Scaling Limits for the Gibbs States on Distance-Regular Graphs with Classical
Parameters

Scaling Limits for the Gibbs States
on Distance-Regular Graphs
with Classical Parameters

Masoumeh KOOHESTANI a, Nobuaki OBATA b and Hajime TANAKA b

M. Koohestani, N. Obata and H. Tanaka

a) Department of Mathematics, K.N. Toosi University of Technology, Tehran
16765-3381, Iran<EMAIL_ADDRESS>
b) Research Center for Pure and Applied Mathematics, Graduate School of
Information Sciences,
b) Tohoku University, Sendai 980-8579, Japan<EMAIL_ADDRESS><EMAIL_ADDRESS>https://www.math.is.tohoku.ac.jp/~obata/,
https://www.math.is.tohoku.ac.jp/~htanaka/

Received July 19, 2021, in final form November 22, 2021; Published online
November 26, 2021

We determine the possible scaling limits in the quantum central limit theorem
with respect to the Gibbs state, for a growing distance-regular graph that has
so-called _classical parameters_ with base unequal to one. We also describe
explicitly the corresponding weak limits of the normalized spectral
distribution of the adjacency matrix. We demonstrate our results with the
known infinite families of distance-regular graphs having classical parameters
and with unbounded diameter.

quantum probability; quantum central limit theorem; distance-regular graph;
Gibbs state; classical parameters

46L53; 60F05; 05E30

##  1 Introduction

_Quantum probability theory_ is a non-commutative extension of classical
probability theory; see, e.g., [1, 12, 27, 28, 31, 42]. This paper is a
contribution to the spectral analysis of growing graphs from the viewpoint of
this theory. The adjacency matrix of a graph is regarded as a random variable
in this context. (Formal definitions will be given in Section 2.) As in many
previous works on this topic, our focus will be on central limit theorems
(CLTs) for growing graphs. Of particular interest are growing Cayley graphs.
For example, generators of free groups give rise to _free independent_ random
variables in the sense of Voiculescu, and we obtain the Wigner semicircle law;
cf. [42]. Another important class of graphs to consider here is that of
_distance-regular graphs_ [5, 6, 8, 10], which generalize distance-transitive
(i.e., two-point homogeneous) graphs. Among many other applications and links,
these graphs have been often used as test instances for problems related to
random walks on general graphs; see [10] and the references therein. Hora [16]
proved CLTs for several families of distance-regular graphs, including the
Hamming graphs (which are also Cayley graphs) and the Johnson graphs, and
obtained various distributions, such as the Gaussian, Poisson, geometric, and
the exponential distributions. The CLTs in [16] are with respect to the vacuum
state, and Hora [17] then considered the _Gibbs state_ , also known as the
_deformed vacuum state_ , and extended the CLT for the Johnson graphs and the
Hamming graphs. Later it turned out that distance-regular graphs are
particularly well-suited for the method of _quantum decomposition_ of the
adjacency matrix, which has been playing a key role in obtaining CLTs. This
method was first introduced by Hashimoto [13] for certain growing Cayley
graphs, and then developed and reformulated further by Hora, Obata, and
others; see, e.g., [14, 15, 18, 19, 20, 29]. It not only made the whole theory
transparent, but also lead to _quantum_ central limit theorems (QCLTs), in
which we take into account the three non-commuting components in the quantum
decomposition.

Our goal in this paper is to establish QCLTs for those distance-regular graphs
said to have _classical parameters_. For such a graph, the structure of the
adjacency algebra is described by just three parameters denoted by $q$,
$\alpha$, and $\beta$, together with the diameter $d$. The parameter $q$,
called the _base_ , is known to be an integer distinct from $0$ and $-1$,
provided that $d\geqslant 3$. Having classical parameters may look like quite
a strong restriction, but it is in some sense rather common among distance-
regular graphs. In fact, except the cycles which we view as trivial, all the
_known_ infinite families of distance-regular graphs with unbounded diameter
either have classical parameters or are closely related to those having
classical parameters (by means of halving, folding, doubling, twisting, etc.);
cf. [10, Section 3]. See also [33] and [32, Theorem 6.3] for geometric
characterizations of this property. The classification of distance-regular
graphs having classical parameters with $q=1$ is already complete, and there
exist only four infinite families: the Hamming graphs, Doob graphs, halved
cubes, and the Johnson graphs. These graphs were discussed by Hora [16, 17]
(see also [19]) in detail,111The Doob graphs were not mentioned in [16, 17],
but they have the same classical parameters as certain Hamming graphs, and
thus separate discussions are not necessary. so we will consider the graphs
with $q\neq 1$ in this paper. Our main result is Theorem 3.5 in Section 3,
where we let $d\rightarrow\infty$ and determine the possible scaling limits
for QCLTs in the Gibbs state, in terms of the behaviors of the classical
parameters and one other parameter associated with the Gibbs state. The
corresponding weak limits of the normalized spectral distributions will be
described explicitly in Section 5. Currently there are fifteen _known_
infinite families of distance-regular graphs having classical parameters with
unbounded diameter, eleven of which are such that $q\neq 1$. Our results apply
to these eleven families with $q\neq 1$, but we stress that they will also be
equally applicable whenever we find a new such infinite family in the
future.222The _twisted Grassmann graphs_ are the last of these fifteen
families and were discovered by Van Dam and Koolen [9] in 2005. We may also
remark that four out of the eleven families are Cayley graphs.

The contents of the other sections are as follows. In Section 2, we review
basic definitions and concepts about algebraic probability spaces and
distance-regular graphs, and then recall the QCLT for distance-regular graphs.
Our account follows [19]. We will also sharpen the QCLT slightly (Proposition
2.6), by showing that some assumption is redundant. Section 4 is another
preliminary section and is devoted to establishing formulas which are needed
in Section 5. Section 6 discusses concrete examples from the eleven infinite
families with $q\neq 1$ mentioned above.

##  2 Algebraic probability spaces, distance-regular graphs,
and the quantum central limit theorem

An _algebraic probability space_ is a pair $(\mathcal{A},\varphi)$, where
$\mathcal{A}$ is a $*$-algebra over $\mathbb{C}$ and
$\varphi\colon\mathcal{A}\rightarrow\mathbb{C}$ is a _state_ , i.e., a linear
map such that $\varphi(1_{\mathcal{A}})=1$ and that $\varphi(a^{*}a)\geqslant
0$ for every $a\in\mathcal{A}$, where $1_{\mathcal{A}}$ denotes the identity
of $\mathcal{A}$; cf. [19, Section 1.1]. The elements of $\mathcal{A}$ are
called (_algebraic_) _random variables_. We call $a\in\mathcal{A}$ _real_ if
$a^{*}=a$. For every real random variable $a\in\mathcal{A}$, there exists a
Borel probability measure $\mu$ on $\mathbb{R}$ (cf. [7, Section 1.3]) such
that

$\displaystyle\varphi(a^{i})=\int_{-\infty}^{+\infty}\xi^{i}\mu({\rm
d}\xi),\qquad i=0,1,2,\dots.$ (2.1)

We note that such a measure $\mu$ may not be unique. Several sufficient
conditions are known on its uniqueness, such as Carleman’s moment test; cf.
[19, Theorem 1.36].

We are interested in algebraic probability spaces arising from graphs. All the
graphs we consider in this paper are finite and simple. Thus, by a _graph_ we
mean a pair $\Gamma=(X,R)$ consisting of a non-empty finite set $X$ and a
subset $R$ of $\binom{X}{2}$, the set of two-element subsets of $X$. The
elements of $X$ are _vertices_ of $\Gamma$, and the elements of $R$ are
_edges_ of $\Gamma$. Two vertices $x,y\in X$ are called _adjacent_ (and
written $x\sim y$) if $\\{x,y\\}\in R$. The _degree_ (or _valency_) $k(x)$ of
$x\in X$ is the number of vertices adjacent to $x$. We call $\Gamma$
$k$-_regular_ if $k(x)=k$ for all $x\in X$. A _path of length $n$ joining_
$x,y\in X$ is a sequence of vertices $x=x_{0}$, $x_{1}$, …, $x_{n}=y$ such
that $x_{j-1}\sim x_{j}$ for $j=1,2,\dots,n$. We will only consider
_connected_ graphs, i.e., those graphs in which any two vertices are joined by
a path. The _distance_ $\partial(x,y)$ of $x,y\in X$ is the length of a
shortest path joining them. The _diameter_ of $\Gamma$ is defined by
$d=\max\\{\partial(x,y)\colon x,y\in X\\}$. Let $M_{X}(\mathbb{C})$ denote the
$\mathbb{C}$-algebra consisting of complex matrices with rows and columns
indexed by $X$. The _adjacency matrix_ $A$ of $\Gamma$ is the matrix in
$M_{X}(\mathbb{C})$ defined by

$\displaystyle A_{x,y}=\begin{cases}1&\text{if}\ x\sim y,\\\
0&\text{otherwise},\end{cases}\qquad x,y\in X.$

By an _eigenvalue_ of $\Gamma$, we mean an eigenvalue of $A$. Likewise, we
speak of the _spectrum_ of $\Gamma$.

As usual, we view $M_{X}(\mathbb{C})$ as a $*$-algebra by letting $*$ mean
adjoint (i.e., conjugate-transpose). Associated with the graph $\Gamma$ above
is the _adjacency algebra_ $\mathcal{A}(\Gamma)$, i.e., the commutative
$*$-subalgebra of $M_{X}(\mathbb{C})$ generated by $A$. Below we give three
examples of states for $\mathcal{A}(\Gamma)$.

The tracial state. This is defined by

$\displaystyle\varphi_{\operatorname{tr}}(B)=\frac{1}{|X|}\operatorname{tr}(B),\qquad
B\in\mathcal{A}(\Gamma).$

For this state, the Borel probability measure $\mu$ from (2.1) for the random
variable $A$ is unique and is the _spectral distribution_ of $\Gamma$ given by

$\displaystyle\mu(\theta_{i})=\frac{m_{i}}{|X|},\qquad i=0,1,\dots,e,$

where $\theta_{0},\theta_{1},\dots,\theta_{e}$ are the distinct eigenvalues of
$\Gamma$, and $m_{i}$ denotes the multiplicity of $\theta_{i}$ in the spectrum
of $\Gamma$.

The vacuum state. Fix a “base vertex” $o\in X$, and let

$\displaystyle\varphi_{0}(B)=\langle\hat{o},B\hat{o}\rangle=B_{o,o},\qquad
B\in\mathcal{A}(\Gamma),$

where $\hat{o}$ denotes the column vector indexed by $X$ with a $1$ in the $o$
coordinate and $0$ in all other coordinates, and $\langle\cdot,\cdot\rangle$
denotes the standard Hermitian inner product.

The Gibbs state. This generalizes $\varphi_{0}$ above. Let $t\in\mathbb{R}$,
and let

$\displaystyle\varphi_{t}(B)=\sum_{x\in
X}t^{\partial(x,o)}\langle\hat{x},B\hat{o}\rangle=\sum_{x\in
X}t^{\partial(x,o)}B_{x,o},\qquad B\in\mathcal{A}(\Gamma),$

where $0^{0}:=1$. The Gibbs state is also called the _deformed vacuum state_.
The scalar $-\log t$ (when $t\geqslant 0$) corresponds to the inverse
temperature parameter in the case of the Gibbs state on a canonical ensemble.
It should be remarked however that, unlike the first two examples, the Gibbs
state is not always a state.333For this reason, it would be more appropriate
to call $\varphi_{t}$, say, the Gibbs _functional_. See Lemma 2.1 below.

Recall that $\Gamma=(X,R)$ is assumed to be connected with diameter $d$. For
every $i=0,1,\dots,d$, let $A_{i}$ be the $i^{\mathrm{th}}$ _distance matrix_
of $\Gamma$, i.e.,

$\displaystyle(A_{i})_{x,y}=\begin{cases}1&\text{if}\ \partial(x,y)=i,\\\
0&\text{otherwise},\end{cases}\qquad x,y\in X.$

In particular, $A_{0}=I$ (the identity matrix) and $A_{1}=A$. We call $\Gamma$
_distance-regular_ if there exist non-negative integers $a_{i}$, $b_{i}$,
$c_{i}$, $i=0,1,\dots,d$, such that $b_{d}=c_{0}=0$, and that

$\displaystyle AA_{i}=b_{i-1}A_{i-1}+a_{i}A_{i}+c_{i+1}A_{i+1},\qquad
i=0,1,\dots,d,$ (2.2)

where $b_{-1}A_{-1}=c_{d+1}A_{d+1}:=0$. We note in this case that $\Gamma$ is
$k$-regular with $k=b_{0}$, $a_{0}=0$, $c_{1}=1$,

$\displaystyle a_{i}+b_{i}+c_{i}=k,\qquad i=0,1,\dots,d,$ (2.3)

and that $b_{i-1}c_{i}\neq 0$, $i=1,2,\dots,d$. Moreover, the matrix $A_{i}$
has constant row and column sum $k_{i}$ (which is the number of vertices at
distance $i$ from any given vertex), where

$\displaystyle k_{i}=\frac{b_{0}b_{1}\cdots b_{i-1}}{c_{1}c_{2}\cdots
c_{i}},\qquad i=0,1,\dots,d.$ (2.4)

Note that $k_{0}=1$ and $k_{1}=k$.

From now on, suppose that $\Gamma$ is distance-regular. It follows from (2.2)
that

$\displaystyle\mathcal{A}(\Gamma)=\operatorname{span}\\{A_{0},A_{1},\dots,A_{d}\\},$

from which it follows that $\dim\mathcal{A}(\Gamma)=d+1$, and hence that
$\Gamma$ has exactly $d+1$ distinct eigenvalues
$k=\theta_{0},\theta_{1},\dots,\theta_{d}$. Moreover, every matrix in
$\mathcal{A}(\Gamma)$ has constant diagonal entries, and hence
$\varphi_{\operatorname{tr}}=\varphi_{0}$. The Gibbs state $\varphi_{t}$ is
also independent of the base vertex $o\in X$, and we have

$\displaystyle\varphi_{t}(B)=\frac{1}{|X|}\operatorname{tr}(K_{t}B),\qquad
B\in\mathcal{A}(\Gamma),$ (2.5)

where

$\displaystyle K_{t}=\bigl{(}t^{\partial(x,y)}\bigr{)}_{x,y\in
X}=A_{0}+tA_{1}+t^{2}A_{2}+\dots+t^{d}A_{d}\in\mathcal{A}(\Gamma).$

From this observation we immediately see that

######  Lemma 2.1.

If $\Gamma$ is distance-regular, then the Gibbs state $\varphi_{t}$ is a state
on $\mathcal{A}(\Gamma)$ if and only if the matrix $K_{t}$ is positive
semidefinite.

It also follows from (2.2)–(2.5) that the mean and the variance of the random
variable $A$ in the Gibbs state $\varphi_{t}$ are given respectively by (cf.
[19, Lemma 3.25])

$\displaystyle\varphi_{t}(A)=tk,\qquad\Sigma_{t}^{2}(A)=\varphi_{t}\big{(}(A-tkI)^{2}\big{)}=k(1-t)(1+t+ta_{1}).$
(2.6)

In view of Lemma 2.1, we consider the following subset of $\mathbb{R}$:

$\displaystyle\pi(\Gamma)=\\{t\in\mathbb{R}\colon K_{t}\ \text{is positive
semidefinite}\\}.$ (2.7)

We always have $0,1\in\pi(\Gamma)$, so that $\pi(\Gamma)\neq\varnothing$.
Moreover, by looking at the $2\times 2$ principal submatrices of $K_{t}$, it
follows that

$\displaystyle\pi(\Gamma)\subset[-1,1].$

With respect to the base vertex $o\in X$, we have the following _quantum
decomposition_ of $A$:

$\displaystyle A=A^{+}+A^{-}+A^{\circ},$

where

$\displaystyle(A^{\epsilon})_{x,y}=\begin{cases}1&\text{if}\ x\sim y,\
\partial(x,o)=\partial(y,o)+i_{\epsilon},\\\
0&\text{otherwise},\end{cases}\qquad x,y\in X$

for $\epsilon\in\\{+,-,\circ\\}$, with $i_{+}=1$, $i_{-}=-1$, and
$i_{\circ}=0$. The matrices $A^{+}$, $A^{-}$, and $A^{\circ}$ are called the
_quantum components_ of $A$ with respect to $o$. Consider the
$\mathbb{C}$-vector space

$\displaystyle
W(\Gamma)=\operatorname{span}\\{\Phi_{0},\Phi_{1},\dots,\Phi_{d}\\},$ (2.8)

where the $\Phi_{i}$ are the unit column vectors given by

$\displaystyle\Phi_{i}=\frac{1}{\sqrt{k_{i}}}A_{i}\hat{o},\qquad
i=0,1,\dots,d.$

Note that $\Phi_{0}=\hat{o}$. With this notation, we have

$\displaystyle\varphi_{t}(B)=\sum_{i=0}^{d}t^{i}\sqrt{k_{i}}\langle\Phi_{i},B\Phi_{0}\rangle,\qquad
B\in\mathcal{A}(\Gamma).$ (2.9)

It follows from (2.2) and (2.4) that

$\displaystyle A^{+}\Phi_{i}=\sqrt{c_{i+1}b_{i}}\Phi_{i+1},\qquad
A^{-}\Phi_{i}=\sqrt{c_{i}b_{i-1}}\Phi_{i-1},\qquad
A^{\circ}\Phi_{i}=a_{i}\Phi_{i}$ (2.10)

for $i=0,1,\dots,d$, where
$\sqrt{c_{d+1}b_{d}}\Phi_{d+1}=\sqrt{c_{0}b_{-1}}\Phi_{-1}:=0$. In particular,
we observe that the actions of these quantum components on $W(\Gamma)$ are
independent of the base vertex $o\in X$.

######  Remark 2.2.

The subalgebra $\tilde{\mathcal{A}}(\Gamma)$ of $M_{X}(\mathbb{C})$ generated
by the quantum components $A^{+}$, $A^{-}$, and $A^{\circ}$ of $A$ is non-
commutative unless $|X|=1$, and is contained in the _Terwilliger algebra_ of
$\Gamma$ with respect to $o$; cf. [34, 35, 36]. See [41] for discussions on
when the two algebras are equal. The space $W(\Gamma)$ is an irreducible
module of the Terwilliger algebra, called the _primary module_.

We now recall the QCLT for a growing distance-regular graph in the Gibbs state
$\varphi_{t}$ established in [19, Section 3.4]. For the rest of this paper,
let $\Lambda$ be an infinite directed set, and let
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ be a net of distance-regular graphs;
see, e.g., [23, Chapter 2]. To simplify the notation, we will mostly omit the
subscript “$\lambda$”. We will view $X$, $d$, $k$, $a_{i}$, $b_{i}$, $c_{i}$,
etc., as functions of $\Gamma$. The scalar $t\in\pi(\Gamma)$ is chosen and
fixed for each of the $\Gamma$ so that the variance $\Sigma_{t}^{2}(A)>0$ (cf.
(2.6)), and we will also think of $t$ as a function of $\Gamma$. In this paper
we are mainly interested in limit distributions with infinite supports, and
hence we will assume that

$\displaystyle d\rightarrow\infty.$ (2.11)

(That is, $\lim\limits_{\lambda\in\Lambda}d(\Gamma_{\lambda})=\infty$.)

In view of (2.6), we work with the following normalization when taking the
limit:

$\displaystyle\frac{A-tkI}{\Sigma_{t}(A)}=\tilde{A}^{+}+\tilde{A}^{-}+\tilde{A}^{\circ},$
(2.12)

where

$\displaystyle\tilde{A}^{\pm}=\frac{A^{\pm}}{\Sigma_{t}(A)},\qquad\tilde{A}^{\circ}=\frac{A^{\circ}-tkI}{\Sigma_{t}(A)}.$

From (2.10) it follows that

$\displaystyle\tilde{A}^{+}\Phi_{i}=\sqrt{\overline{\omega}_{i+1}}\Phi_{i+1},\qquad\tilde{A}^{-}\Phi_{i}=\sqrt{\overline{\omega}_{i}}\Phi_{i-1},\qquad\tilde{A}^{\circ}\Phi_{i}=\overline{\alpha}_{i+1}\Phi_{i}$

for $i=0,1,\dots,d$, where
$\sqrt{\overline{\omega}_{d+1}}\Phi_{d+1}=\sqrt{\overline{\omega}_{0}}\Phi_{-1}:=0$,
and

$\displaystyle\overline{\omega}_{i}=\frac{c_{i}b_{i-1}}{\Sigma_{t}^{2}(A)},\qquad
i=1,2,\dots,d,\qquad\overline{\alpha}_{i}=\frac{a_{i-1}-tk}{\Sigma_{t}(A)},\qquad
i=1,2,\dots,d+1.$

We also define the scalar $\overline{\gamma}_{i}$ by (cf. (2.9))

$\displaystyle\overline{\gamma}_{i}=t^{i}\sqrt{k_{i}},\qquad i=0,1,\dots,d.$

Consider the following limits:

$\displaystyle\overline{\omega}_{i}\rightarrow\omega_{i},\qquad\overline{\alpha}_{i}\rightarrow\alpha_{i},\qquad
i=1,2,\dots,\qquad\overline{\gamma}_{i}\rightarrow\gamma_{i},\qquad
i=0,1,\dots.$

These limits do not necessarily exist in general, and we impose the following:

######  Assumption 2.3.

With the above situation, we assume that the limits $\omega_{i}$,
$\alpha_{i}$, and $\gamma_{i}$ exist and that $\omega_{i}>0$ for all $i$. We
note that $\gamma_{0}=1$.

With reference to Assumption 2.3, let $\mathcal{W}$ be an infinite-dimensional
$\mathbb{C}$-vector space with a fixed basis $\\{\Psi_{i}\colon
i=0,1,\dots\\}$, where we equip $\mathcal{W}$ with the Hermitian inner product
$\langle\cdot,\cdot\rangle$ for which the $\Psi_{i}$ are orthonormal. We
define the linear operators $B^{+}$, $B^{-}$, and $B^{\circ}$ on $\mathcal{W}$
by

$\displaystyle B^{+}\Psi_{i}=\sqrt{\omega_{i+1}}\Psi_{i+1},\qquad
B^{-}\Psi_{i}=\sqrt{\omega_{i}}\Psi_{i-1},\qquad
B^{\circ}\Psi_{i}=\alpha_{i+1}\Psi_{i}$

for $i=0,1,\dots$, where $\sqrt{\omega_{-1}}\Psi_{-1}:=0$. Note that $B^{+}$
and $B^{-}$ are adjoints of each other. The quadruple
$(\mathcal{W},\\{\Psi_{i}\\},B^{+},B^{-})$ is called the _interacting Fock
space_ (of one mode) _associated with Jacobi sequence_ $\\{\omega_{i}\\}$.

Recall the non-commutative algebra $\tilde{\mathcal{A}}(\Gamma)$ from Remark
2.2, and observe that $\tilde{A}^{+}$, $\tilde{A}^{-}$, and
$\tilde{A}^{\circ}$ generate $\tilde{\mathcal{A}}(\Gamma)$. We now extend the
domain of the Gibbs state $\varphi_{t}$ to $\tilde{\mathcal{A}}(\Gamma)$; cf.
(2.9). This extension is again independent of the base vertex $o\in X$, but it
should be remarked that it may fail to be a state on
$\tilde{\mathcal{A}}(\Gamma)$ (though it is indeed a state on
$\mathcal{A}(\Gamma)$ by Lemma 2.1). The QCLT in the Gibbs state is stated as
follows:

######  Theorem 2.4 ([19, Theorem 3.29]).

With reference to Assumption 2.3, we have

$\displaystyle\varphi_{t}\big{(}\tilde{A}^{\epsilon_{m}}\cdots\tilde{A}^{\epsilon_{1}}\big{)}\rightarrow\sum_{i=0}^{\infty}\gamma_{i}\langle\Psi_{i},B^{\epsilon_{m}}\cdots
B^{\epsilon_{1}}\Psi_{0}\rangle$

for any $\epsilon_{1},\dots,\epsilon_{m}\in\\{+,-,\circ\\}$ and $m=1,2,\dots$.

######  Remark 2.5.

There exists a Borel probability measure $\mu_{\infty}$ on $\mathbb{R}$ such
that (cf. (2.1))

$\displaystyle\sum_{i=0}^{\infty}\gamma_{i}\langle\Psi_{i},(B^{+}+B^{-}+B^{\circ})^{m}\Psi_{0}\rangle=\int_{-\infty}^{+\infty}\\!\\!\xi^{m}\mu_{\infty}({\rm
d}\xi),\qquad m=1,2,\dots.$

This $\mu_{\infty}$ is called the _asymptotic normalized spectral
distribution_ of $A$ in the Gibbs state, and we are interested in finding and
describing it. See Section 5.

We end this section with some comments. In [19, Section 3.4], Hora and Obata
also considered the case when $\omega_{m}=0$ for some $m$, so that the
probability measure $\mu_{\infty}$ above has finite support. However, if we
stick to the case when $\omega_{i}>0$ for all $i$ as in Assumption 2.3, then
assuming the existence of the $\gamma_{i}$ turns out to be redundant. To be
more precise, we show the following:

######  Proposition 2.6.

Suppose that the $\omega_{i}$ and the $\alpha_{i}$ exist and that
$\omega_{i}>0$ for all $i$. Then the $\gamma_{i}$ exist as well. In
particular, Theorem 2.4 holds true under this weaker assumption.

Proposition 2.6 is a consequence of Claims 2.8–2.11 below. For the rest of
this section, we assume the existence of the $\omega_{i}>0$ and that of the
$\alpha_{i}$.

First, observe that

$\displaystyle(1-t)(1+t+ta_{1})=\frac{\Sigma_{t}^{2}(A)}{k}=\frac{1}{\overline{\omega}_{1}}\rightarrow\frac{1}{\omega_{1}}>0.$
(2.13)

Since $a_{0}=0$, the existence of $\gamma_{1}$ follows from that of
$\alpha_{1}$ and (2.13): $\gamma_{1}=-\alpha_{1}/\sqrt{\omega_{1}}$. Note that
if $k=2$ then $\Gamma$ is either the $2d$-cycle or the $(2d+1)$-cycle. Bang,
Dubickas, Koolen, and Moulton [4] proved the _Bannai–Ito conjecture_ :

######  Theorem 2.7 ([4]).

There exist only finitely many distance-regular graphs for each fixed degree
$k\geqslant 3$.

Since we are letting $d\rightarrow\infty$ (cf. (2.11)), it follows that

######  Claim 2.8.

If $\xi$ is an accumulation point of $1/k\in(0,1/2]$, then
$\xi\in\\{0,1/2\\}$.

We next handle each of the two possible accumulation points of $1/k$.

######  Claim 2.9.

Suppose that $1/2$ is an accumulation point of $1/k$, and consider a subnet of
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which $k=2$ eventually. Then the
$\gamma_{i}$ exist on this subnet. Moreover, we have $\omega_{i}=\omega_{1}/2$
and $\alpha_{i}=\alpha_{1}$ for $i=2,3,\dots$.

###### Proof.

Recall that $\gamma_{1}$ exists. Since $k=2$ eventually, this means that $t$
is convergent on this subnet. For the cycles, we have $k_{i}=2$,
$i=1,2,\dots,d-1$. Since $d\rightarrow\infty$, it follows that the
$\gamma_{i}$ all exist on this subnet. The last statement also follows from
(2.13) and since the cycles satisfy $(a_{i},b_{i},c_{i})=(0,1,1)$ for
$i=1,2,\dots,d-1$. ∎

######  Claim 2.10.

Suppose that $0$ is an accumulation point of $1/k$, and consider a subnet of
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which $k\rightarrow\infty$. Then
the $\gamma_{i}$ exist on this subnet. Moreover, we have
$a_{i}/\sqrt{k}\rightarrow\alpha_{i+1}/\sqrt{\omega_{1}}+\gamma_{1}$,
$b_{i}/k\rightarrow 1$, and $c_{i}\rightarrow\omega_{i}/\omega_{1}$ for
$i=1,2,\dots$ on this subnet.

###### Proof.

That $a_{i}/\sqrt{k}\rightarrow\alpha_{i+1}/\sqrt{\omega_{1}}+\gamma_{1}$,
$i=1,2,\dots$, is immediate from (2.13). We next show that $b_{i}/k\rightarrow
1$ and $c_{i}\rightarrow\omega_{i}/\omega_{1}$ for $i=1,2,\dots$ on this
subnet. Suppose by induction that $c_{i}\rightarrow\omega_{i}/\omega_{1}$ for
some $i$. We have $a_{i}=o(k)$ and $c_{i}=o(k)$ since $k\rightarrow\infty$,
and hence $b_{i}/k\rightarrow 1$ by (2.3). Then it follows in turn that
$c_{i+1}\sim\overline{\omega_{i+1}}/\overline{\omega_{1}}\rightarrow\omega_{i+1}/\omega_{1}$.
The existence of the $\gamma_{i}$ on this subnet now follows from these
comments, (2.4), and the existence of $\gamma_{1}$. ∎

Finally, we show that the above two cases do not coexist.

######  Claim 2.11.

There exists exactly one accumulation point of $1/k\in(0,1/2]$. More
precisely, we have either $k=2$ eventually, or $k\rightarrow\infty$.

###### Proof.

In view of Claim 2.8, suppose on the contrary that both $0$ and $1/2$ are
accumulation points of $1/k$. On the one hand, we have
$\omega_{i}=\omega_{1}/2$, $i=2,3,\dots$, by Claim 2.9. On the other hand, we
have $\omega_{i}\geqslant\omega_{1}$, $i=2,3,\dots$, by Claim 2.10 and since
$c_{i}\geqslant 1$. This is a contradiction, and the result follows. ∎

###### Proof of Proposition 2.6.

Immediate from Claims 2.8–2.11. ∎

The following is another important consequence of the above discussions:

######  Claim 2.12.

Each of the $c_{i}$ is eventually constant.

###### Proof.

Follows from Claims 2.8–2.11 and since the $c_{i}$ are integers. ∎

######  Remark 2.13.

In [19, Chapter 7], Hora and Obata extended the method of the quantum
decomposition and the QCLT to more general growing regular graphs, and
obtained some sufficient conditions for the theorem to hold. See also [20]. In
particular, these conditions can be applied to Cayley graphs on Coxeter
groups, such as the symmetric groups. For distance-regular graphs, these
conditions turn out to be reduced to the following (besides that
$\Sigma_{t}^{2}(A)>0$): (i) $k\rightarrow\infty$; (ii) each of the $c_{i}$ is
eventually constant; (iii) the $a_{i}/\sqrt{k}$ are convergent; (iv)
$\gamma_{1}$ exists. See [19, Theorem 7.14 and Proposition 7.17]. Therefore,
if we focus only on distance-regular graphs with $k\geqslant 3$, then it
follows from Claims 2.8–2.12 that these sufficient conditions are in fact
equivalent to Assumption 2.3 (or the existence of the $\omega_{i}>0$ and that
of the $\alpha_{i}$, by virtue of Proposition 2.6).

##  3 Distance-regular graphs with classical parameters

A distance-regular graph $\Gamma$ with diameter $d$ is said to have _classical
parameters_ $(d,q,\alpha,\beta)$ (cf. [8, Section 6.1]) whenever the $b_{i}$
and the $c_{i}$ are expressed as

$\displaystyle
b_{i}=\left(\genfrac{[}{]}{0.0pt}{}{d}{1}-\genfrac{[}{]}{0.0pt}{}{i}{1}\right)\\!\\!\left(\beta-\alpha\genfrac{[}{]}{0.0pt}{}{i}{1}\right),\qquad
c_{i}=\genfrac{[}{]}{0.0pt}{}{i}{1}\\!\left(1+\alpha\genfrac{[}{]}{0.0pt}{}{i-1}{1}\right)$
(3.1)

for $i=0,1,\dots,d$, where

$\displaystyle\genfrac{[}{]}{0.0pt}{}{i}{1}=1+q+q^{2}+\dots+q^{i-1}$

is a Gaussian binomial coefficient. We call $q$ the _base_. In particular,

$\displaystyle k=b_{0}=\genfrac{[}{]}{0.0pt}{}{d}{1}\beta,$ (3.2)

and by (2.3) we have

$\displaystyle
a_{i}=\genfrac{[}{]}{0.0pt}{}{i}{1}\\!\left(\beta-1+\alpha\left(\genfrac{[}{]}{0.0pt}{}{d}{1}-\genfrac{[}{]}{0.0pt}{}{i}{1}-\genfrac{[}{]}{0.0pt}{}{i-1}{1}\right)\\!\right),\qquad
i=0,1,\dots,d.$ (3.3)

It is known (see [8, Proposition 6.2.1]) that

$\displaystyle q\in\mathbb{Z}\setminus\\{0,-1\\}\qquad\text{if}\quad
d\geqslant 3.$

As mentioned in Section 1, all the graphs with $q=1$ are known, and the QCLTs
for them have been obtained, so our aim in this paper is to discuss the case
where $q\in\\{\pm 2,\pm 3,\dots\\}$.

Suppose that $\Gamma$ has classical parameters $(d,q,\alpha,\beta)$. By [8,
Corollary 8.4.2], the $d+1$ distinct eigenvalues of $\Gamma$ are given by

$\displaystyle\theta_{i}=\frac{b_{i}}{q^{i}}-\genfrac{[}{]}{0.0pt}{}{i}{1}=\genfrac{[}{]}{0.0pt}{}{d-i}{1}\\!\left(\beta-\alpha\genfrac{[}{]}{0.0pt}{}{i}{1}\right)-\genfrac{[}{]}{0.0pt}{}{i}{1},\qquad
i=0,1,\dots,d.$ (3.4)

For $i=0,1,\dots,d$, let $E_{i}$ denote the orthogonal projection onto the
eigenspace of $A$ for $\theta_{i}$. The $E_{i}$ are polynomials in $A$, and we
have

$\displaystyle\mathcal{A}(\Gamma)=\operatorname{span}\\{E_{0},E_{1},\dots,E_{d}\\}.$
(3.5)

Note by (3.2) that $\theta_{0}=k$. Since $\Gamma$ is regular and connected, it
follows that (cf. [8, p. 45])

$\displaystyle E_{0}=\frac{1}{|X|}J,$ (3.6)

where $J$ denotes the all-ones matrix.

Recall the set $\pi(\Gamma)$ from (2.7). It seems to be a difficult problem to
determine $\pi(\Gamma)$ in general. For the Hamming graphs and the Johnson
graphs, which have classical parameters with $q=1$, it is shown (see [19,
Propositions 5.16 and 6.27]) that this set contains the interval $[0,1]$, as
consequences of Bożejko’s quadratic embedding test; cf. [19, Proposition
2.14]. For the case $q\neq 1$, the following result again finds infinitely
many elements of $\pi(\Gamma)$:

######  Proposition 3.1.

Suppose that $\Gamma$ has classical parameters $(d,q,\alpha,\beta)$ with
$d\geqslant 3$ and $q\in\\{\pm 2,\pm 3,\dots\\}$. Then $q^{-i}\in\pi(\Gamma)$
for $i=0,1,2,\dots$.

###### Proof.

We already mentioned that $q^{0}=1\in\pi(\Gamma)$. By [8, Corollary 8.4.2],
the first projection matrix $E_{1}$ is of the form

$\displaystyle E_{1}=\frac{1}{|X|}\sum_{i=0}^{d}\big{(}\zeta+\eta
q^{-i}\big{)}A_{i}=\zeta E_{0}+\frac{\eta}{|X|}K_{q^{-1}}$

for some $\zeta,\eta\in\mathbb{R}$, where we have used (3.6). It is customary
to write444The $*$-notation here is used to mean “dual” objects, and is
standard in the theory of distance-regular graphs. The $\theta_{i}^{*}$ are
referred to as the _dual eigenvalues_ of $\Gamma$. $\theta_{i}^{*}=\zeta+\eta
q^{-i}$, $i=0,1,\dots,d$. Note that
$\theta_{0}^{*}=\operatorname{tr}(E_{1})=m_{1}$, the multiplicity of
$\theta_{1}$ in the spectrum of $\Gamma$. It is known (see [8, Lemma 2.2.1])
that $|\theta_{i}^{*}|\leqslant m_{1}$ for $i=0,1,\dots,d$. We have $\eta\neq
0$, for otherwise $E_{1}$ would be a scalar multiple of $E_{0}$, a
contradiction. If $\eta<0$ then $\eta q^{-1}>\eta$, so that
$\theta_{1}^{*}>m_{1}$, again a contradiction. Hence $\eta>0$.

We next show that $\zeta\leqslant 0$. If $\zeta>0$ and $q\geqslant 2$ then
$E_{1}$ would be a non-zero non-negative matrix and thus
$\operatorname{tr}(E_{0}E_{1})>0$ by (3.6), which is absurd. Hence suppose
that $q\leqslant-2$. We observe that $\zeta\leqslant 0$ if and only if
$\theta_{1}^{*}\leqslant m_{1}/q$. By [8, Lemma 2.2.1], we have
$\theta_{1}/k=\theta_{1}^{*}/m_{1}$. Using this, (3.2), (3.3), and (3.4), we
easily verify that $\zeta\leqslant 0$ if and only if $\theta_{1}\leqslant k/q$
if and only if $a_{1}+q+1\leqslant 0$. A _kite of length two_ in $\Gamma$ is a
quadruple $(x,y,z,w)$ of vertices such that $x\sim y\sim w$, $x\sim z\sim w$,
$y\sim z$, and $x\not\sim w$:

$x$$y$$z$$w$

A kite of length two is also called a _parallelogram of length two_. By [37,
Theorem 2.12], $\Gamma$ has no kite of length two. By [43, Lemma 3.6], we then
have $a_{1}+q+1\leqslant 0$. It follows that $\zeta\leqslant 0$.

Since

$\displaystyle K_{q^{-1}}=\frac{|X|}{\eta}(E_{1}-\zeta E_{0}),$

it follows that $K_{q^{-1}}$ is positive semidefinite, i.e.,
$q^{-1}\in\pi(\Gamma)$. For $i=2,3,\dots$, we observe that the matrix
$K_{q^{-i}}$ is a principal submatrix of $(K_{q^{-1}})^{\otimes i}$ since
$(K_{q^{-i}})_{x,y}=((K_{q^{-1}})_{x,y})^{i}$ for all $x,y\in X$, and it is
therefore positive semidefinite as well. This completes the proof. ∎

We comment on the uniqueness of the classical parameters for $\Gamma$. By [8,
Corollary 6.2.2], the classical parameters $(d,q,\alpha,\beta)$ for $\Gamma$
are uniquely determined provided that $d\geqslant 3$, with the exception of
the pairs

$\displaystyle\big{(}d,\ell^{2},0,\ell\big{)},\qquad\bigg{(}d,-\ell,\frac{\ell(\ell+1)}{1-\ell},\frac{\ell(1+(-\ell)^{d})}{1-\ell}\bigg{)},$
(3.7)

where $\ell\geqslant 2$. Ivanov and Shpectorov [22] showed that if $\Gamma$
has the above classical parameters then $\ell$ is a prime power and $\Gamma$
is the Hermitian dual polar graph ${}^{2}\\!A_{2d-1}(\ell)$ (cf. [8, Section
9.4]).

######  Assumption 3.2.

Recall Assumption 2.3. We moreover assume that the graph
$\Gamma=\Gamma_{\lambda}$ has classical parameters $(d,q,\alpha,\beta)$ with
$d\geqslant 3$ and $q\in\\{\pm 2,\pm 3,\dots\\}$. We will view $q$, $\alpha$,
and $\beta$ as functions of $\Gamma$. For the classical parameters in (3.7),
we understand that we may choose either set of them.

Recall that we are assuming that $d\rightarrow\infty$; cf. (2.11). Our goal is
to describe the limit behaviors of the other parameters $q$, $\alpha$, and
$\beta$. In the following discussions, we will freely use the expressions
(3.1), (3.2), and (3.3). In particular, we note that

$\displaystyle\alpha=\frac{c_{2}}{q+1}-1\in\frac{1}{q+1}\mathbb{Z}.$ (3.8)

The cycles ($k=2$) with $d\geqslant 3$ do not have classical parameters, so
that it follows from Claim 2.11 that

$\displaystyle k\rightarrow\infty.$ (3.9)

######  Claim 3.3.

With reference to Assumption 3.2, $\limsup|q|<\infty$. In particular, $q$
eventually takes only finitely many values.

###### Proof.

Suppose that $\limsup|q|=\infty$, or equivalently, $0$ is an accumulation
point of $1/q$. Then there exists a subnet of
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which $|q|\rightarrow\infty$. By
Claim 2.12, $c_{2}$ is eventually constant, so that it follows from (3.8) that
$\alpha\rightarrow-1$ on this subnet. This in turn implies that
$c_{3}\rightarrow\infty$ on this subnet, but this is impossible since $c_{3}$
is also eventually constant by Claim 2.12. It follows that
$\limsup|q|<\infty$. ∎

######  Claim 3.4.

With reference to Assumption 3.2, suppose that $q$ is not convergent. Then the
set of accumulation points of $q$ is of the form
$\big{\\{}\ell,\ell^{2}\big{\\}}$ or $\big{\\{}\ell,-\ell,\ell^{2}\big{\\}}$
for some $\ell\in\\{\pm 2,\pm 3,\dots\\}$, where $\alpha\neq 0$ when
$q=\pm\ell$, and $\alpha=0$ when $q=\ell^{2}$. Moreover, we have
$a_{i}/\sqrt{k}\rightarrow 0$ for every $i=1,2,\dots$.

###### Proof.

Recall Claim 3.3. Since $c_{2}$ is eventually constant by Claim 2.12, it
follows from (3.8) that $\alpha$ is eventually determined by $q$. We have

$\displaystyle\frac{c_{i+1}}{c_{i}}=\frac{q^{i+1}-1}{q^{i}-1}\cdot\frac{q-1+\alpha(q^{i}-1)}{q-1+\alpha(q^{i-1}-1)},\qquad
i=1,2,\dots.$

For sufficiently large $i$ (cf. (2.11)), the RHS can be arbitrarily close to
$q^{2}$ when $\alpha\neq 0$ and $q$ when $\alpha=0$. Let $\ell$ and
$\ell^{\prime}$ be two distinct accumulation points of $q$, where
$|\ell^{\prime}|\geqslant|\ell|(\geqslant 2)$. Since the LHS above is
eventually constant for every $i$ by Claim 2.12, it follows that
$\ell^{\prime}\in\big{\\{}{-}\ell,\ell^{2}\big{\\}}$, where $\alpha\neq 0$
when $q=\pm\ell$, and $\alpha=0$ when $q=\ell^{2}$.

We next show that $a_{i}/\sqrt{k}\rightarrow 0$ for every $i$. Recall that
$k\rightarrow\infty$ (cf. (3.9)). The $a_{i}/\sqrt{k}$ are convergent by
Claims 2.10 and 2.11. Suppose that $a_{i}/\sqrt{k}\nrightarrow 0$ for some
$i$. Then we have $a_{i}\rightarrow\infty$ for this $i$, and since $q$ and
$\alpha$ are eventually bounded, it follows that
$\bigl{|}\beta+\alpha\genfrac{[}{]}{0.0pt}{}{d}{1}\bigr{|}=\Theta(a_{i})\rightarrow\infty$,
and hence that
$a_{j}\sim\genfrac{[}{]}{0.0pt}{}{j}{1}\bigl{(}\beta+\alpha\genfrac{[}{]}{0.0pt}{}{d}{1}\bigr{)}\sim\genfrac{[}{]}{0.0pt}{}{j}{1}a_{1}$
for all $j$. On the one hand, this shows that $a_{1}/\sqrt{k}\nrightarrow 0$.
On the other hand, this also shows that $a_{j}/\sqrt{k}$ cannot converge
whenever $j\geqslant 2$ since $\genfrac{[}{]}{0.0pt}{}{j}{1}$ takes at least
two values depending on $q$. Hence we must have $a_{i}/\sqrt{k}\rightarrow 0$
for every $i$, as desired.

It remains to show that $\ell^{2}$ is an accumulation point of $q$. There is
nothing to prove if $\ell^{\prime}=\ell^{2}$, so that we assume that
$\ell^{\prime}=-\ell$. We have $c_{3}=\big{(}q^{2}+q+1\big{)}(c_{2}-q)$ by
(3.8). By setting $q=\pm\ell$ in this expression and then equating, we find
that eventually $c_{2}=\ell^{2}+1$. Choose $q\in\\{\ell,-\ell\\}$ with $q<0$,
and recall that $\alpha\neq 0$ in this case. In particular, we have
$a_{2}\neq\genfrac{[}{]}{0.0pt}{}{2}{1}a_{1}$. Suppose that $a_{1}=0$. Then
$a_{2}\neq 0$, and we have $c_{2}\leqslant 2$ by [30, Theorem 2.1], but this
is impossible since $c_{2}=\ell^{2}+1\geqslant 5$. Hence $a_{1}\neq 0$. Then
it follows from [44, Main Theorem] that, provided that $d\geqslant 4$, either
(i) $\Gamma$ is the dual polar graph ${}^{2}\\!A_{2d-1}(-q)$ with
$\alpha=q(q-1)/(q+1)$ (cf. [8, Section 9.4]), or (ii) $\Gamma$ is the
Hermitian forms graph $\operatorname{Her}\big{(}d,q^{2}\big{)}$ with
$\alpha=q-1$ (cf. [8, Section 9.5]), or (iii) we have $\alpha=(q-1)/2$ and
$\beta=-\big{(}q^{d}+1\big{)}/2$. Since $c_{2}=q^{2}+1$, it follows from (3.8)
that we are in (i) above. However, the graph ${}^{2}\\!A_{2d-1}(-q)$ has
another set of classical parameters $\big{(}d,q^{2},0,-q\big{)}$ (cf. (3.7)),
and therefore $\ell^{2}=q^{2}$ must also be an accumulation point. ∎

######  Theorem 3.5.

With reference to Assumption 3.2, $q$ eventually takes at most three values.
Suppose that $q$ is eventually constant. Then so is $\alpha$, and the
following hold:

  1. $(i)$

If $\alpha\neq 0$, then $\beta/\sqrt{k}$ is eventually bounded, and there
exist scalars $\gamma$ and $\rho$ with $\rho>0$ and
$\gamma(\rho+\alpha/\rho)>-1$, such that $t\sqrt{k}\rightarrow\gamma$ and the
accumulation points of $\beta/\sqrt{k}$ are in $\\{\rho,\alpha/\rho\\}$.
Moreover, we have $\rho=\sqrt{-\alpha}$ if $q<0$.

  2. $(ii)$

If $\alpha=0$, then there exist scalars $\gamma$ and $\rho$ with
$\rho\geqslant 0$ and $\gamma\rho>-1$, such that $t\sqrt{k}\rightarrow\gamma$
and $\beta/\sqrt{k}\rightarrow\rho$.

Suppose that $q$ is not convergent. Then there exists a subnet of
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which $q$ is eventually constant
and (ii) holds above with $\rho=0$.

Conversely, if $(\Gamma_{\lambda})_{\lambda\in\Lambda}$ is a net of distance-
regular graphs having classical parameters with $d\geqslant 3$ and $q\in\\{\pm
2,\pm 3,\dots\\}$, where $q$ and $\alpha$ are eventually constant, such that
$d\rightarrow\infty$ and $(i)$ or $(ii)$ holds above with respect to a
suitable function $t\in\pi(\Gamma)$ with $\Sigma_{t}^{2}(A)>0$, then
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ satisfies Assumption 2.3 $($and thus
Assumption 3.2 as well$)$.

###### Proof.

The first statement follows from Claims 3.3 and 3.4. We also mentioned earlier
(cf. (3.9)) that $k\rightarrow\infty$.

Suppose that $q$ is eventually constant. That $\alpha$ is eventually constant
follows from Claim 2.12 and (3.8). Set $\gamma=\gamma_{1}$. Since it exists,
we have $t\rightarrow 0$. Recall again that the $a_{i}/\sqrt{k}$ are
convergent by Claims 2.10 and 2.11, and observe that this is equivalent to
saying that
$\bigr{(}\beta+\alpha\genfrac{[}{]}{0.0pt}{}{d}{1}\bigr{)}/\sqrt{k}$
converges, say, to $\sigma$. Assume that $\alpha\neq 0$, and let $\xi=\rho$ be
a root of the equation $\xi+\alpha/\xi=\sigma$ in the variable $\xi$. Then the
other root is $\xi=\alpha/\rho$. Since $\beta/\sqrt{k}$ and
$\genfrac{[}{]}{0.0pt}{}{d}{1}/\sqrt{k}$ are inverses of each other, it
follows that $\beta/\sqrt{k}$ is eventually bounded, and that $\rho$ and
$\alpha/\rho$ are its only possible accumulation points. If $q>0$ then
$\alpha>0$ and $\beta>0$, so that we must have $\rho>0$. If $q<0$ then
$\alpha<0$, and since
$a_{i}/\sqrt{k}\rightarrow\genfrac{[}{]}{0.0pt}{}{i}{1}\sigma$ for every $i$
and the $\genfrac{[}{]}{0.0pt}{}{i}{1}$ alternate in sign, it follows that
$\sigma=0$, so that we may take $\rho=\sqrt{-\alpha}>0$ (and thus
$\alpha/\rho=-\sqrt{-\alpha}<0$). By (2.13) and since $ta_{1}=t\sqrt{k}\cdot
a_{1}/\sqrt{k}\rightarrow\gamma\sigma=\gamma(\rho+\alpha/\rho)$, we have
$\gamma(\rho+\alpha/\rho)>-1$. Assume next that $\alpha=0$. We have $q>0$ and
$\beta>0$ in this case, and set $\rho=\sigma\geqslant 0$.

Suppose that $q$ is not convergent, and let the integer $\ell$ be as in Claim
3.4. Then $\ell^{2}$ is an accumulation point of $q$, so that there is a
subnet of $(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which eventually
$q=\ell^{2}$. Recall by Claim 3.4 that eventually $\alpha=0$, and that
$a_{i}/\sqrt{k}\rightarrow 0$ for every $i$. Hence we are in the second case
above with $\rho=\sigma=0$.

Finally, let $(\Gamma_{\lambda})_{\lambda\in\Lambda}$ be a net of distance-
regular graphs as described in the last paragraph of the theorem. Note that
$k\geqslant 3$ since the cycles with $d\geqslant 3$ do not have classical
parameters. Hence it follows from Theorem 2.7 that $k\rightarrow\infty$. Since
$t\sqrt{k}\rightarrow\gamma$, we then have $t\rightarrow 0$. If $\alpha\neq 0$
then we must also have $|\beta|\rightarrow\infty$ since $\rho$ and
$\alpha/\rho$ are non-zero. Observe that
$a_{i}/\sqrt{k}\rightarrow\genfrac{[}{]}{0.0pt}{}{i}{1}(\rho+\alpha/\rho)$ for
every $i$, where we set $0/0:=0$ for brevity. In particular, we have
$ta_{1}\rightarrow\gamma(\rho+\alpha/\rho)>-1$, from which it follows that
$\omega_{1}$ exists and is positive. It is now immediate to verify that the
$\omega_{i}$ all exist and are positive, and that the $\alpha_{i}$ exist. From
Proposition 2.6 it also follows that the $\gamma_{i}$ exist. ∎

Consider the case when $q$ is eventually constant in Theorem 3.5, and recall
that so is $\alpha$ in this case. Recall (cf. (3.1)) also the formula for the
$c_{i}$. The scalars $\omega_{i}$, $\alpha_{i}$, and $\gamma_{i}$ are
expressed in terms of $q$, $\alpha$, and the two scalars $\gamma$ and $\rho$
in Theorem 3.5 $(i)$ and $(ii)$ as

$\displaystyle\omega_{i}=\frac{c_{i}}{1+\gamma(\rho+\alpha/\rho)},\qquad\alpha_{i}=\frac{\genfrac{[}{]}{0.0pt}{}{i-1}{1}(\rho+\alpha/\rho)-\gamma}{\sqrt{1+\gamma(\rho+\alpha/\rho)}},\qquad
i=1,2,\dots,$

and

$\displaystyle\gamma_{i}=\frac{\gamma^{i}}{\sqrt{c_{i}\cdots c_{1}}},\qquad
i=0,1,\dots,$

where we set $0/0:=0$ and $0^{0}:=1$.

##  4 More background on graphs with classical parameters

In order to describe the asymptotic normalized spectral distributions
corresponding to Theorem 3.5 $(i)$ and $(ii)$, we collect in this section
necessary formulas for distance-regular graphs with classical parameters.
Thus, throughout this section, we let $\Gamma=(X,R)$ denote a (fixed)
distance-regular graph with classical parameters $(d,q,\alpha,\beta)$, where
$d\geqslant 3$ and $q\in\\{\pm 2,\pm 3,\dots\\}$.

Recall the eigenvalues $\theta_{i}$, $i=0,1,\dots,d$, of $\Gamma$ and the
corresponding orthogonal projections $E_{i}$, $i=0,1,\dots,d$; cf. (3.4) and
(3.5). It is known (see [8, Corollary 8.4.2]) that the ordering
$(E_{0},E_{1},\dots,E_{d})$ is $Q$-_polynomial_ ; that is to say, there exist
scalars555See footnote 4. $a_{i}^{*},b_{i}^{*},c_{i}^{*}\in\mathbb{R}$,
$i=0,1,\dots,d$, such that $b_{d}^{*}=c_{0}^{*}=0$, $b_{i-1}^{*}c_{i}^{*}\neq
0$, $i=1,2,\dots,d$, and that

$\displaystyle E_{1}\circ
E_{i}=\frac{1}{|X|}(b_{i-1}^{*}E_{i-1}+a_{i}^{*}E_{i}+c_{i+1}^{*}E_{i+1}),\qquad
i=0,1,\dots,d,$ (4.1)

where $\circ$ denotes the entrywise (or _Hadamard_ or _Schur_) product of
matrices, and $b_{-1}^{*}E_{-1}=c_{d+1}^{*}E_{d+1}:=0$. Note that this
property is dual to (2.2). See [10, Section 5] for recent updates on the study
of $Q$-polynomial distance-regular graphs.

From (2.2) it follows that there exist polynomials
$v_{i}(\xi)\in\mathbb{R}[\xi]$, $i=0,1,\dots,d$, such that $\deg v_{i}(\xi)=i$
and $A_{i}=v_{i}(A)$. Set $u_{i}(\xi)=v_{i}(\xi)/k_{i}$, $i=0,1,\dots,d$. In
general, by Leonard’s theorem (see [26], [5, Section III.5]), the polynomials
$u_{i}$ associated with every $Q$-polynomial distance-regular graph are
expressed in terms of the $q$-_Racah polynomials_ (cf. [25, Section 3.2]) and
their special/limit cases in the _Askey scheme_ of (basic) hypergeometric
orthogonal polynomials [24, 25]. In the most general (i.e., $q$-Racah) case,
the $u_{i}$ are of the form

$\displaystyle
u_{i}(\theta_{j})={}_{4}\phi_{3}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,s^{*}q^{i+1},\,q^{-j},\,sq^{j+1}\\\
r_{1}q,\,r_{2}q,\,q^{-d}\end{array}\\!\\!\right|q;q\right)\\!,\qquad
i,j=0,1,\dots,d,$

where the parameters $r_{1}$, $r_{2},s$, and $s^{*}$ satisfy
$r_{1}r_{2}=ss^{*}q^{d+1}\neq 0$, and we are using the standard notation for a
basic hypergeometric series ${}_{m}\phi_{n}$:

$\displaystyle{}_{m}\phi_{n}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}\mathfrak{a}_{1},\dots,\mathfrak{a}_{m}\\\
\mathfrak{b}_{1},\dots,\mathfrak{b}_{n}\end{array}\\!\\!\right|q;\mathfrak{x}\right)=\sum_{h=0}^{\infty}\frac{(\mathfrak{a}_{1};q)_{h}\cdots(\mathfrak{a}_{m};q)_{h}}{(\mathfrak{b}_{1};q)_{h}\cdots(\mathfrak{b}_{n};q)_{h}}\,\frac{(-1)^{(m-n-1)h}\mathfrak{x}^{h}}{(q;q)_{h}q^{(m-n-1)\binom{h}{2}}},$

where $(\mathfrak{a};q)_{h}$ denotes the $q$-shifted factorial defined by

$\displaystyle(\mathfrak{a};q)_{h}=(1-\mathfrak{a})(1-\mathfrak{a}q)\cdots\big{(}1-\mathfrak{a}q^{h-1}\big{)},\qquad
h=0,1,\dots.$

To get the $u_{i}$ for our $\Gamma$, first fix $s,r_{2}\neq 0$ and let
$s^{*}\rightarrow 0$ (so $r_{1}\rightarrow 0$), and then set

$\displaystyle s=\frac{\alpha+1-q}{(\alpha-\beta+\beta q)q^{d+1}},\qquad
r_{2}=\frac{\alpha}{(\alpha-\beta+\beta q)q},$ (4.2)

or equivalently,

$\displaystyle\alpha=\frac{r_{2}(1-q)}{sq^{d}-r_{2}},\qquad\beta=\frac{r_{2}q-1}{q(sq^{d}-r_{2})}.$

See [32, Proposition 6.2]. The $u_{i}$ are the _dual $q$-Hahn polynomials_
(cf. [25, Section 3.7]) for $s\neq 0$ and $r_{2}\neq 0$, the _affine
$q$-Krawtchouk polynomials_ (cf. [25, Section 3.16]) for $s=0$ and $r_{2}\neq
0$, and the _dual $q$-Krawtchouk polynomials_ (cf. [25, Section 3.17]) for
$s\neq 0$ and $r_{2}=0$. See also [40, Examples 5.3–5.9]. We note that, in
[32, Proposition 6.2], there is mentioned another case, called IA, which also
corresponds to classical parameters with $q\neq 1$. The $u_{i}$ are then the
_quantum $q$-Krawtchouk polynomials_ (cf. [25, Section 3.14]). However, it is
known (see [10, Proposition 5.8]) that there exists no actual $\Gamma$ in this
case.

For later use, we now establish another basic hypergeometric expression for
the polynomials $u_{i}$. For the moment, fix $i,j=0,1,\dots,d$. Recall Sear’s
transformation formula for a terminating balanced ${}_{4}\phi_{3}$ series (cf.
[25, Section 0.6]):

$\displaystyle{}_{4}\phi_{3}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,\mathfrak{x},\,\mathfrak{y},\,\mathfrak{z}\\\
\mathfrak{l},\,\mathfrak{m},\,\mathfrak{n}\end{array}\\!\\!\right|q;q\right)={}_{4}\phi_{3}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,\mathfrak{x},\,\mathfrak{l}/\mathfrak{y},\,\mathfrak{l}/\mathfrak{z}\\\
\mathfrak{l},\,\mathfrak{x}q^{1-i}/\mathfrak{m},\,\mathfrak{x}q^{1-i}/\mathfrak{n}\end{array}\\!\\!\right|q;q\right)\frac{(\mathfrak{m}/\mathfrak{x};q)_{i}(\mathfrak{n}/\mathfrak{x};q)_{i}}{(\mathfrak{m};q)_{i}(\mathfrak{n};q)_{i}}\mathfrak{x}^{i},$

where $\mathfrak{xyz}q^{1-i}=\mathfrak{lmn}\neq 0$. Applying this formula
twice and then simplifying a bit, we obtain

$\displaystyle{}_{4}\phi_{3}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,\mathfrak{x},\,\mathfrak{y},\,\mathfrak{z}\\\
\mathfrak{l},\,\mathfrak{m},\,\mathfrak{n}\end{array}\\!\\!\right|q;q\right)={}_{4}\phi_{3}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,\mathfrak{x},\,\mathfrak{xy}q^{1-i}/\mathfrak{ln},\,\mathfrak{xz}q^{1-i}/\mathfrak{ln}\\\
\mathfrak{x}q^{1-i}/\mathfrak{n},\,\mathfrak{x}q^{1-i}/\mathfrak{l},\,\mathfrak{m}\end{array}\\!\\!\right|q;q\right)\frac{(\mathfrak{n}/\mathfrak{x};q)_{i}(\mathfrak{l}/\mathfrak{x};q)_{i}}{(\mathfrak{n};q)_{i}(\mathfrak{l};q)_{i}}\mathfrak{x}^{i}.$

Set

$\displaystyle\mathfrak{x}=q^{-j},\qquad\mathfrak{y}=s^{*}q^{i+1},\qquad\mathfrak{z}=sq^{j+1},\qquad\mathfrak{l}=q^{-d},\qquad\mathfrak{m}=r_{1}q,\qquad\mathfrak{n}=r_{2}q$

in this result. Then we obtain the following expression for the $u_{i}$ for
the $q$-Racah case:

$\displaystyle
u_{i}(\theta_{j})={}_{4}\phi_{3}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,q^{-j},\,s^{*}q^{d-j+1}/r_{2},\,sq^{d-i+1}/r_{2}\\\
q^{-i-j}/r_{2},\,q^{d-i-j+1},\,r_{1}q\end{array}\\!\\!\right|q;q\right)\frac{(r_{2}q^{j+1};q)_{i}(q^{j-d};q)_{i}}{(r_{2}q;q)_{i}(q^{-d};q)_{i}q^{ij}}.$

By letting $s^{*}\rightarrow 0$, the RHS becomes

$\displaystyle{}_{3}\phi_{2}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-i},\,q^{-j},\,sq^{d-i+1}/r_{2}\\\
q^{-i-j}/r_{2},\,q^{d-i-j+1}\end{array}\\!\\!\right|q;q\right)\frac{(r_{2}q^{j+1};q)_{i}(q^{j-d};q)_{i}}{(r_{2}q;q)_{i}(q^{-d};q)_{i}q^{ij}}$
$\displaystyle\qquad{}=\sum_{h=0}^{i}\frac{(q^{-i};q)_{h}(q^{-j};q)_{h}(r_{2};sq^{d-i+1};q)_{h}q^{h}}{(r_{2};q^{-i-j};q)_{h}(q^{d-i-j+1};q)_{h}(q;q)_{h}}\,\frac{(r_{2}q^{j+1};q)_{i}(q^{j-d};q)_{i}}{(r_{2}q;q)_{i}(q^{-d};q)_{i}q^{ij}},$

where we write

$\displaystyle(\mathfrak{x};\mathfrak{y};q)_{h}=(\mathfrak{x}-\mathfrak{y})(\mathfrak{x}-\mathfrak{y}q)\cdots\big{(}\mathfrak{x}-\mathfrak{y}q^{h-1}\big{)},\qquad
h=0,1,\dots$

for convenience. We then set $s$ and $r_{2}$ as in (4.2), and obtain the
$u_{i}$ for $\Gamma$ as follows:

$\displaystyle
u_{i}(\theta_{j})=\sum_{h=0}^{i}\frac{(q^{-i};q)_{h}(q^{-j};q)_{h}(\alpha;(\alpha+1-q)q^{1-i};q)_{h}q^{h}}{(\alpha;(\alpha-\beta+\beta
q)q^{1-i-j};q)_{h}(q^{d-i-j+1};q)_{h}(q;q)_{h}}\frac{(\alpha\\!-\beta+\beta
q;\alpha q^{j};q)_{i}(q^{j-d};q)_{i}}{(\alpha\\!-\beta+\beta
q;\alpha;q)_{i}(q^{-d};q)_{i}q^{ij}}.$

Using (2.4) and (3.1) we have

$\displaystyle k_{i}=\frac{(q^{-d};q)_{i}(\alpha-\beta+\beta
q;\alpha;q)_{i}q^{di}}{(\alpha+1-q;\alpha;q)_{i}(q;q)_{i}},$

from which it follows that

$\displaystyle
v_{i}(\theta_{j})=k_{i}u_{i}(\theta_{j})=\sum_{h=0}^{i}\frac{(q^{-j};q)_{h}(q^{j-d};q)_{i-h}(\alpha-\beta+\beta
q;\alpha
q^{j};q)_{i-h}q^{(i-h)(d-j)+jh}}{(q;q)_{h}(\alpha+1-q;\alpha;q)_{i-h}(q;q)_{i-h}}.$
(4.3)

Let $m_{i}=\operatorname{tr}(E_{i})$, the multiplicity of $\theta_{i}$ in the
spectrum of $\Gamma$. This value is computed in [8, Theorem 8.4.3]:

$\displaystyle m_{i}={}$
$\displaystyle\frac{\prod_{h=0}^{i-1}\genfrac{[}{]}{0.0pt}{}{d-h}{1}\bigl{(}\beta-\genfrac{[}{]}{0.0pt}{}{h}{1}\alpha\bigr{)}\bigl{(}1+\genfrac{[}{]}{0.0pt}{}{d-h}{1}\alpha+q^{d-h}\beta\bigr{)}}{\prod_{h=1}^{i}\genfrac{[}{]}{0.0pt}{}{h}{1}\bigl{(}\beta-\genfrac{[}{]}{0.0pt}{}{h}{1}\alpha+q^{h}\bigr{)}\bigl{(}1+\genfrac{[}{]}{0.0pt}{}{d-h}{1}\alpha\bigr{)}}\,\frac{\bigl{(}1+\genfrac{[}{]}{0.0pt}{}{d-2i}{1}\alpha+q^{d-2i}\beta\bigr{)}q^{i}}{1+\genfrac{[}{]}{0.0pt}{}{d}{1}\alpha+q^{d}\beta}$
$\displaystyle={}$ $\displaystyle\frac{(q^{-d};q)_{i}(\alpha-\beta+\beta
q;\alpha;q)_{i}(\alpha-\beta+\beta
q;(\alpha+1-q)q^{-d};q)_{i}}{(q;q)_{i}(\alpha-\beta+\beta
q;(\alpha+1-q)q;q)_{i}(\alpha+1-q;\alpha q^{d-i};q)_{i}}$
$\displaystyle\times\frac{(\alpha-\beta+\beta
q-(\alpha+1-q)q^{2i-d})q^{2di-i^{2}}}{\alpha-\beta+\beta
q-(\alpha+1-q)q^{-d}}.$ (4.4)

Finally, we obtain a closed formula for $|X|$, the number of vertices of
$\Gamma$. Recall the $\mathbb{C}$-vector space $W(\Gamma)$ from (2.8). In view
of (3.5), $W(\Gamma)$ has another orthonormal basis
$\Psi_{0},\Psi_{1},\dots,\Psi_{d}$ defined by

$\displaystyle\Psi_{i}=\sqrt{\frac{|X|}{m_{i}}}E_{i}\hat{o},\qquad
i=0,1,\dots,d.$

As in the proof of Proposition 3.1, write

$\displaystyle E_{1}=\frac{1}{|X|}\sum_{i=0}^{d}\theta_{i}^{*}A_{i}.$

Now, let

$\displaystyle D=|X|\operatorname{diag}E_{1}\hat{o}.$

Then we have

$\displaystyle A\Psi_{i}=\theta_{i}\Psi_{i},\qquad
D\Phi_{i}=\theta_{i}^{*}\Phi_{i},\qquad i=0,1,\dots,d.$

Moreover, it follows from (2.2) and (4.1) that the matrix representing the
action of $A$ (resp. $D$) on $W(\Gamma)$ with respect to the $\Phi_{i}$ (resp.
the $\Psi_{i}$) is tridiagonal with non-zero superdiagonal and subdiagonal
entries. This means that $A$ and $D$ act on $W(\Gamma)$ as a _Leonard pair_ in
the sense of [38, Definition 1.1]. In the theory of Leonard pairs, there is a
scalar denoted by $\nu$ [39, Definition 9.3], and it is easy to see that
$\nu=\langle\Phi_{0},\Psi_{0}\rangle^{-2}=|X|$ for the above Leonard pair on
$W(\Gamma)$. For the $q$-Racah case, the scalar $\nu$ is given in [39, p. 273]
as follows:

$\displaystyle\nu=\frac{\big{(}sq^{2};q\big{)}_{d}\big{(}s^{*}q^{2};q\big{)}_{d}}{r_{1}^{d}q^{d}(sq/r_{1};q)_{d}(s^{*}q/r_{1};q)_{d}}.$

Again by letting $s^{*}\rightarrow 0$ and then setting $s$ and $r_{2}$ as in
(4.2), it follows that

$\displaystyle|X|=\frac{(-1)^{d}(\alpha-\beta+\beta
q;(\alpha+1-q)q^{1-d};q)_{d}q^{\binom{d}{2}}}{(\alpha+1-q;\alpha;q)_{d}}.$
(4.5)

##  5 Description of asymptotic normalized spectral distributions

In this section, we describe the asymptotic normalized spectral distributions
(cf. Remark 2.5) corresponding to Theorem 3.5 $(i)$ and $(ii)$, following
[17].

We retain the notation of the previous section. The Borel probability measure
$\mu$ on $\mathbb{R}$ from (2.1) associated with the normalized adjacency
matrix (2.12) is given by

$\displaystyle\mu\bigg{(}\frac{\theta_{j}-tk}{\Sigma_{t}(A)}\bigg{)}=\sum_{i=0}^{d}t^{i}v_{i}(\theta_{j})\,\frac{m_{j}}{|X|},\qquad
j=0,1,\dots,d,$ (5.1)

which follows from (2.5) and since

$\displaystyle\bigg{(}\frac{A-tkI}{\Sigma_{t}(A)}\bigg{)}^{\\!\\!\ell}\\!E_{j}=\bigg{(}\frac{\theta_{j}-tk}{\Sigma_{t}(A)}\bigg{)}^{\\!\\!\ell}\\!E_{j},\qquad
A_{i}E_{j}=v_{i}(\theta_{j})E_{j}$

for $\ell=0,1,\dots$ and $i,j=0,1,\dots,d$. From (4.4) and (4.5) it follows
that

$\displaystyle\frac{m_{j}}{|X|}={}$
$\displaystyle\frac{(q^{-d};q)_{j}(\alpha-\beta+\beta
q;\alpha;q)_{j}(\alpha+1-q;\alpha;q)_{d-j}}{(q;q)_{j}(\alpha-\beta+\beta
q;(\alpha+1-q)q^{j-d};q)_{d+1}}$
$\displaystyle\times(-1)^{d}(\alpha-\beta+\beta
q-(\alpha+1-q)q^{2j-d})q^{2dj-j^{2}-\binom{d}{2}}$ (5.2)

for $j=0,1,\dots,d$. From (4.3) it follows that

$\displaystyle\sum_{i=0}^{d}t^{i}v_{i}(\theta_{j})$
$\displaystyle=\sum_{h=0}^{d}\frac{(q^{-j};q)_{h}q^{jh}t^{h}}{(q;q)_{h}}\,\sum_{i=h}^{d}\frac{(q^{j-d};q)_{i-h}(\alpha-\beta+\beta
q;\alpha
q^{j};q)_{i-h}q^{(i-h)(d-j)}t^{i-h}}{(\alpha+1-q;\alpha;q)_{i-h}(q;q)_{i-h}}$
$\displaystyle={}_{1}\phi_{0}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-j}\\\
-\end{array}\\!\\!\right|q;q^{j}t\right)\sum_{\ell=0}^{d-j}\frac{(q^{j-d};q)_{\ell}(\alpha-\beta+\beta
q;\alpha
q^{j};q)_{\ell}q^{\ell(d-j)}t^{\ell}}{(\alpha+1-q;\alpha;q)_{\ell}(q;q)_{\ell}}$
(5.5)
$\displaystyle=(t;q)_{j}\sum_{\ell=0}^{d-j}\frac{(q^{j-d};q)_{\ell}(\alpha-\beta+\beta
q;\alpha
q^{j};q)_{\ell}q^{\ell(d-j)}t^{\ell}}{(\alpha+1-q;\alpha;q)_{\ell}(q;q)_{\ell}}$
(5.6)

for $j=0,1,\dots,d$, where we have used the $q$-binomial theorem (cf. [25,
Section 0.5])

$\displaystyle{}_{1}\phi_{0}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-n}\\\
-\end{array}\\!\\!\right|q;\mathfrak{x}\right)=(\mathfrak{x}q^{-n};q)_{n},\qquad
n=0,1,2,\dots.$ (5.9)

Note that the last sum in (5.6) is a ${}_{2}\phi_{1}$ in general.

###  5.1 Case $\boldsymbol{\beta/\sqrt{k}\rightarrow\rho>0}$

With reference to Assumption 3.2, suppose that we are in Theorem 3.5 $(i)$, or
$(ii)$ with $\rho>0$. For $(i)$, we moreover assume that $\rho$ is indeed an
accumulation point of $\beta/\sqrt{k}$, and will consider a subnet of
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which
$\beta/\sqrt{k}\rightarrow\rho$ if necessary (i.e., if $\alpha/\rho$ is also
an accumulation point). We note that $k\rightarrow\infty$ (cf. (3.9)),
$|\beta|\rightarrow\infty$, $t\rightarrow 0$, and that

$\displaystyle\frac{\beta(q-1)}{q^{d}}\sim\frac{\beta^{2}}{k}\rightarrow\rho^{2},\qquad
t\beta=t\sqrt{k}\frac{\beta}{\sqrt{k}}\rightarrow\gamma\rho,\qquad
tq^{d}\rightarrow\frac{\gamma(q-1)}{\rho}.$

Using this, (3.4), (5.2), and (5.6), we routinely compute

$\displaystyle\frac{\theta_{d-j}-tk}{\Sigma_{t}(A)}\rightarrow\frac{\genfrac{[}{]}{0.0pt}{}{j}{1}(\rho-\alpha/\rho
q^{j})-1/\rho q^{j}-\gamma}{\sqrt{1+\gamma(\rho+\alpha/\rho)}},$ (5.10)
$\displaystyle\frac{m_{d-j}}{|X|}\rightarrow\frac{\big{(}\alpha/\rho^{2}q^{j+1};q^{-1}\big{)}_{\infty}(\alpha+1-q;\alpha;q)_{j}\big{(}1-(\alpha+1-q)/\rho^{2}q^{2j}\big{)}}{\big{(}(\alpha+1-q)/\rho^{2}q^{j};q^{-1}\big{)}_{\infty}(q;q)_{j}\rho^{2j}q^{j^{2}-j}},$
(5.11)
$\displaystyle\sum_{i=0}^{d}t^{i}v_{i}(\theta_{d-j})\rightarrow\big{(}\gamma(q-1)/\rho
q^{j+1};q^{-1}\big{)}_{\infty}\sum_{\ell=0}^{j}\frac{(q^{-j};q)_{\ell}\big{(}\alpha/\rho^{2}q^{j};q\big{)}_{\ell}\gamma^{\ell}\rho^{\ell}q^{j\ell}(q-1)^{\ell}}{(\alpha+1-q;\alpha;q)_{\ell}(q;q)_{\ell}}$
(5.12)

for $j=0,1,2,\dots$. The measure (5.1) converges weakly to the discrete
measure $\mu_{\infty}$ on $\mathbb{R}$ defined on the limit points in (5.10),
where the masses are given by the products of the limits in (5.11) and
(5.12).666This follows for example from the observation that
$\mu((a,b))\rightarrow\mu_{\infty}((a,b))$ for every bounded open interval
$(a,b)$ in $\mathbb{R}$ and [7, Theorem 8.2.17]. When $q>0$, it is also
immediate to check that the distribution function of $\mu$ converges to that
of $\mu_{\infty}$ at the points of continuity of the latter.

###  5.2 Case $\boldsymbol{\alpha\neq 0}$,
$\boldsymbol{\beta/\sqrt{k}\rightarrow\alpha/\rho}$

With reference to Assumption 3.2, suppose that we are in Theorem 3.5 $(i)$,
and that $\alpha/\rho$ is an accumulation point of $\beta/\sqrt{k}$. We will
consider a subnet of $(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which
$\beta/\sqrt{k}\rightarrow\alpha/\rho$ if necessary. The formulas for the
limit distribution are simply obtained by replacing $\rho$ by $\alpha/\rho$ in
those of the previous case:

$\displaystyle\frac{\theta_{d-j}-tk}{\Sigma_{t}(A)}\rightarrow\frac{\genfrac{[}{]}{0.0pt}{}{j}{1}\big{(}\alpha/\rho-\rho/q^{j}\big{)}-\rho/\alpha
q^{j}-\gamma}{\sqrt{1+\gamma(\rho+\alpha/\rho)}},$
$\displaystyle\frac{m_{d-j}}{|X|}\rightarrow\frac{\big{(}\rho^{2}/\alpha
q^{j+1};q^{-1}\big{)}_{\infty}(\alpha+1-q;\alpha;q)_{j}\big{(}1-(\alpha+1-q)\rho^{2}/\alpha^{2}q^{2j}\big{)}\rho^{2j}}{\big{(}(\alpha+1-q)\rho^{2}/\alpha^{2}q^{j};q^{-1}\big{)}_{\infty}(q;q)_{j}\alpha^{2j}q^{j^{2}-j}},$
$\displaystyle\sum_{i=0}^{d}t^{i}v_{i}(\theta_{d-j})\rightarrow\big{(}\gamma\rho(q-1)/\alpha
q^{j+1};q^{-1}\big{)}_{\infty}\sum_{\ell=0}^{j}\frac{(q^{-j};q)_{\ell}\big{(}\rho^{2}/\alpha
q^{j};q\big{)}_{\ell}\alpha^{\ell}\gamma^{\ell}q^{j\ell}(q-1)^{\ell}}{(\alpha+1-q;\alpha;q)_{\ell}(q;q)_{\ell}\rho^{\ell}}$

for $j=0,1,2,\dots$. We note that, while the roles of $\rho$ and $\alpha/\rho$
are interchangeable when $q>0$, their distinction is essential when $q<0$, as
$\rho=\sqrt{-\alpha}$ and $\alpha/\rho=-\sqrt{-\alpha}$.

###  5.3 Case $\boldsymbol{\alpha=0}$, $\boldsymbol{\beta/\sqrt{k}\rightarrow
0}$

With reference to Assumption 3.2, suppose that we are in Theorem 3.5 $(ii)$
with $\rho=0$. Note that $q>0$ in this case, and let

$\displaystyle c=\bigl{\lfloor}\log_{q}\sqrt{k}\bigr{\rfloor}.$

Then we have $\sqrt{k}/q^{c}\in[1,q)$. Let $\eta/\sqrt{q-1}\in[1,q]$ be an
accumulation point of $\sqrt{k}/q^{c}$, and consider a subnet of
$(\Gamma_{\lambda})_{\lambda\in\Lambda}$ for which
$\sqrt{k}/q^{c}\rightarrow\eta/\sqrt{q-1}$ if $\eta$ is not unique. We note
that $k\rightarrow\infty$, $t\rightarrow 0$, $c\rightarrow\infty$,
$d-c\rightarrow\infty$, and that

$\displaystyle\beta
q^{d-2c}\sim\frac{k(q-1)}{q^{2c}}\rightarrow\eta^{2},\qquad t\beta\rightarrow
0,\qquad tq^{c}\rightarrow\frac{\gamma\sqrt{q-1}}{\eta}.$

Using this, (3.4), (5.2), and (5.6), we obtain

$\displaystyle\frac{\theta_{c-j}-tk}{\Sigma_{t}(A)}\rightarrow\frac{\eta
q^{j}-1/\eta q^{j}}{\sqrt{q-1}}-\gamma,$
$\displaystyle\frac{m_{c-j}}{|X|}\rightarrow\frac{\big{(}1+1/\eta^{2}q^{2j}\big{)}q^{-2j^{2}+j}}{\big{(}q^{-1};q^{-1}\big{)}_{\infty}\big{(}{-}1/\eta^{2};q^{-1}\big{)}_{\infty}\big{(}{-}\eta^{2}/q;q^{-1}\big{)}_{\infty}\eta^{4j}},$
$\displaystyle\sum_{i=0}^{d}t^{i}v_{i}(\theta_{c-j})\rightarrow\big{(}\gamma\sqrt{q-1}/\eta
q^{j+1};q^{-1}\big{)}_{\infty}\big{(}{-}\gamma\eta
q^{j-1}\sqrt{q-1};q^{-1}\big{)}_{\infty}$

for $j=0,\pm 1,\pm 2,\dots$, where we have used

$\displaystyle\big{(}{-}q^{c-j-d}/\beta;q\big{)}_{d+1}=\big{(}{-}q^{c-j-d}/\beta;q\big{)}_{c+j+1}\big{(}{-}q^{2c-d+1}/\beta;q\big{)}_{d-c-j}$

for the second formula, and (5.9) for the third one. Again, the measure (5.1)
converges weakly to the discrete measure $\mu_{\infty}$ on $\mathbb{R}$
defined by the above limits.

##  6 Examples

There are currently eleven known infinite families of distance-regular graphs
having classical parameters with unbounded diameter and such that $q\neq 1$.
In this section, we apply the results of the previous sections to these eleven
families. For more detailed information on these families, see the references
given. It should be remarked that, by virtue of Proposition 3.1, there exist
infinitely many choices for the scalar $\gamma$ in Theorem 3.5 $(i)$ and
$(ii)$.

###  6.1 Grassmann graphs and twisted Grassmann graphs

The _Grassmann graph_ $J_{q}(n,d)$ has as vertices the $d$-dimensional
subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over the
finite field $\mathbb{F}_{q}$ with $q$ elements, where two vertices $x$ and
$y$ are adjacent when $\dim x\cap y=d-1$; cf. [8, Section 9.1]. We always
assume that $n\geqslant 2d$, as $J_{q}(n,d)$ and $J_{q}(n,n-d)$ are
isomorphic. The graph $J_{q}(n,d)$ has classical parameters
$(d,q,\alpha,\beta)$, where

$\displaystyle\alpha=q,\qquad\beta=q\genfrac{[}{]}{0.0pt}{}{n-d}{1}.$

Fix $q$ and let $d\rightarrow\infty$, $t\sqrt{k}\rightarrow\gamma$, and let
also $n-2d+1\rightarrow 2\delta$ for some $\delta\in\frac{1}{2}\mathbb{Z}$, so
that we have $\beta/\sqrt{k}\rightarrow\rho:=q^{\delta}$. We are in Theorem
3.5 $(i)$, and from the results of Section 5.1 it follows that the measure
(5.1) converges weakly to $\mu_{\infty}$, where

$\displaystyle\mu_{\infty}\bigg{(}\frac{q^{\delta+j}+q^{-\delta-j}-q^{\delta}-q^{1-\delta}-\gamma(q-1)}{(q-1)\sqrt{1+\gamma(q^{\delta}+q^{1-\delta})}}\bigg{)}$
$\displaystyle\qquad{}=\bigg{(}\frac{1}{q^{j(2\delta+j-1)}}-\frac{1}{q^{(j+1)(2\delta+j)}}\bigg{)}\big{(}\gamma(q-1)/q^{\delta+j+1};q^{-1}\big{)}_{\infty}$
$\displaystyle\qquad\hphantom{=}{}\times{}_{2}\phi_{1}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-j},q^{1-2\delta-j}\\\
q\end{array}\\!\\!\right|q;\gamma(q-1)q^{\delta+j}\right)$

for $j=0,1,2,\dots$. Note that $\delta\geqslant 1/2$ and
$\alpha/\rho=q^{1-\delta}$, from which it follows that a different choice of
$\delta$ gives rise to a different limit in Theorem 3.5 $(i)$. Hora [16]
previously obtained $\mu_{\infty}$ for the vacuum state $\varphi_{0}$, i.e.,
for $\gamma=0$.

The _twisted Grassmann graph_ $\tilde{J}_{q}(2d+1,d)$ is defined as follows.
Fix a hyperplane $H$ of $\mathbb{F}_{q}^{2d+1}$. The vertex set consists of
the $(d+1)$-dimensional subspaces of $\mathbb{F}_{q}^{2d+1}$ which are not
contained in $H$, together with the $(d-1)$-dimensional subspaces of $H$. Two
vertices $x$ and $y$ are adjacent when $\dim x+\dim y-2\dim x\cap y=2$. See
[9]. The graph $\tilde{J}_{q}(2d+1,d)$ has the same classical parameters as
$J_{q}(2d+1,d)$, and hence we obtain the above measure $\mu_{\infty}$ with
$\delta=1$.

###  6.2 Dual polar graphs and Hemmeter graphs

Consider one of the following vector spaces $V$ endowed with a non-degenerate
form:

> $C_{d}(q)$: $V=\mathbb{F}_{q}^{2d}$ with a symplectic form;
>  $B_{d}(q)$: $V=\mathbb{F}_{q}^{2d+1}$ with a quadratic form;
>  $D_{d}(q)$: $V=\mathbb{F}_{q}^{2d}$ with a quadratic form of Witt index
> $d$;
>  ${}^{2}\\!D_{d+1}(q)$: $V=\mathbb{F}_{q}^{2d+2}$ with a quadratic form of
> Witt index $d$;
>  ${}^{2}\\!A_{2d}(r)$: $V=\mathbb{F}_{q}^{2d+1}$ with a Hermitian form
> ($q=r^{2}$);
>  ${}^{2}\\!A_{2d-1}(r)$: $V=\mathbb{F}_{q}^{2d}$ with a Hermitian form
> ($q=r^{2}$).

We note that maximal isotropic subspaces of $V$ have dimension $d$. The _dual
polar graph_ on $V$ has as vertices these maximal isotropic subspaces, where
two vertices $x$ and $y$ are adjacent when $\dim x\cap y=d-1$; cf. [8, Section
9.4]. This graph has classical parameters $(d,q,0,q^{e})$, where we let $e$ be
1, 1, 0, 2, 3/2, 1/2 in the respective types $C_{d}(q)$, $B_{d}(q)$,
$D_{d}(q)$, ${}^{2}\\!D_{d+1}(q)$, ${}^{2}\\!A_{2d}(r)$, and
${}^{2}\\!A_{2d-1}(r)$. Fix one of these types as well as $q$, and let
$d\rightarrow\infty$, $t\sqrt{k}\rightarrow\gamma$. We have
$\beta/\sqrt{k}\rightarrow 0$, and hence we are in Theorem 3.5 $(ii)$ with
$\rho=0$. Let the scalar $c$ be as in Section 5.3. Note that

$\displaystyle
c=\bigg{\lfloor}\frac{\log_{q}(q^{d}-1)-\log_{q}(q-1)+e}{2}\bigg{\rfloor}.$

Using $d\sim\log_{q}(q^{d}-1)<d$ and $0\leqslant\log_{q}(q-1)<1$, we obtain
the value of $c$ for sufficiently large $d$ as follows:

| $e=1$ | $e=0$ | $e=2$ | $e=3/2$ | $e=1/2$
---|---|---|---|---|---
$d$ even | $d/2$ | $d/2-1$ | $d/2$ | $d/2$ | $d/2-1$
$d$ odd | $(d-1)/2$ | $(d-1)/2$ | $(d+1)/2$ | $(d-1)/2$ | $(d-1)/2$

For the last two cases ($e\in\\{3/2,1/2\\}$), we have also used $q\geqslant 4$
(as $q=r^{2}$ is a square) and $\log_{4}3=0.792\dots$. It follows that
$\sqrt{k}/q^{c}\in[1,q)$ has two accumulation points, and considering limits
for even $d$ and odd $d$ separately, the scalar $\eta$ from Section 5.3 is
given as in the following table:

| $e=1$ | $e=0$ | $e=2$ | $e=3/2$ | $e=1/2$
---|---|---|---|---|---
$d$ even | $q^{1/2}$ | $q$ | $q$ | $q^{3/4}$ | $q^{5/4}$
$d$ odd | $q$ | $q^{1/2}$ | $q^{1/2}$ | $q^{5/4}$ | $q^{3/4}$

The measure (5.1) converges weakly to the measure $\mu_{\infty}$ as described
in Section 5.3. (We do not write down the result here, since there are four
values of $\eta$ and also since the substitution of these values does not seem
to simplify the formula significantly.)

Note that the graphs $C_{d}(q)$ and $B_{d}(q)$ share the same classical
parameters $(d,q,0,q)$. The _extended bipartite double_ of a graph
$\Gamma=(X,R)$ is the graph with vertex set $\mathbb{F}_{2}\times X$, where a
vertex $(\epsilon,x)$ is adjacent to $(\epsilon+1,x)$ and all the vertices
$(\epsilon+1,y)$ with $x\sim y$. The graph $D_{d}(q)$ is shown to be
isomorphic to the extended bipartite double of $B_{d-1}(q)$. The _Hemmeter
graph_ $\operatorname{Hem}_{d}(q)$ is then defined as the extended bipartite
double of $C_{d-1}(q)$; cf. [8, Section 9.4C]. It has the same classical
parameters $(d,q,0,1)$ as $D_{d}(q)$, so that we obtain the above
$\mu_{\infty}$ for $e=0$.

###  6.3 Half dual polar graphs and Ustimenko graphs

Recall that a graph $\Gamma=(X,R)$ is said to be _bipartite_ whenever there is
a bipartition $X=X^{+}\sqcup X^{-}$ such that no edge is contained in $X^{+}$
or $X^{-}$. In this case, a _halved graph_ of $\Gamma$ has as vertex set
either $X^{+}$ or $X^{-}$, where two distinct vertices are adjacent when there
is a path of length two joining them in $\Gamma$. The dual polar graph
$D_{n}(r)$ and the Hemmeter graph $\operatorname{Hem}_{n}(r)$ are bipartite,
and their halved graphs are called the _half dual polar graph_ $D_{n,n}(r)$
and the _Ustimenko graph_ $\operatorname{Ust}_{n}(r)$, respectively; cf. [8,
Section 9.4C]. These graphs have classical parameters $(d,q,\alpha,\beta)$,
where

$\displaystyle d=\left\lfloor\frac{n}{2}\right\rfloor,\qquad
q=r^{2},\qquad\alpha=r(r+1),\qquad\beta=\frac{r(r^{m}-1)}{r-1},$

where $m=2\lceil n/2\rceil-1$. Fix $r$, and let $d\rightarrow\infty$,
$t\sqrt{k}\rightarrow\gamma$. Note that $\beta/\sqrt{k}$ has two accumulation
points $\sqrt{r+1}$ and $r\sqrt{r+1}=\alpha/\sqrt{r+1}$, where
$\beta/\sqrt{k}\rightarrow\sqrt{r+1}$ for even $n$, and
$\beta/\sqrt{k}\rightarrow r\sqrt{r+1}$ for odd $n$. We consider limits for
even $n$ and odd $n$ separately. Set $\epsilon=0$ for even $n$, and
$\epsilon=1$ for odd $n$. From the results of Sections 5.1 and 5.2 it follows
that the measure (5.1) converges weakly to $\mu_{\infty}$, where

$\displaystyle\mu_{\infty}\bigg{(}\frac{r^{\epsilon+2j}+r^{-\epsilon-2j}-r-1-\gamma(r-1)\sqrt{r+1}}{(r-1)\sqrt{r+1+\gamma(r+1)^{5/2}}}\bigg{)}$
$\displaystyle\qquad{}=\frac{r^{2\epsilon+4j}-1}{r^{\epsilon+2j}-1}\,\frac{(r^{-1};r^{-2})_{\infty}}{r^{(\epsilon+j)(2j+1)}(r^{-2};r^{-2})_{\infty}}\,\big{(}\gamma(r-1)\sqrt{r+1}/r^{\epsilon+2j+2};r^{-2}\big{)}_{\infty}$
$\displaystyle\qquad\hphantom{=}{}\times{}_{2}\phi_{1}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}r^{-2j},r^{1-2\epsilon-2j}\\\
r\end{array}\\!\\!\right|r^{2};\gamma r^{\epsilon+2j}(r-1)\sqrt{r+1}\right)$

for $j=0,1,2,\dots$, where we understand that
$\big{(}r^{0}-1\big{)}/\big{(}r^{0}-1\big{)}=1$ when $\epsilon=j=0$.

###  6.4 Second classical parameters for dual polar graphs
$\boldsymbol{{}^{2}\\!A_{2d-1}(r)}$

The Hermitian dual polar graph ${}^{2}\\!A_{2d-1}(r)$ has another set of
classical parameters $(d,q,\alpha,\beta)$, where (cf. (3.7))

$\displaystyle
q=-r,\qquad\alpha=\frac{r(r+1)}{1-r},\qquad\beta=\frac{r(1+(-r)^{d})}{1-r}.$

Fix $r$, and let $d\rightarrow\infty$, $t\sqrt{k}\rightarrow\gamma$. Note that
$\beta/\sqrt{k}$ has two accumulation points $\pm\sqrt{-\alpha}$, where
$\beta/\sqrt{k}\rightarrow-\sqrt{-\alpha}$ for even $d$, and
$\beta/\sqrt{k}\rightarrow\sqrt{-\alpha}$ for odd $d$. We consider limits for
even $d$ and odd $d$ separately. According to whether
$\beta/\sqrt{k}\rightarrow\pm\sqrt{-\alpha}$, the measure (5.1) converges
weakly to $\mu_{\infty}$, where

$\displaystyle\mu_{\infty}\bigg{(}\frac{\mp\sqrt{r}(-r)^{j}\pm\sqrt{r}^{-1}(-r)^{-j}}{\sqrt{r^{2}-1}}-\gamma\bigg{)}$
$\displaystyle\qquad{}=\frac{(r^{2j+1}\\!+\\!1)(r^{-1};-r^{-1})_{\infty}}{r^{(j+1)^{2}}(-r^{-1};-r^{-1})_{\infty}}\bigg{(}\\!\\!\mp\gamma(-r)^{-j-1}\sqrt{\frac{r^{2}\\!-\\!1}{r}};-r^{-1}\\!\bigg{)}_{\\!\\!\infty}\\!\\!\bigg{(}\\!\\!\mp\gamma(-r)^{j-1}\sqrt{\frac{r^{2}\\!-\\!1}{r}};r^{-2}\\!\bigg{)}_{\\!\\!j}$

for $j=0,1,2,\dots$. Here we have again used (5.9) to get the result. We may
routinely verify that this measure is identical to the one in Section 6.2 with
$e=1/2$, using

$\displaystyle\big{(}r^{-1};-r^{-1}\big{)}_{\infty}\big{(}{-}r^{-1};r^{-2}\big{)}_{\infty}=1,$

which is a special case of Lebesgue’s identity; see, e.g., [11].

###  6.5 Sesquilinear forms graphs

There are four infinite families of sesquilinear forms graphs, all of which
are Cayley graphs.

The _bilinear forms graph_ $\operatorname{Bil}(d\times e,q)$ has as vertices
the $d\times e$ matrices over $\mathbb{F}_{q}$, where two vertices $x$ and $y$
are adjacent when $\operatorname{rank}(x-y)=1$; cf. [8, Section 9.5A]. We
always assume that $d\leqslant e$, as $\operatorname{Bil}(d\times e,q)$ and
$\operatorname{Bil}(e\times d,q)$ are isomorphic. The graph
$\operatorname{Bil}(d\times e,q)$ has classical parameters
$(d,q,q-1,q^{e}-1)$. Fix $q$ and let $d\rightarrow\infty$,
$t\sqrt{k}\rightarrow\gamma$, and $e-d\rightarrow 2\delta$ for some
$\delta\in\frac{1}{2}\mathbb{Z}$, so that
$\beta/\sqrt{k}\rightarrow\rho:=q^{\delta}\sqrt{q-1}$. The measure (5.1)
converges weakly to $\mu_{\infty}$, where

$\displaystyle\mu_{\infty}\bigg{(}\frac{q^{\delta+j}-q^{\delta}-q^{-\delta}-\gamma\sqrt{q-1}}{\sqrt{q-1+\gamma(q^{\delta}+q^{-\delta})(q-1)^{3/2}}}\bigg{)}$
$\displaystyle\qquad{}=\frac{(-1)^{j}(q^{-2\delta-j-1};q^{-1})_{\infty}}{(q;q)_{j}q^{2\delta
j+\binom{j}{2}}}\,\bigl{(}\gamma\sqrt{q\\!-\\!1}/q^{\delta+j+1};q^{-1}\bigr{)}_{\infty}{}_{2}\phi_{0}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}q^{-j},q^{-2\delta-j}\\\
-\end{array}\\!\\!\right|q;\gamma q^{\delta+j}\sqrt{q\\!-\\!1}\right)$

for $j=0,1,2,\dots$. Since $\delta\geqslant 0$ and
$\alpha/\rho=q^{-\delta}\sqrt{q-1}$, it follows that a different choice of
$\delta$ gives rise to a different limit in Theorem 3.5 $(i)$.

The _alternating forms graph_ $\operatorname{Alt}(n,r)$ has as vertices the
$n\times n$ skew-symmetric matrices with zero diagonal over $\mathbb{F}_{r}$,
where two vertices $x$ and $y$ are adjacent when $\operatorname{rank}(x-y)=2$;
cf. [8, Section 9.5B]. It has classical parameters $(d,q,\alpha,\beta)$, where

$\displaystyle d=\bigg{\lfloor}\frac{n}{2}\bigg{\rfloor},\qquad
q=r^{2},\qquad\alpha=r^{2}-1,\qquad\beta=r^{m}-1,$

where $m=2\lceil n/2\rceil-1$. Fix $r$, and let $d\rightarrow\infty$,
$t\sqrt{k}\rightarrow\gamma$. Note that we can apply the above computation
with $e:=m/2$. Hence we consider limits for even $n$ and odd $n$ separately,
and set $\delta=-1/4$ for even $n$ and $\delta=1/4$ for odd $n$. We then
obtain the above measure $\mu_{\infty}$ with $q=r^{2}$.

The _quadratic forms graph_ $\operatorname{Qua}(n-1,r)$ has as vertices the
quadratic forms on $\mathbb{F}_{r}^{n-1}$, where two vertices $x$ and $y$ are
adjacent when $\operatorname{rank}(x-y)\in\\{1,2\\}$. See [8, Section 9.6] for
a precise description. This graph has the same classical parameters as
$\operatorname{Alt}(n,r)$, and thus the result is the same as above.

Finally, the _Hermitian forms graph_ $\operatorname{Her}\big{(}d,r^{2}\big{)}$
has as vertices the $d\times d$ Hermitian matrices over $\mathbb{F}_{r^{2}}$,
where two vertices $x$ and $y$ are adjacent when $\operatorname{rank}(x-y)=1$;
cf. [8, Section 9.5C]. It has classical parameters $(d,-r,-r-1,-(-r)^{d}-1)$.
Fix $r$, and let $d\rightarrow\infty$, $t\sqrt{k}\rightarrow\gamma$. We
consider limits for even $d$ and odd $d$ separately, where
$\beta/\sqrt{k}\rightarrow-\sqrt{r+1}$ for even $d$, and
$\beta/\sqrt{k}\rightarrow\sqrt{r+1}$ for odd $d$. According to whether
$\beta/\sqrt{k}\rightarrow\pm\sqrt{r+1}$, the measure (5.1) converges weakly
to $\mu_{\infty}$, where

$\displaystyle\mu_{\infty}\bigg{(}\\!\\!\mp\frac{(-r)^{j}}{\sqrt{r+1}}-\gamma\bigg{)}={}$
$\displaystyle\frac{\big{(}{-}(-r)^{-j-1};-r^{-1}\big{)}_{\infty}}{(-r;-r)_{j}(-r)^{\binom{j}{2}}}\,\bigl{(}{}\mp\gamma\sqrt{r+1}/(-r)^{j+1};-r^{-1}\bigr{)}_{\infty}$
$\displaystyle{}\times{}_{2}\phi_{0}\\!\left(\left.\\!\\!\\!\begin{array}[]{c}(-r)^{-j},-(-r)^{-j}\\\
-\end{array}\\!\\!\right|-r;\pm\gamma(-r)^{j}\sqrt{r+1}\right)$

for $j=0,1,2,\dots$.

######  Remark 6.1.

Set $\gamma=0$ in the above examples, which is the case when we take scaling
limits of the vacuum state $\varphi_{0}$. The measure $\mu_{\infty}$ is then
an affine transformation of the discrete orthogonality measure of the _Al-
Salam–Chihara polynomials_ (cf. [25, Section 3.8]) for Sections 6.1 and 6.3,
that of the _continuous_ $q^{-1}$-_Hermite polynomials_ (cf. [25, Section
3.26]) for Sections 6.2 and 6.4, and that of the _Al-Salam–Carlitz II
polynomials_ (cf. [25, Section 3.25]) with base $q^{-1}$ for Section 6.5. For
the discrete orthogonality measures of the first two families of orthogonal
polynomials, see, e.g., [2, equation (3.82)], [3, equation (3.18)], and [21,
equation (6.31)].

### Acknowledgements

The authors thank the anonymous referees for valuable comments. HT thanks
Professor Tom Koornwinder for letting him know that the measure $\mu_{\infty}$
with $\gamma=0$ in Section 6.1 corresponds to the Al-Salam–Chihara
polynomials, and for providing relevant references. Part of this work was done
while MK was visiting Tohoku University from February to July 2020, supported
by K.N. Toosi University of Technology, Office of Vice-Chancellor for Global
Strategies and International Affairs. NO and HT were supported by JSPS KAKENHI
Grant Number JP19H01789. HT was also supported by JSPS KAKENHI Grant Numbers
JP17K05156 and JP20K03551. This work was also partially supported by the
Research Institute for Mathematical Sciences at Kyoto University.

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