|=r}\left|\left|\mathbf{f}_{0,L}(1-z)^{-1}p_{M+N}(1-z)\right|\right|$
$\displaystyle=\mathcal{O}\left(\frac{1}{\left(\inf_{k\in\mathbb{Z}_{+}}a_{k}\right)^{N}}\left(\sup_{|z|=r}\sup_{k\in\mathbb{Z}_{+}}|z+a_{k}-1|\right)^{N}\right)$
$\displaystyle=\mathcal{O}\left(\frac{1}{(1-\mathfrak{c})^{N}}\left(\sup_{|z|=r}\sup_{x\in[-\mathfrak{c},\mathfrak{c}]}|z+x|\right)^{N}\right)$
$\displaystyle=\mathcal{O}\left(\left(\frac{\mathfrak{c}+r}{1-\mathfrak{c}}\right)^{N}\right),$

and similarly,

$\displaystyle\sup_{|z|=r^{-1}}\left|\left|\mathbf{f}_{0,L}(1-z)^{-1}p_{M}^{2}(1-z)p_{M+N}(1-z)^{-1}\right|\right|$
$\displaystyle=\mathcal{O}\left(\left(\sup_{k\in\mathbb{Z}_{+}}a_{k}\right)^{N}\left(\sup_{|z|=r^{-1}}\sup_{k\in\mathbb{Z}_{+}}\frac{1}{|z+a_{k}-1|}\right)^{N}\right)$
$\displaystyle=\mathcal{O}\left((1+\mathfrak{c})^{N}\left(\frac{1}{\inf_{|z|=r^{-1}}\inf_{x\in[-\mathfrak{c},\mathfrak{c}]}|z+x|}\right)^{N}\right)$
$\displaystyle=\mathcal{O}\left(\left(\frac{1+\mathfrak{c}}{r^{-1}-\mathfrak{c}}\right)^{N}\right).$

Observe that, for $r\in(\mathfrak{c},1-2\mathfrak{c})$ with
$0\leq\mathfrak{c}<\frac{1}{3}$, we have

$\mathfrak{c}_{r}=\frac{\mathfrak{c}+r}{1-\mathfrak{c}}=\max\left\\{\frac{\mathfrak{c}+r}{1-\mathfrak{c}},\frac{1+\mathfrak{c}}{r^{-1}-\mathfrak{c}}\right\\}<1.$

Now, consider the function $\mathbf{J}(z)$ given by

$\mathbf{J}(z)=\mathbf{V}(z)\tilde{\mathbf{V}}(z)^{-1}.$

Then, observe that $\mathbf{J}(z)$, which only has jumps on the circles
$|z|=r$ and $|z|=r^{-1}$ which are exponentially small, can be solved in terms
of a Neumann series, see for example [56], to give

$\mathbf{J}(z)=\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right),\textnormal{
as }N\to\infty,$

uniformly on compact sets of
$\mathbb{C}\setminus\left(\\{|z|=r\\}\cup\\{|z|=r^{-1}\\}\right)$. This
completes this step.

Step 5 We finally undo the transformations to obtain the statement of the
proposition. For $|z|<1-2\mathfrak{c}$ we can choose
$r\in(\mathfrak{c},1-2\mathfrak{c})$ such that $\max\\{|z|,\eta\\}<r$, to
obtain

$\displaystyle\tilde{\mathbf{Y}}(z)=\mathbf{X}(z)=\mathbf{V}(z)$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right)\right)\tilde{\mathbf{V}}(z)$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right)\right)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{+}(z)\\\
-\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{+}^{-1}(z)&\mathbf{0}_{\mathfrak{p}}\end{pmatrix}.$

For $|z|>(1-2\mathfrak{c})^{-1}$ we can choose
$r\in(\mathfrak{c},1-2\mathfrak{c})$ such that
$r^{-1}<\min\\{|z|,\eta^{-1}\\}$, to obtain

$\displaystyle\tilde{\mathbf{Y}}(z)$
$\displaystyle=\mathbf{X}(z)\begin{pmatrix}p_{M}(1-z)p_{M+N}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&p_{M+N}(1-z)p_{M}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix}$
$\displaystyle=\mathbf{V}(z)\begin{pmatrix}p_{M}(1-z)p_{M+N}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&p_{M+N}(1-z)p_{M}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix}$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right)\right)\tilde{\mathbf{V}}(z)\begin{pmatrix}p_{M}(1-z)p_{M+N}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&p_{M+N}(1-z)p_{M}(1-z)^{-1}\mathbf{I}_{\mathfrak{p}}\end{pmatrix}$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{r}^{N}\right)\right)\begin{pmatrix}\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{-}^{-1}(z)p_{M+N}(1-z)&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{-}(z)p_{M+N}(1-z)^{-1}\end{pmatrix}.$

This completes the proof. ∎

We can finally prove Theorem 13.7.

###### Proof of Theorem 13.7.

Note that, by our assumption on $\mathbf{a}$ we can pick the
$\mathsf{C}_{\mathbf{a}}$ contour to be the circle $|z-1|=1$. After the change
of variables $z\mapsto 1-z$, $w\mapsto 1-w$, we can then write the kernel
$\mathsf{K}_{N}$ presented in block matrix form in (132) as follows:

$\displaystyle\left[\mathsf{K}_{N}\left[(r,m\mathfrak{p}+j);(r^{\prime},k\mathfrak{p}+i)\right]\right]_{i,j=0}^{\mathfrak{p}-1}=-\mathbf{1}_{r>r^{\prime}}\frac{1}{2\pi\textnormal{i}}\frac{1}{a_{m}}\oint_{|z|=1}\mathbf{f}_{r^{\prime},r}(1-z)\frac{p_{k}(1-z)}{p_{m+1}(1-z)}dz$
$\displaystyle+\frac{1}{(2\pi\textnormal{i})^{2}}\frac{1}{a_{k}}\oint_{|z|=1}\oint_{|w|=1}\mathbf{f}_{r^{\prime},L}(1-w)\frac{1}{z-w}\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}\
\mathbf{I}_{\mathfrak{p}}\end{pmatrix}\tilde{\mathbf{Y}}^{-1}(w)\tilde{\mathbf{Y}}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}\end{pmatrix}\mathbf{f}_{0,r}(1-z)\times$
$\displaystyle\times\frac{p_{k}(1-w)}{p_{M+N}(1-w)p_{m+1}(1-z)}dzdw,$

where $\tilde{\mathbf{Y}}$ is the Riemann-Hilbert problem given in Proposition
13.16. We then deform the contour for $w$ to a circle of radius at least
$(1-2\mathfrak{c})^{-1}$ but still within the annulus of analyticity
$\mathfrak{A}_{\eta}$ and similarly deform the contour for $z$ to a circle of
radius at most $(1-2\mathfrak{c})$ but also contained in
$\mathfrak{A}_{\eta}$. Note that, by our assumption this is possible. Hence,
from Proposition 13.16 we have for any
$\delta\in(\mathfrak{c},1-2\mathfrak{c})$ such that
$\delta^{-1}<\min\\{|w|,\eta^{-1}\\}$

$\displaystyle\frac{1}{p_{M+N}(1-w)}\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}\
\mathbf{I}_{\mathfrak{p}}\end{pmatrix}\tilde{\mathbf{Y}}^{-1}(w)=\frac{1}{p_{M+N}(1-w)}\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}\
\mathbf{I}_{\mathfrak{p}}\end{pmatrix}\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{\delta}^{N}\right)\right)\times$
$\displaystyle\times\begin{pmatrix}\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\tilde{\mathbf{S}}_{-}(w)p_{M+N}(1-w)^{-1}&\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}\right)\mathbf{S}_{-}^{-1}(w)p_{M+N}(1-w)\end{pmatrix}$
$\displaystyle=\left(\prod_{k=0}^{M+N-1}a_{k}\right)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}\
\mathbf{S}_{-}^{-1}(w)\end{pmatrix}\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{\delta}^{N}\right)\right),$

and similarly for any $\delta\in(\mathfrak{c},1-2\mathfrak{c})$ such that
$\max\\{|z|,\eta\\}<\delta$,

$\displaystyle\tilde{\mathbf{Y}}(z)\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}\end{pmatrix}$
$\displaystyle=\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{\delta}^{N}\right)\right)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}&\left(\prod_{k=0}^{M+N-1}a_{k}\right)\tilde{\mathbf{S}}_{+}(z)\\\
-\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\mathbf{S}_{+}^{-1}(z)&\mathbf{0}_{\mathfrak{p}}\end{pmatrix}\begin{pmatrix}\mathbf{I}_{\mathfrak{p}}\\\
\mathbf{0}_{\mathfrak{p}}\end{pmatrix}$
$\displaystyle=-\left(\prod_{k=0}^{M+N-1}a_{k}^{-1}\right)\left(\mathbf{I}_{2\mathfrak{p}}+\mathcal{O}\left(\mathfrak{c}_{\delta}^{N}\right)\right)\begin{pmatrix}\mathbf{0}_{\mathfrak{p}}\\\
\mathbf{S}_{+}^{-1}(z)\end{pmatrix}.$

Plugging these expressions in the correlation kernel $\mathsf{K}_{N}$, noting
that $\prod_{k=0}^{M+N-1}a_{k}^{-1}$ and $\prod_{k=0}^{M+N-1}a_{k}$ cancel out
exactly, and taking the $N\to\infty$ limit first (recall
$\mathfrak{c}_{\delta}<1$), and then deforming the contours to the circles
$|z|=1^{-}$ and $|w|=1^{+}$ we get

$\lim_{N\to\infty}\left[\mathsf{K}_{N}\left[(r,m\mathfrak{p}+j);(r^{\prime},k\mathfrak{p}+i)\right]\right]_{i,j=0}^{\mathfrak{p}-1}=\left[\mathsf{K}_{\infty}\left[(r,m\mathfrak{p}+j);(r^{\prime},k\mathfrak{p}+i)\right]\right]_{i,j=0}^{\mathfrak{p}-1}.$

This implies the convergence of all the correlation functions and since the
state space is discrete it implies convergence in distribution of the
corresponding point processes, and thus of the minimal particles, and gives
the statement of the theorem. ∎

We finally give an explicit example where all the conditions of Theorem 13.7
are satisfied. We focus on the scalar case $\mathfrak{p}=1$ with functions
corresponding to the three types of non-intersecting processes (pure-birth,
Bernoulli, geometric) we have been studying in this paper. Recall that for
$\mathfrak{p}=1$ we can just take
$\tilde{\mathbf{S}}_{\pm}(z)=\mathbf{S}_{\pm}(z)$. For a detailed study of the
factorisation in the matrix case $\mathfrak{p}\geq 2$, see [16], also [22,
45].

######  Proposition 13.17.

Let $\mathfrak{p}=1$. Assume that the parameter sequence $\mathbf{a}$
satisfies

$\inf_{x\in\mathbb{Z}_{+}}a_{x}\geq 1-\mathfrak{c}\textnormal{ and
}\sup_{x\in\mathbb{Z}_{+}}a_{x}\leq 1+\mathfrak{c},$ (145)

for some $0\leq\mathfrak{c}<\frac{1}{3}$. Suppose the functions
$\mathbf{f}_{0}(z),\dots,\mathbf{f}_{L-1}(z)$ are a product of factors of the
form $(1-\alpha z),\ (1+\beta z)^{-1},\ e^{-tz}$ so that in particular the
function $\mathbf{f}_{0,L}(1-z)$ is given by

$\mathbf{f}_{0,L}(1-z)=\prod_{r=1}^{L_{1}}(1-\alpha_{r}+\alpha_{r}z)\prod_{r=1}^{L_{2}}\frac{1}{1+\beta_{r}-\beta_{r}z}\prod_{r=1}^{L_{3}}e^{t_{r}z-t_{r}},$

for some $L_{1},L_{2},L_{3}\in\mathbb{Z}_{+}$ and constants
$\alpha_{r},\beta_{r},t_{r}\in\mathbb{R}_{+}$. Moreover, suppose there exist
exactly $M$ indices $i_{1},\dots,i_{M}$ such that

$(2-2\mathfrak{c})^{-1}<\alpha_{i_{j}}<(1+\mathfrak{c})^{-1},\ \ j=1,\dots,M,$

and for $r\neq i_{j}$ we have
$\alpha_{r}<\frac{1-2\mathfrak{c}}{2-2\mathfrak{c}}$. Finally, assume that
$\beta_{r}<\frac{1}{2\mathfrak{c}}-1$. Then, with the factorisation
$\mathbf{f}_{0,L}(1-z)=\mathbf{S}_{+}(z)\mathbf{S}_{-}(z)$ chosen as follows

$\displaystyle\mathbf{S}_{-}(z)$
$\displaystyle=\prod_{j=1}^{M}\left(z+\frac{1-\alpha_{i_{j}}}{\alpha_{i_{j}}}\right),$
$\displaystyle\mathbf{S}_{+}(z)$
$\displaystyle=\prod_{j=1}^{M}\alpha_{i_{j}}\prod_{r=1,r\neq
i_{j}}^{L_{1}}(1-\alpha_{r}+\alpha_{r}z)\prod_{r=1}^{L_{2}}\frac{1}{1+\beta_{r}-\beta_{r}z}\prod_{r=1}^{L_{3}}e^{t_{r}z-t_{r}},$

all the conditions of Theorem 13.7 are satisfied.

###### Proof.

Observe that, $\mathbf{f}_{i}(1-z)^{\pm 1}$ can have poles only at points
$(1-\alpha_{r}^{-1})$ or $(1+\beta_{r}^{-1})$ for some $r$. Then, elementary
computations show that the conditions above on the $\alpha_{r},\beta_{r}$
ensure that $\mathbf{f}_{0,L}(z)\in\mathsf{Hol}\left(\mathbb{H}_{-R}\right)$
for some
$R>R(\mathbf{a})=\sup_{x\in\mathbb{Z}_{+}}a_{x}-\inf_{x\in\mathbb{Z}_{+}}a_{x}$
and that we can find $\eta<1-2\mathfrak{c}$ such that none of the points
$(1-\alpha_{r}^{-1})$ or $(1+\beta_{r}^{-1})$ are in the intervals
$(-\eta^{-1},-\eta)\cup(\eta,\eta^{-1})$. Finally, it is easy to see by the
assumptions on the $\alpha_{r},\beta_{r}$’s the functions $\mathbf{S}^{\pm
1}_{-}(z)$, $\mathbf{S}^{\pm 1}_{+}(z)$ are analytic for $|z|>1$ and $|z|<1$
respectively (and continuous up to the boundary) and we have the correct
growth condition as $z\to\infty$ for $\mathbf{S}_{-}(z)$. ∎

###### Proof of Theorem 2.50.

Note that, the induced measure on $\mathbb{W}_{N}^{L-1}$ by the non-
intersecting random walks is given by (129) with $\mathfrak{p}=1$ and
$\mathbf{f}_{r}=f_{r,r+1}$ as in the statement of Theorem 2.50. Then, the
result follows by combining Theorem 13.7 and Proposition 13.17 with
$\mathsf{K}_{\infty}^{L,M}=\mathsf{K}_{\infty}$. ∎

##  14 Convergence to the discrete Bessel point process

###### Proof of Theorem 2.20.

Let us denote by $\tilde{K}_{t}^{(N)}(y_{1},y_{2})$ the correlation kernel of
the point process with the origin shifted to $N$, namely
$\left\\{\mathsf{X}_{i}^{(N)}(t)-N\right\\}_{i}$, which is then given by from
Theorem 2.14, where we make a couple of changes of variables in the contour
integrals,

$\displaystyle\tilde{K}_{t}^{(N)}(y_{1},y_{2})$
$\displaystyle=-\frac{1}{a_{y_{1}+N}}\frac{1}{(2\pi\textnormal{i})^{2}}\oint_{\mathsf{C}_{\mathbf{a},0}}dw\oint_{\mathsf{C}_{0}}du\frac{p_{y_{2}+N}(u)e^{-tw}}{p_{y_{1}+N+1}(w)e^{-tu}}\frac{w^{N}}{u^{N}}\frac{1}{w-u}$
$\displaystyle=\frac{1}{a_{y_{1}+N}}\frac{1}{(2\pi\textnormal{i})^{2}}\oint_{1-\mathsf{C}_{\mathbf{a},0}}dw\oint_{1-\mathsf{C}_{0}}du\frac{p_{y_{2}+N}(1-u)e^{tw}}{p_{y_{1}+N+1}(1-w)e^{tu}}\frac{(1-w)^{N}}{(1-u)^{N}}\frac{1}{w-u}$
$\displaystyle=\frac{1}{a_{y_{1}+N}}\frac{1}{(2\pi\textnormal{i})^{2}}\oint_{|w|=\epsilon^{-1}}dw\oint_{|u|=(\epsilon^{\prime})^{-1}}du\frac{p_{y_{2}+N}(1-u)e^{tw}}{p_{y_{1}+N+1}(1-w)e^{tu}}\frac{(1-w)^{N}}{(1-u)^{N}}\frac{1}{w-u}$
$\displaystyle=\frac{1}{a_{{y_{1}+N}}}\frac{1}{\left(2\pi\textnormal{i}\right)^{2}}\oint_{|w|=\epsilon}dw\oint_{|u|=\epsilon^{\prime}}du\frac{p_{y_{2}+N}\left(1-\frac{1}{u}\right)}{p_{y_{1}+N+1}\left(1-\frac{1}{w}\right)}\frac{e^{tw^{-1}}}{e^{tu^{-1}}}\frac{u^{N-1}}{w^{N+1}}\frac{\left(1-w\right)^{N}}{\left(1-u\right)^{N}}\frac{1}{u-w},$

with the radii $0<\epsilon<\epsilon^{\prime}$ chosen appropriately small so
that the circles $|w|=\epsilon^{-1}$ and $|z|=(\epsilon^{\prime})^{-1}$
contain all the relevant poles. Let us pick, for concreteness, for $N$ large
enough, $\epsilon=\frac{\zeta}{N}$ and $\epsilon^{\prime}=\frac{2\zeta}{N}$
and make the change of variables $v=\frac{N}{\zeta}u$ and
$z=\frac{N}{\zeta}w$. We then obtain that the kernel is given by, for $N$
large enough,

$\displaystyle\tilde{K}^{(N)}_{t}\left(y_{1},y_{2}\right)=\frac{1}{a_{{y_{1}+N}}}\frac{1}{\left(2\pi\textnormal{i}\right)^{2}}\oint_{|z|=1}dz\oint_{|v|=2}dv\frac{p_{y_{2}+N}\left(1-\frac{N}{\zeta
v}\right)}{p_{y_{1}+N+1}\left(1-\frac{N}{\zeta
z}\right)}\frac{e^{\frac{tN}{\zeta}z^{-1}}}{e^{\frac{tN}{\zeta}v^{-1}}}\frac{v^{N-1}}{z^{N+1}}\frac{\left(1-\frac{\zeta
z}{N}\right)^{N}}{\left(1-\frac{\zeta
v}{N}\right)^{N}}\frac{1}{v-z}\frac{N}{\zeta}.$

Consider the ratio of characteristic polynomials

$\frac{p_{y_{2}+N}\left(1-\frac{N}{\zeta
v}\right)}{p_{y_{1}+N+1}\left(1-\frac{N}{\zeta
z}\right)}=\frac{\prod_{k=0}^{N-1}\left(1-\frac{\left(1-\frac{N}{\zeta
v}\right)}{a_{k}}\right)}{\prod_{k=0}^{N-1}\left(1-\frac{\left(1-\frac{N}{\zeta
z}\right)}{a_{k}}\right)}\times\frac{\prod_{k=N}^{y_{2}+N-1}\left(1-\frac{\left(1-\frac{N}{\zeta
v}\right)}{a_{k}}\right)}{\prod_{k=N}^{y_{1}+N}\left(1-\frac{\left(1-\frac{N}{\zeta
z}\right)}{a_{k}}\right)}.$ (146)

The second factor above satisfies, where as usual
$\mathfrak{f}_{N}\sim\mathfrak{g}_{N}$ means
$\lim_{N\to\infty}\frac{\mathfrak{f}_{N}}{\mathfrak{g}_{N}}=1$,

$\frac{\prod_{k=N}^{y_{2}+N-1}\left(1-\frac{\left(1-\frac{N}{\zeta
v}\right)}{a_{k}}\right)}{\prod_{k=N}^{y_{1}+N}\left(1-\frac{\left(1-\frac{N}{\zeta
z}\right)}{a_{k}}\right)}\sim\frac{\left(\frac{N}{\zeta
v}\right)^{y_{2}}}{\left(\frac{N}{\zeta
z}\right)^{y_{1}+1}}\frac{\prod_{k=N}^{y_{1}+N}a_{k}}{\prod_{k=N}^{y_{2}+N-1}a_{k}}=\frac{\left(\frac{N}{\zeta
v}\right)^{y_{2}}}{\left(\frac{N}{\zeta
z}\right)^{y_{1}+1}}\frac{\prod_{k=0}^{y_{1}+N}a_{k}}{\prod_{k=0}^{y_{2}+N-1}a_{k}}.$

Recall that $\\{a_{x}\\}_{x=0}^{\infty}$ is uniformly bounded. If we write
$a_{i}^{(N)}=N^{-1}\left(\zeta a_{i}-N\right)$, for $1\leq i\leq N$, then we
have $a_{i}^{(N)}\to 0$ for all $i$,
$\sum_{i=1}^{N}a_{i}^{(N)}\to\zeta(\bar{a}-1)$ and
$\sum_{i=1}^{N}(a_{i}^{(N)})^{2}\to 0$. We then note that, uniformly on
compact sets in $\mathbb{C}$, we have as $N\to\infty$,

$\prod_{k=0}^{N-1}\left(1+v\left(\frac{\zeta
a_{k}-\zeta}{N}\right)\right)=\prod_{k=0}^{N-1}\left(1+va_{i}^{(N)}\right)\to
e^{\zeta(\bar{a}-1)v}.$ (147)

Hence, we obtain that the first factor in (146) satisfies, as $N\to\infty$,

$\frac{\prod_{k=0}^{N-1}\left(1-\frac{\left(1-\frac{N}{\zeta
v}\right)}{a_{k}}\right)}{\prod_{k=0}^{N-1}\left(1-\frac{\left(1-\frac{N}{\zeta
z}\right)}{a_{k}}\right)}=\frac{z^{N}}{v^{N}}\frac{\prod_{k=0}^{N-1}\left(1+v\left(\frac{\zeta
a_{k}-\zeta}{N}\right)\right)}{\prod_{k=0}^{N-1}\left(1+z\left(\frac{\zeta
a_{k}-\zeta}{N}\right)\right)}\sim\frac{z^{N}}{v^{N}}\frac{e^{\zeta(\bar{a}-1)v}}{e^{\zeta(\bar{a}-1)z}}.$

Thus, we get the asymptotics as $N\to\infty$,

$\frac{v^{N-1}}{z^{N+1}}\frac{p_{y_{2}+N}\left(1-\frac{N}{\zeta
v}\right)}{p_{y_{1}+N+1}\left(1-\frac{N}{\zeta
z}\right)}\frac{N}{\zeta}\sim\frac{z^{y_{1}}}{v^{y_{2}+1}}\frac{e^{\zeta(\bar{a}-1)v}}{e^{\zeta(\bar{a-1})z}}\left(\frac{N}{\zeta}\right)^{y_{2}-y_{1}}\frac{\prod_{k=0}^{y_{1}+N}a_{k}}{\prod_{k=0}^{y_{2}+N-1}a_{k}}.$

Hence, putting everything together we obtain, as $N\to\infty$,

$\displaystyle\tilde{K}_{\frac{\zeta}{N}}^{(N)}\left(y_{1},y_{2}\right)\sim\left(\frac{N}{\zeta}\right)^{y_{2}-y_{1}}\frac{\prod_{k=0}^{y_{1}+N-1}a_{k}}{\prod_{k=0}^{y_{2}+N-1}a_{k}}\frac{1}{\left(2\pi\textnormal{i}\right)^{2}}\oint_{|z|=1}dz\oint_{|v|=2}dv\frac{z^{y_{1}}}{v^{y_{2}+1}}e^{z^{-1}-v^{-1}+\zeta\bar{a}v-\zeta\bar{a}z}\frac{1}{v-z}.$

Observe that the ratio of factors in front of the contour integral cancel out
when we take the determinant. Moreover, we can deform the $z$ and $v$ contours
as long as the $z$-contour is contained in the $v$-contour and contains $0$.
Hence, for any $m\geq 1$,

$\displaystyle\det\left(\tilde{K}_{\frac{\zeta}{N}}^{(N)}\left(y_{i},y_{j}\right)\right)_{i,j=1}^{m}\overset{N\to\infty}{\longrightarrow}\det\left(\mathbf{J}_{\zeta\bar{a}}\left(y_{i},y_{j}\right)\right)_{i,j=1}^{m}.$

This implies the convergence of all the correlation functions and since the
state space is discrete it implies convergence in distribution of the
corresponding point processes, and thus of the maximal particles, and gives
the statement of the theorem.

∎

## References

  * [1] Aggarwal, A. Universality for lozenge tiling local statistics.  arXiv preprint arXiv:1907.09991 (2019).
  * [2] Aggarwal, A., Borodin, A., Petrov, L., and Wheeler, M. Free fermion six vertex model: symmetric functions and random domino tilings.  Selecta Math. (N.S.) 29, 3 (2023), Paper No. 36, 138.
  * [3] Aldous, D. J. Exchangeability and related topics.  In École d’été de probabilités de Saint-Flour, XIII—1983, vol. 1117 of Lecture Notes in Math. Springer, Berlin, 1985, pp. 1–198.
  * [4] Arai, Y. The kpz fixed point for discrete time taseps.  Journal of Physics A: Mathematical and Theoretical 53, 41 (2020), 415202.
  * [5] Arista, J., Bisi, E., and O’Connell, N. Matrix whittaker processes.  Probability Theory and Related Fields (2023), 1–55.
  * [6] Assiotis, T. Random surface growth and Karlin-McGregor polynomials.  Electron. J. Probab. 23 (2018), Paper No. 106, 81.
  * [7] Assiotis, T. On a gateway between the Laguerre process and dynamics on partitions.  ALEA Lat. Am. J. Probab. Math. Stat. 16, 2 (2019), 1055–1076.
  * [8] Assiotis, T. Determinantal structures in space-inhomogeneous dynamics on interlacing arrays.  Ann. Henri Poincaré 21, 3 (2020), 909–940.
  * [9] Assiotis, T. Exact solution of interacting particle systems related to random matrices.  Communications in Mathematical Physics (2023), 1–50.
  * [10] Assiotis, T. On the singular values of complex matrix Brownian motion with a matrix drift.  Bernoulli 29, 2 (2023), 1686–1713.
  * [11] Assiotis, T., and Najnudel, J. The boundary of the orbital beta process.  Mosc. Math. J. 21, 4 (2021), 659–694.
  * [12] Assiotis, T., O’Connell, N., and Warren, J. Interlacing Diffusions.  Sémin. Probab. L (2019), 301–380.
  * [13] Basu, R., Sidoravicius, V., and Sly, A. Last passage percolation with a defect line and the solution of the slow bond problem.  arXiv preprint arXiv:1408.3464 (2014).
  * [14] Berggren, T. Domino tilings of the Aztec diamond with doubly periodic weightings.  Ann. Probab. 49, 4 (2021), 1965–2011.
  * [15] Berggren, T., and Borodin, A. Geometry of the doubly periodic aztec dimer model.  arXiv preprint arXiv:2306.07482 (2023).
  * [16] Berggren, T., and Duits, M. Correlation functions for determinantal processes defined by infinite block Toeplitz minors.  Adv. Math. 356 (2019), 106766, 48.
  * [17] Biane, P., Bougerol, P., and O’Connell, N. Littelmann paths and Brownian paths.  Duke Math. J. 130, 1 (2005), 127–167.
  * [18] Bisi, E., Liao, Y., Saenz, A., and Zygouras, N. Non-intersecting path constructions for tasep with inhomogeneous rates and the kpz fixed point.  arXiv preprint arXiv:2208.13580 (2022).
  * [19] Borodin, A. Determinantal point processes.  In The Oxford handbook of random matrix theory. Oxford Univ. Press, Oxford, 2011, pp. 231–249.
  * [20] Borodin, A. Schur dynamics of the Schur processes.  Adv. Math. 228, 4 (2011), 2268–2291.
  * [21] Borodin, A., and Corwin, I. Macdonald processes.  Probab. Theory Related Fields 158, 1-2 (2014), 225–400.
  * [22] Borodin, A., and Duits, M. Biased $2\times 2$ periodic aztec diamond and an elliptic curve.  arXiv preprint arXiv:2203.11885 (2022).
  * [23] Borodin, A., and Ferrari, P. L. Large time asymptotics of growth models on space-like paths. I. PushASEP.  Electron. J. Probab. 13 (2008), no. 50, 1380–1418.
  * [24] Borodin, A., and Ferrari, P. L. Anisotropic growth of random surfaces in $2+1$ dimensions.  Comm. Math. Phys. 325, 2 (2014), 603–684.
  * [25] Borodin, A., and Ferrari, P. L. Random tilings and Markov chains for interlacing particles.  Markov Process. Related Fields 24, 3 (2018), 419–451.
  * [26] Borodin, A., Ferrari, P. L., Prähofer, M., and Sasamoto, T. Fluctuation properties of the TASEP with periodic initial configuration.  J. Stat. Phys. 129, 5-6 (2007), 1055–1080.
  * [27] Borodin, A., Ferrari, P. L., and Sasamoto, T. Large time asymptotics of growth models on space-like paths. II. PNG and parallel TASEP.  Comm. Math. Phys. 283, 2 (2008), 417–449.
  * [28] Borodin, A., and Kuan, J. Asymptotics of Plancherel measures for the infinite-dimensional unitary group.  Adv. Math. 219, 3 (2008), 894–931.
  * [29] Borodin, A., Okounkov, A., and Olshanski, G. Asymptotics of Plancherel measures for symmetric groups.  J. Amer. Math. Soc. 13, 3 (2000), 481–515.
  * [30] Borodin, A., and Olshanski, G. Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes.  Ann. of Math. (2) 161, 3 (2005), 1319–1422.
  * [31] Borodin, A., and Olshanski, G. The boundary of the Gelfand-Tsetlin graph: a new approach.  Adv. Math. 230, 4-6 (2012), 1738–1779.
  * [32] Borodin, A., and Olshanski, G. Markov processes on the path space of the Gelfand-Tsetlin graph and on its boundary.  J. Funct. Anal. 263, 1 (2012), 248–303.
  * [33] Borodin, A., and Olshanski, G. The Young bouquet and its boundary.  Mosc. Math. J. 13, 2 (2013), 193–232, 363.
  * [34] Borodin, A., and Olshanski, G. Representations of the infinite symmetric group, vol. 160 of Cambridge Studies in Advanced Mathematics.  Cambridge University Press, Cambridge, 2017.
  * [35] Borodin, A., and Petrov, L. Nearest neighbor Markov dynamics on Macdonald processes.  Adv. Math. 300 (2016), 71–155.
  * [36] Borodin, A., and Petrov, L. Inhomogeneous exponential jump model.  Probab. Theory Related Fields 172, 1-2 (2018), 323–385.
  * [37] Borodin, A., and Rains, E. M. Eynard-Mehta theorem, Schur process, and their Pfaffian analogs.  J. Stat. Phys. 121, 3-4 (2005), 291–317.
  * [38] Boutillier, C., Bouttier, J., Chapuy, G., Corteel, S., and Ramassamy, S. Dimers on rail yard graphs.  Ann. Inst. Henri Poincaré D 4, 4 (2017), 479–539.
  * [39] Bouttier, J., Chapuy, G., and Corteel, S. From Aztec diamonds to pyramids: steep tilings.  Trans. Amer. Math. Soc. 369, 8 (2017), 5921–5959.
  * [40] Bufetov, A., and Gorin, V. Representations of classical Lie groups and quantized free convolution.  Geom. Funct. Anal. 25, 3 (2015), 763–814.
  * [41] Bufetov, A., and Gorin, V. Fluctuations of particle systems determined by Schur generating functions.  Adv. Math. 338 (2018), 702–781.
  * [42] Bufetov, A., and Gorin, V. Fourier transform on high-dimensional unitary groups with applications to random tilings.  Duke Math. J. 168, 13 (2019), 2559–2649.
  * [43] Bufetov, A., Mucciconi, M., and Petrov, L. Yang-Baxter random fields and stochastic vertex models.  Adv. Math. 388 (2021), Paper No. 107865, 94.
  * [44] Bufetov, A., and Petrov, L. Yang-Baxter field for spin Hall-Littlewood symmetric functions.  Forum Math. Sigma 7 (2019), Paper No. e39, 70.
  * [45] Chhita, S., and Duits, M. On the domino shuffle and matrix refactorizations.  Communications in Mathematical Physics (2023), 1–51.
  * [46] Chhita, S., and Ferrari, P. A combinatorial identity for the speed of growth in an anisotropic KPZ model.  Ann. Inst. Henri Poincaré D 4, 4 (2017), 453–477.
  * [47] Chhita, S., Ferrari, P. L., and Toninelli, F. L. Speed and fluctuations for some driven dimer models.  Ann. Inst. Henri Poincaré D 6, 4 (2019), 489–532.
  * [48] Chhita, S., and Johansson, K. Domino statistics of the two-periodic Aztec diamond.  Adv. Math. 294 (2016), 37–149.
  * [49] Chhita, S., and Toninelli, F. The domino shuffling algorithm and anisotropic KPZ stochastic growth.  Ann. H. Lebesgue 4 (2021), 1005–1034.
  * [50] Chhita, S., and Toninelli, F. L. A $(2+1)$-dimensional anisotropic KPZ growth model with a smooth phase.  Comm. Math. Phys. 367, 2 (2019), 483–516.
  * [51] Chhita, S., and Young, B. Coupling functions for domino tilings of Aztec diamonds.  Adv. Math. 259 (2014), 173–251.
  * [52] Corwin, I. The Kardar-Parisi-Zhang equation and universality class.  Random Matrices Theory Appl. 1, 1 (2012), 1130001, 76.
  * [53] Corwin, I., and Hammond, A. Brownian Gibbs property for Airy line ensembles.  Invent. Math. 195, 2 (2014), 441–508.
  * [54] Dauvergne, D., Nica, M., and Virág, B. RSK in last passage percolation: a unified approach.  Probab. Surv. 19 (2022), 65–112.
  * [55] Dauvergne, D., and Virág, B. Bulk properties of the Airy line ensemble.  Ann. Probab. 49, 4 (2021), 1738–1777.
  * [56] Deift, P. A. Orthogonal polynomials and random matrices: a Riemann-Hilbert approach, vol. 3 of Courant Lecture Notes in Mathematics.  New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.
  * [57] Delvaux, S. Average characteristic polynomials for multiple orthogonal polynomial ensembles.  J. Approx. Theory 162, 5 (2010), 1033–1067.
  * [58] Denisov, D., and Wachtel, V. Random walks in cones.  Ann. Probab. 43, 3 (2015), 992–1044.
  * [59] Diaconis, P., and Fill, J. A. Strong stationary times via a new form of duality.  Ann. Probab. 18, 4 (1990), 1483–1522.
  * [60] Dieker, A. B., and Warren, J. Determinantal transition kernels for some interacting particles on the line.  Ann. Inst. Henri Poincaré Probab. Stat. 44, 6 (2008), 1162–1172.
  * [61] Doob, J. L. Classical potential theory and its probabilistic counterpart, vol. 262 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, New York, 1984.
  * [62] Duits, M., and Kuijlaars, A. B. J. The two-periodic Aztec diamond and matrix valued orthogonal polynomials.  J. Eur. Math. Soc. (JEMS) 23, 4 (2021), 1075–1131.
  * [63] Eichelsbacher, P., and König, W. Ordered random walks.  Electron. J. Probab. 13 (2008), no. 46, 1307–1336.
  * [64] Elkies, N., Kuperberg, G., Larsen, M., and Propp, J. Alternating-sign matrices and domino tilings. I.  J. Algebraic Combin. 1, 2 (1992), 111–132.
  * [65] Elkies, N., Kuperberg, G., Larsen, M., and Propp, J. Alternating-sign matrices and domino tilings. II.  J. Algebraic Combin. 1, 3 (1992), 219–234.
  * [66] Eynard, B., and Mehta, M. L. Matrices coupled in a chain. I. Eigenvalue correlations.  J. Phys. A 31, 19 (1998), 4449–4456.
  * [67] Fill, J. A. Strong stationary duality for continuous-time Markov chains. I. Theory.  J. Theoret. Probab. 5, 1 (1992), 45–70.
  * [68] Fulton, W. Young tableaux, vol. 35 of London Mathematical Society Student Texts.  Cambridge University Press, Cambridge, 1997.  With applications to representation theory and geometry.
  * [69] Gessel, I., and Viennot, G. Binomial determinants, paths, and hook length formulae.  Adv. in Math. 58, 3 (1985), 300–321.
  * [70] Goncharov, A. B., and Kenyon, R. Dimers and cluster integrable systems.  Ann. Sci. Éc. Norm. Supér. (4) 46, 5 (2013), 747–813.
  * [71] Gorin, V. The $q$-Gelfand-Tsetlin graph, Gibbs measures and $q$-Toeplitz matrices.  Adv. Math. 229, 1 (2012), 201–266.
  * [72] Gorin, V., and Olshanski, G. A quantization of the harmonic analysis on the infinite-dimensional unitary group.  J. Funct. Anal. 270, 1 (2016), 375–418.
  * [73] Gorin, V., and Petrov, L. Universality of local statistics for noncolliding random walks.  Ann. Probab. 47, 5 (2019), 2686–2753.
  * [74] Gorin, V. E. Nonintersecting paths and the Hahn orthogonal polynomial ensemble.  Funktsional. Anal. i Prilozhen. 42, 3 (2008), 23–44, 96.
  * [75] Iwao, S., Motegi, K., and Scrimshaw, T. Free fermionic probability theory and k-theoretic schubert calculus.  arXiv preprint arXiv:2311.01116 (2023).
  * [76] Jansen, S., and Kurt, N. On the notion(s) of duality for Markov processes.  Probab. Surv. 11 (2014), 59–120.
  * [77] Johansson, K. Discrete orthogonal polynomial ensembles and the Plancherel measure.  Ann. of Math. (2) 153, 1 (2001), 259–296.
  * [78] Johansson, K. Non-intersecting paths, random tilings and random matrices.  Probab. Theory Related Fields 123, 2 (2002), 225–280.
  * [79] Johansson, K. The arctic circle boundary and the Airy process.  Ann. Probab. 33, 1 (2005), 1–30.
  * [80] Johansson, K. Non-intersecting, simple, symmetric random walks and the extended Hahn kernel.  Ann. Inst. Fourier (Grenoble) 55, 6 (2005), 2129–2145.
  * [81] Johansson, K. Random matrices and determinantal processes.  In Mathematical statistical physics. Elsevier B. V., Amsterdam, 2006, pp. 1–55.
  * [82] Johansson, K. Random matrices and determinantal processes.  In Mathematical statistical physics. Elsevier B. V., Amsterdam, 2006, pp. 1–55.
  * [83] Johansson, K. A multi-dimensional Markov chain and the Meixner ensemble.  Ark. Mat. 48, 1 (2010), 79–95.
  * [84] Johansson, K. Two time distribution in Brownian directed percolation.  Comm. Math. Phys. 351, 2 (2017), 441–492.
  * [85] Johansson, K. Edge fluctuations of limit shapes.  In Current developments in mathematics 2016. Int. Press, Somerville, MA, 2018, pp. 47–110.
  * [86] Johansson, K. The two-time distribution in geometric last-passage percolation.  Probab. Theory Related Fields 175, 3-4 (2019), 849–895.
  * [87] Johansson, K. Long and short time asymptotics of the two-time distribution in local random growth.  Math. Phys. Anal. Geom. 23, 4 (2020), Paper No. 43, 34.
  * [88] Johansson, K., and Rahman, M. Multitime distribution in discrete polynuclear growth.  Communications on Pure and Applied Mathematics 74, 12 (2021), 2561–2627.
  * [89] Johansson, K., and Rahman, M. On inhomogeneous polynuclear growth.  Ann. Probab. 50, 2 (2022), 559–590.
  * [90] Karlin, S., and McGregor, J. Coincidence probabilities.  Pacific J. Math. 9 (1959), 1141–1164.
  * [91] Kasteleyn, P. W. Dimer statistics and phase transitions.  J. Mathematical Phys. 4 (1963), 287–293.
  * [92] Kenyon, R. Lectures on dimers.  In Statistical mechanics, vol. 16 of IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2009, pp. 191–230.
  * [93] Knizel, A., Petrov, L., and Saenz, A. Generalizations of TASEP in discrete and continuous inhomogeneous space.  Comm. Math. Phys. 372, 3 (2019), 797–864.
  * [94] König, W., O’Connell, N., and Roch, S. Non-colliding random walks, tandem queues, and discrete orthogonal polynomial ensembles.  Electron. J. Probab. 7 (2002), no. 5, 24.
  * [95] König, W., and Schmid, P. Random walks conditioned to stay in Weyl chambers of type C and D.  Electron. Commun. Probab. 15 (2010), 286–296.
  * [96] Korotkikh, S. Hidden diagonal integrability of $q$-Hahn vertex model and Beta polymer model.  Probab. Theory Related Fields 184, 1-2 (2022), 493–570.
  * [97] Lam, T., and Pylyavskyy, P. Total positivity in loop groups, I: Whirls and curls.  Adv. Math. 230, 3 (2012), 1222–1271.
  * [98] Lascoux, A., and Schützenberger, M.-P. Structure de Hopf de l’anneau de cohomologie et de l’anneau de Grothendieck d’une variété de drapeaux.  C. R. Acad. Sci. Paris Sér. I Math. 295, 11 (1982), 629–633.
  * [99] Lascoux, A., and Schützenberger, M.-P. Symmetry and flag manifolds.  In Invariant theory (Montecatini, 1982), vol. 996 of Lecture Notes in Math. Springer, Berlin, 1983, pp. 118–144.
  * [100] Macdonald, I. G. Schur functions: theme and variations.  In Séminaire Lotharingien de Combinatoire (Saint-Nabor, 1992), vol. 498 of Publ. Inst. Rech. Math. Av. Univ. Louis Pasteur, Strasbourg, 1992, pp. 5–39.
  * [101] Macdonald, I. G. Symmetric functions and Hall polynomials, second ed.  Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1995.  With contributions by A. Zelevinsky, Oxford Science Publications.
  * [102] Matetski, K., Quastel, J., and Remenik, D. Polynuclear growth and the toda lattice.  arXiv preprint arXiv:2209.02643 (2022).
  * [103] Matetski, K., and Remenik, D. TASEP and generalizations: method for exact solution.  Probability Theory and Related Fields 185, 1-2 (apr 2022), 615–698.
  * [104] Matetski, K., and Remenik, D. Exact solution of tasep and variants with inhomogeneous speeds and memory lengths, 2023.
  * [105] Matetski, K., Remenik, D., and Jeremy, Q. The KPZ fixed point.  Acta Math. 227 (2021), 115–203.
  * [106] Matveev, K. Macdonald-positive specializations of the algebra of symmetric functions: proof of the Kerov conjecture.  Ann. of Math. (2) 189, 1 (2019), 277–316.
  * [107] Miclo, L. Strong stationary times for one-dimensional diffusions.  Ann. Inst. Henri Poincaré Probab. Stat. 53, 2 (2017), 957–996.
  * [108] Miclo, L. On the construction of measure-valued dual processes.  Electron. J. Probab. 25 (2020), Paper No. 6, 64.
  * [109] Miclo, L., and Patie, P. On interweaving relations.  J. Funct. Anal. 280, 3 (2021), Paper No. 108816, 53.
  * [110] Nica, M., Quastel, J., and Remenik, D. One-sided reflected brownian motions and the kpz fixed point.  Forum of Mathematics, Sigma 8 (2020), e63.
  * [111] Nordenstam, E. On the shuffling algorithm for domino tilings.  Electron. J. Probab. 15 (2010), no. 3, 75–95.
  * [112] O’Connell, N. Conditioned random walks and the RSK correspondence.  J. Phys. A 36, 12 (2003), 3049–3066.
  * [113] O’Connell, N. A path-transformation for random walks and the Robinson-Schensted correspondence.  Trans. Amer. Math. Soc. 355, 9 (2003), 3669–3697.
  * [114] O’Connell, N. Directed polymers and the quantum Toda lattice.  Ann. Probab. 40, 2 (2012), 437–458.
  * [115] O’Connell, N., and Yor, M. A representation for non-colliding random walks.  Electron. Comm. Probab. 7 (2002), 1–12.
  * [116] Okounkov, A., and Olshanski, G. Asymptotics of Jack polynomials as the number of variables goes to infinity.  Internat. Math. Res. Notices, 13 (1998), 641–682.
  * [117] Okounkov, A., and Reshetikhin, N. Correlation function of Schur process with application to local geometry of a random 3-dimensional Young diagram.  J. Amer. Math. Soc. 16, 3 (2003), 581–603.
  * [118] Olshanski, G. The problem of harmonic analysis on the infinite-dimensional unitary group.  J. Funct. Anal. 205, 2 (2003), 464–524.
  * [119] Olshanski, G. Macdonald polynomials and extended Gelfand-Tsetlin graph.  Selecta Math. (N.S.) 27, 3 (2021), Paper No. 41, 61.
  * [120] Olshanski, G., and Vershik, A. Ergodic unitarily invariant measures on the space of infinite Hermitian matrices.  In Contemporary mathematical physics, vol. 175 of Amer. Math. Soc. Transl. Ser. 2. Amer. Math. Soc., Providence, RI, 1996, pp. 137–175.
  * [121] O’Connell, N. Interacting diffusions on positive definite matrices.  Probability Theory and Related Fields 180, 3-4 (2021), 679–726.
  * [122] Petrov, L. PushTASEP in inhomogeneous space.  Electron. J. Probab. 25 (2020), Paper No. 114, 25.
  * [123] Propp, J. Generalized domino-shuffling.  vol. 303. 2003, pp. 267–301.  Tilings of the plane.
  * [124] Quastel, J., and Sarkar, S. Convergence of exclusion processes and the KPZ equation to the KPZ fixed point.  J. Amer. Math. Soc. 36, 1 (2023), 251–289.
  * [125] Revuz, D., and Yor, M. Continuous martingales and Brownian motion, third ed., vol. 293 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences].  Springer-Verlag, Berlin, 1999.
  * [126] Rogers, L. C. G., and Pitman, J. W. Markov functions.  Ann. Probab. 9, 4 (1981), 573–582.
  * [127] Sasamoto, T. Spatial correlations of the 1D KPZ surface on a flat substrate.  J. Phys. A 38, 33 (2005), L549–L556.
  * [128] Schütz, G. M. Exact solution of the master equation for the asymmetric exclusion process.  J. Statist. Phys. 88, 1-2 (1997), 427–445.
  * [129] Siegmund, D. The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes.  Ann. Probability 4, 6 (1976), 914–924.
  * [130] Van Peski, R. Hall–Littlewood Polynomials, Boundaries, and p-Adic Random Matrices.  International Mathematics Research Notices (06 2022).  rnac143.
  * [131] Vershik, A. M., and Kerov, S. V. Asymptotic theory of the characters of a symmetric group.  Funktsional. Anal. i Prilozhen. 15, 4 (1981), 15–27, 96.
  * [132] Vershik, A. M., and Kerov, S. V. Characters and factor-representations of the infinite unitary group.  Dokl. Akad. Nauk SSSR 267, 2 (1982), 272–276.
  * [133] Wang, D., and Waugh, D. The transition probability of the $q$-TAZRP ($q$-bosons) with inhomogeneous jump rates.  SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 037, 16.
  * [134] Warren, J. Dyson’s Brownian motions, intertwining and interlacing.  Electron. J. Probab. 12 (2007), no. 19, 573–590.
  * [135] Warren, J., and Windridge, P. Some examples of dynamics for Gelfand-tsetlin patterns.  Electron. J. Probab. 14 (2009), 1745–1769.
  * [136] Zygouras, N. Some algebraic structures in KPZ universality.  Probab. Surv. 19 (2022), 590–700.

School of Mathematics, University of Edinburgh, James Clerk Maxwell Building,
Peter Guthrie Tait Rd, Edinburgh EH9 3FD, U.K.
<EMAIL_ADDRESS>