$\mathbb{C}\setminus\\{0\\}$, and if $\nu_{1}\geq 0$, then it is uniform on
compact subsets of $\mathbb{C}$.

######  Remark 5.3.

The theorem is a direct application of the integral representation in Theorem
3.1. If we use instead Theorems 3.2 and 3.3, we could possibly extend the
theorem for general $q,t\in(0,1)$. Since our main application does require
$\theta\in\mathbb{N}$ and $t=q^{\theta}$, we do not bother to pursue the more
general case $q,t\in(0,1)$.

###### Proof.

Let us first prove that the limit (5.1) holds uniformly for $x$ in compact
subsets of $\mathcal{U}$. We use the integral representation for Macdonald
characters of one variable, Theorem 3.1, for the signature
$\lambda(N)\in\mathbb{GT}_{N}$, and with $x$ replaced by $xt^{N}$,
$x\in\mathcal{U}$. Since $P_{\lambda(N)}$ is a homogeneous polynomial of
degree $|\lambda(N)|$, then the left-hand side is

$\displaystyle
t^{|\lambda(N)|}\frac{P_{\lambda(N)}\big{(}x,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}.$

After simple algebraic manipulations, the right-hand side becomes

$\displaystyle q^{-{\theta N\choose 2}}\frac{(q;q)_{\theta
N-1}}{(xq;q)_{\theta
N-1}}\frac{\ln{q}}{2\pi\sqrt{-1}}\oint_{\mathcal{C}_{0}}{\frac{x^{z}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}q^{-(\lambda_{i}(N)+\theta(N-i)+j)}-q^{-z}\big{)}}}}{\rm
d}z.$

Therefore we conclude (make change of variables $i\mapsto N-i+1$ in the inner
product)

$\displaystyle\frac{P_{\lambda(N)}\big{(}x,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}$
$\displaystyle\qquad{}=\frac{(q;q)_{\theta N-1}}{(xq;q)_{\theta
N-1}}\frac{\ln{q}}{2\pi\sqrt{-1}}\oint_{\mathcal{C}_{0}}{\frac{x^{z}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{-z}q^{\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{)}}}{\rm
d}z},$ (5.3)

where $\mathcal{C}_{0}$ is a finite contour encloses all real poles of the
integrand and no other poles. We have the limit

$\displaystyle\lim_{N\rightarrow\infty}{\frac{(q;q)_{\theta
N-1}}{(xq;q)_{\theta N-1}}}=\frac{(q;q)_{\infty}}{(xq;q)_{\infty}},$

uniformly for $x$ belonging to compact subsets of $\mathcal{U}$.

Next modify the contour $\mathcal{C}_{0}$ into an infinite contour
$\mathcal{C}^{+}$ as described in the statement of the theorem. This is
possible to do because all the poles of the integrands of (5.3) belong to the
interior of $\mathcal{C}^{+}$, as $N$ grows. Moreover the resulting integral
is well-posed for large enough $N$ since the integrand is of order
$\big{(}|x|q^{\theta N}\big{)}^{\Re z}$ and for any compact set
$K\subset\mathbb{C}$, there exists $N_{0}\in\mathbb{N}$ such that
$\sup\limits_{N>N_{0}}\sup\limits_{x\in K}{\big{\\{}|x|q^{\theta
N}\big{\\}}}<1$.

We now look at the asymptotics of the integral in (5.3), with
$\mathcal{C}_{0}$ replaced by $\mathcal{C}^{+}$. The denominator in the
integrand has the following limit

$\displaystyle\lim_{N\rightarrow\infty}{\prod_{i=1}^{N}\prod_{j=0}^{\theta-1}{\big{(}1-q^{-z+\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{)}}}$
$\displaystyle\qquad{}=\prod_{i=1}^{\infty}\prod_{j=0}^{\theta-1}{\big{(}1-q^{-z+\nu_{i}+\theta(i-1)+j}\big{)}}=\prod_{i=1}^{\infty}{\frac{\big{(}q^{-z+\nu_{i}}t^{i-1};q\big{)}_{\infty}}{\big{(}q^{-z+\nu_{i}}t^{i};q\big{)}_{\infty}}}.$

The limit above can be justified properly by using the dominated convergence
theorem and the estimate

$\displaystyle\sum_{i=1}^{N}\sum_{j=0}^{\theta-1}{\big{|}q^{-z+\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{|}}$
$\displaystyle\qquad{}\leq\sum_{i=1}^{N}\sum_{j=0}^{\theta-1}{q^{-\Re
z+\lambda_{N}(N)+\theta(i-1)+j}}\leq
c\sum_{i=1}^{N}\sum_{j=0}^{\theta-1}{q^{\theta(i-1)+j}}<\frac{c}{1-q},$

valid for some constant $c>0$ such that $q^{-\Re z+\lambda_{N}(N)}<c$ for all
$N\geq 1$ (it exists because
$\lim\limits_{N\rightarrow\infty}{\lambda_{N}(N)}=\nu_{1}$).

To prove the limit in the statement of the theorem, we are left to show

$\displaystyle\lim_{N\rightarrow\infty}{\int_{\mathcal{C}^{+}}\frac{x^{z}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{-z}q^{\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{)}}}{\rm
d}z}=\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{\infty}{\frac{\big{(}q^{-z+\nu_{i}}t^{i};q\big{)}_{\infty}}{\big{(}q^{-z+\nu_{i}}t^{i-1};q\big{)}_{\infty}}}{\rm
d}z}.$

We already proved the pointwise convergence; to make use of the dominated
convergence theorem, we simply need estimates on the contribution of the tails
of $\mathcal{C}^{+}$ that are uniform in $N$ and uniform for $x$ belonging to
compact subsets of $\mathbb{C}\setminus\\{0\\}$. Parametrize the tails of
$\mathcal{C}^{+}$ as $z=r+\frac{\pi\sqrt{-1}}{\ln{q}}$ or
$z=r-\frac{\pi\sqrt{-1}}{\ln{q}}$; for $r$ ranging from some large $R>1$ to
$+\infty$, we want to show that the contribution of each of these lines is
small. We have

$\displaystyle\left|\frac{x^{z}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1-q^{-z}q^{\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{)}}}\right|=\frac{\big{|}x^{r\pm\frac{\pi\sqrt{-1}}{\ln{q}}}\big{|}}{\prod\limits_{i=1}^{N}\prod\limits_{j=0}^{\theta-1}{\big{(}1+q^{-r}q^{\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{)}}}$
$\displaystyle\qquad{}\leq
C_{1}\times\frac{|x|^{r}}{\prod\limits_{i=1}^{k}\prod\limits_{j=0}^{\theta-1}{\big{(}1+q^{-r}q^{\lambda_{N-i+1}(N)+\theta(i-1)+j}\big{)}}}$
$\displaystyle\qquad{}\leq
C_{2}\times\frac{|x|^{r}}{\prod\limits_{i=1}^{k}\prod\limits_{j=0}^{\theta-1}{(1+q^{-r})}}\leq
C_{2}\times\frac{|x|^{r}}{(q^{-r})^{\theta
k}}=C_{2}\times\big{(}|x|t^{k}\big{)}^{r},$

for all $1\leq k\leq N$, where the constant in the second line is uniform for
$x$ over compact subsets of $\mathbb{C}\setminus\\{0\\}$, and the constant in
the third line depends on $k$, but not on $N$ (note that to go from the second
to the third line, we needed to use that
$\lim\limits_{N\rightarrow\infty}{\lambda_{N-i+1}(N)}$ exists for
$i=1,\dots,k$ and so the sequence $\\{\lambda_{N-i+1}(N)\\}_{N\geq 1}$ is
uniformly bounded for $i=1,\dots,k$). Now for any compact set
$K\subset\mathbb{C}\setminus\\{0\\}$ and any $M>0$, there exists
$k\in\mathbb{N}$ such that $\sup\limits_{x\in K}{|x|}\cdot t^{k}\leq e^{-M}$.
Since $\int_{R}^{\infty}{e^{-Mr}{\rm
d}r}=e^{-MR}/M\xrightarrow{R\rightarrow\infty}0$, we have shown that the
contribution of the tails of $\mathcal{C}^{+}$ is uniformly small over $x$ in
compact subsets of $\mathbb{C}\setminus\\{0\\}$ and for large enough $N$.

We have proved so far that the limit in the theorem holds uniformly for $x$ in
compact subsets of $\mathcal{U}$. Since the set $\\{q^{-1},q^{-2},\dots\\}$ is
discrete and has no accumulation points, Cauchy’s integral formula allows us
to deduce the uniform convergence in compact subsets of
$\mathbb{C}\setminus\\{0\\}$ as soon as we show that $\Phi^{\nu}(x;q,t)$
admits an analytic continuation to $\mathbb{C}\setminus\\{0\\}$.

By virtue of Riemann’s theorem of removable singularities, it will suffice to
show that $\Phi^{\nu}(x;q,t)$ is uniformly bounded in an open neighborhood of
each pole $q^{-k}$. Let $R>0$, $R\notin\\{q^{n}\colon n\in\mathbb{Z}\\}$, be
arbitrary. From what we have shown so far, it is clear that

$\displaystyle\sup_{\begin{subarray}{c}\frac{1}{R}\leq|x|\leq R\\\
x\in\mathcal{U}\end{subarray}}\big{|}\Phi^{\nu}(x;q,t)\big{|}\leq\sup_{N\geq
1}\sup_{\begin{subarray}{c}\frac{1}{R}\leq|x|\leq R\\\
x\in\mathcal{U}\end{subarray}}{\left|\frac{P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|}$
$\displaystyle\hphantom{\sup_{\begin{subarray}{c}\frac{1}{R}\leq|x|\leq R\\\
x\in\mathcal{U}\end{subarray}}\big{|}\Phi^{\nu}(x;q,t)\big{|}}{}\leq\sup_{N\geq
1}\sup_{\frac{1}{R}\leq|x|\leq
R}{\left|\frac{P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|}.$

Thanks to the branching rule for Macdonald polynomials, Theorem 2.5, the fact
that all the branching coefficients $\psi_{\mu/\nu}(q,t)$ are nonnegative when
$q,t\in(0,1)$, and the fact that each
$P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}$ is a Laurent polynomial
in $x$, we have

$\displaystyle\sup_{\frac{1}{R}\leq|x|\leq
R}{\left|\frac{P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|}\leq\sup_{\frac{1}{R}\leq|x|\leq
R}{\frac{P_{\lambda(N)}\big{(}|x|,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}}$
$\displaystyle\hphantom{\sup_{\frac{1}{R}\leq|x|\leq
R}{\left|\frac{P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|}}{}\leq\frac{P_{\lambda(N)}\big{(}R,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\\!+\\!\frac{P_{\lambda(N)}\big{(}R^{-1},t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}.$

From the pointwise limits

$\displaystyle\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big{(}R^{\pm
1},t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}}=\Phi^{\nu}(R^{\pm
1};q,t),$

we deduce that the sequences $\big{\\{}P_{\lambda(N)}\big{(}R^{\pm
1},t^{-1},\dots,t^{1-N}\big{)}/P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}\big{\\}}_{N\geq
1}$ are uniformly bounded. As a result of the estimates above,

$\displaystyle\sup\limits_{\begin{subarray}{c}\frac{1}{R}\leq|x|\leq R\\\
x\in\mathcal{U}\end{subarray}}{|\Phi^{\nu}(x;q,t)|}<\infty.$

Thus $\Phi^{\nu}(x;q,t)$ admits an analytic continuation to all the poles in
$\big{\\{}z\in\mathbb{C}\colon\frac{1}{R}\leq|z|\leq
R\big{\\}}\cap\big{\\{}q^{-1},q^{-2},\dots\big{\\}}$. Since $R>0$ was
arbitrary (outside of a lattice), we conclude that $\Phi^{\nu}(x;q,t)$ admits
an analytic continuation to $\mathbb{C}\setminus\\{0\\}$.

If $\nu_{1}\geq 0$, then
$\lim\limits_{N\rightarrow\infty}{\lambda_{N}(N)}=\nu_{1}$ shows that
$\lambda_{N}(N)\geq 0$ for all $N>N_{0}$ and $N_{0}\in\mathbb{N}$ large
enough. For all $N>N_{0}$, the functions
${P_{\lambda(N)}\big{(}x,t^{-1},\dots,t^{1-N}\big{)}}/{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}$
are polynomials in $x$ and therefore holomorphic on $\mathbb{C}$. Similar
considerations as above allow us to analytically continue $\Phi^{\nu}(x;q,t)$
to $\mathbb{C}$ in this case, and also show that the limit (5.1) holds
uniformly for $x$ belonging to compact subsets of $\mathbb{C}$. ∎

We end this subsection with a Lemma that will be used in Section 7.

######  Lemma 5.4.

If $\nu,\widetilde{\nu}\in\mathcal{N}$ are such that

$\displaystyle\Phi^{\nu}(x;q,t)=\Phi^{\widetilde{\nu}}(x;q,t)\qquad\forall\,x\in\mathbb{T},$
(5.4)

then $\nu=\widetilde{\nu}$.

###### Proof.

In the integral representation of $\Phi^{\nu}(x;q,t)$, the set of poles of the
integrand is

$\bigcup_{r=1}^{\infty}\bigcup_{s=0}^{\theta-1}{\\{\nu_{r}+\theta(r-1)+s\\}}$

and they are all enclosed by contour $\mathcal{C}^{+}$. We can therefore
expand

$\displaystyle\frac{\ln{q}}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{\infty}{\frac{(q^{-z+\nu_{i}}t^{i};q)_{\infty}}{(q^{-z+\nu_{i}}t^{i-1};q)_{\infty}}}{\rm
d}z},$

at least formally, as the sum of residues

$\displaystyle\sum_{r=1}^{\infty}\sum_{s=0}^{\theta-1}\left\\{x^{\nu_{r}+\theta(r-1)+s}\prod_{\begin{subarray}{c}i\geq
1\\\ i\neq
r\end{subarray}}{\frac{\big{(}q^{-\nu_{r}+\nu_{i}-s}t^{i-r+1};q\big{)}_{\infty}}{\big{(}q^{-\nu_{r}+\nu_{i}-s}t^{i-r};q\big{)}_{\infty}}}\times\prod_{\begin{subarray}{c}\theta>j\geq
0\\\ j\neq s\end{subarray}}{\frac{1}{(1-q^{j-s})}}\right\\}.$ (5.5)

The series above converges absolutely for all
$x\in\mathbb{C}\setminus\\{0\\}$, and uniformly for $x$ in compact subsets of
$\mathbb{C}\setminus\\{0\\}$; in fact, we can argue as in the special case
$\theta=1$ of [24, Section 6.2]. We have the bounds

$\displaystyle\prod_{\begin{subarray}{c}\theta>j\geq 0\\\ j\neq
s\end{subarray}}{\left|\frac{1}{(1-q^{j-s})}\right|}\leq\frac{1}{(1-q)\cdots\big{(}1-q^{\theta}\big{)}\big{(}q^{-1}-1\big{)}\cdots\big{(}q^{-\theta}-1\big{)}};$
$\displaystyle\prod_{i>r}{\left|\frac{\big{(}q^{-\nu_{r}+\nu_{i}-s}t^{i-r+1};q\big{)}_{\infty}}{\big{(}q^{-\nu_{r}+\nu_{i}-s}t^{i-r};q\big{)}_{\infty}}\right|}=\prod_{i>r}\prod_{j=0}^{\theta-1}{\left|\frac{1}{\big{(}1-q^{-\nu_{r}+\nu_{i}-s+\theta(i-r)+j}\big{)}}\right|}$
$\displaystyle\qquad{}=\prod_{i>r}\prod_{j=0}^{\theta-1}{\frac{1}{\big{(}1-q^{-\nu_{r}+\nu_{i}-s+\theta(i-r)+j}\big{)}}}\qquad{}\leq\prod_{i>r}\prod_{j=0}^{\theta-1}{\frac{1}{\big{(}1-q^{-(\theta-1)+\theta(i-r)+j}\big{)}}}=\frac{1}{(q;q)_{\infty}};$
$\displaystyle\prod_{i<r}{\left|\frac{\big{(}q^{-\nu_{r}+\nu_{i}-s}t^{i-r+1};q\big{)}_{\infty}}{\big{(}q^{-\nu_{r}+\nu_{i}-s}t^{i-r};q\big{)}_{\infty}}\right|}=\prod_{i<r}\prod_{j=0}^{\theta-1}{\frac{q^{\nu_{r}+\theta(r-1)+s-\nu_{i}-\theta(i-1)-j}}{1-q^{\nu_{r}-\nu_{i}+\theta(r-i)-j+s}}}$
$\displaystyle\qquad{}\leq\frac{1}{(q;q)_{\infty}}\prod_{i<r}\prod_{j=0}^{\theta-1}{q^{\nu_{r}+\theta(r-1)+s-\nu_{i}-\theta(i-1)-j}}$
$\displaystyle\qquad{}\leq\frac{1}{(q;q)_{\infty}}\prod_{i=1}^{m}\prod_{j=0}^{\theta-1}{q^{\nu_{r}+\theta(r-1)+s-\nu_{i}-\theta(i-1)-j}},\qquad\textrm{
for any}\quad 1\leq m<r,$
$\displaystyle\qquad{}=q^{m\theta(\nu_{r}+\theta(r-1)+s)}\times\frac{q^{-\theta(\nu_{1}+\dots+\nu_{m})}q^{-{\theta
m\choose 2}}}{(q;q)_{\infty}}.$

Choose an arbitrary $m\in\mathbb{N}$ and fix it. For $r>m$, the general term
in brackets at (5.5) has modulus upper bounded by

$\big{(}|x|q^{m\theta}\big{)}^{\nu_{r}+\theta(r-1)+s}\times c(m,\theta;q),$

where $c(m,\theta;q)>0$ is a constant depending on $m,\theta$, as well as
$\nu_{1},\dots,\nu_{m}$, but not on $r$. It follows that the sum (5.5)
converges absolutely for any $x\in\mathbb{C}\setminus\\{0\\}$ with
$|x|<q^{-m\theta}$. Since $m\in\mathbb{N}$ was arbitrary, it follows that
(5.5) converges absolutely for any $x\in\mathbb{C}\setminus\\{0\\}$. The
uniform convergence also follows from the bound above. In particular, (5.5) is
the Fourier expansion of
$\frac{(xq;q)_{\infty}}{(q;q)_{\infty}}\Phi^{\nu}(x;q,t)$. A similar Fourier
expansion can be given for
$\frac{(xq;q)_{\infty}}{(q;q)_{\infty}}\Phi^{\widetilde{\nu}}(x;q,t)$. After
multiplying equality (5.4) by $(xq;q)_{\infty}/(q;q)_{\infty}$, we have

$\displaystyle\frac{\ln{q}}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{\infty}{\frac{\big{(}q^{-z+\nu_{i}}t^{i};q\big{)}_{\infty}}{\big{(}q^{-z+\nu_{i}}t^{i-1};q\big{)}_{\infty}}}{\rm
d}z}=\frac{\ln{q}}{2\pi\sqrt{-1}}\int_{\mathcal{C}^{+}}{x^{z}\prod_{i=1}^{\infty}{\frac{\big{(}q^{-z+\widetilde{\nu}_{i}}t^{i};q\big{)}_{\infty}}{\big{(}q^{-z+\widetilde{\nu}_{i}}t^{i-1};q\big{)}_{\infty}}}{\rm
d}z}.$ (5.6)

We can expand both sides of (5.6) as above, to get an equality of the form
$\sum\limits_{k\in\mathbb{Z}}{c_{k}(\nu)x^{k}}=\sum\limits_{k\in\mathbb{Z}}{c_{k}(\widetilde{\nu})x^{k}}$.
More precisely, (5.5) gives that the set $\\{k\in\mathbb{Z}\colon
c_{k}(\nu)\neq 0\\}$ of indices which appear in the expansion of the left-hand
side of (5.6) is $\\{\nu_{r}+\theta(r-1)+s\colon 1\leq r,\,0\leq
s\leq\theta-1\\}$. Similarly, $\\{k\in\mathbb{Z}\colon
c_{k}(\widetilde{\nu})\neq 0\\}=\\{\widetilde{\nu}_{r}+\theta(r-1)+s\colon
1\leq r,\,0\leq s\leq\theta-1\\}$. The equality of these sets, and the
inequalities $\nu_{1}\leq\nu_{2}\leq\cdots$,
$\widetilde{\nu}_{1}\leq\widetilde{\nu}_{2}\leq\cdots$, imply that
$\nu_{r}=\widetilde{\nu}_{r}$ for all $r\geq 1$, i.e., $\nu=\widetilde{\nu}$.
∎

###  5.2 Asymptotics of Macdonald characters of a fixed number
$\boldsymbol{m}$ of variables

The following theorem is the main result for asymptotics of Macdonald
characters of any given rank $m\in\mathbb{N}$ as $N$ tends to infinity.

######  Theorem 5.5.

Let $\theta\in\mathbb{N}$, $t=q^{\theta}$, and $\\{\lambda(N)\\}_{N\geq 1}$,
$\lambda(N)\in\mathbb{GT}_{N}$, be a sequence of signatures that stabilizes to
$\nu\in\mathcal{N}$. Also let $m\in\mathbb{N}$ be arbitrary. Then

$\displaystyle\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big{(}x_{1},\dots,x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}}=\Phi^{\nu}(x_{1},\dots,x_{m};q,t),$
(5.7)

where

$\displaystyle\Phi^{\nu}(x_{1},\dots,x_{m};q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{q^{-2\theta^{2}{m\choose 3}-{\theta+1\choose 2}{m\choose
2}}}{\prod\limits_{i=1}^{m}{\big{(}x_{i}qt^{m-1};q\big{)}_{\infty}}}\frac{1}{\prod\limits_{\begin{subarray}{c}1\leq
i<j\leq m\\\ 0\leq k<\theta\end{subarray}}{(q^{k}x_{j}-x_{i})}}$
$\displaystyle\hphantom{\Phi^{\nu}(x_{1},\dots,x_{m};q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}}{}\times\widetilde{D}_{q,\theta}^{(m)}\left\\{\prod_{i=1}^{m}{\Phi^{\nu}(x_{i};q,t)(x_{i}q;q)_{\infty}}\right\\},$
$\displaystyle\widetilde{D}_{q,\theta}^{(m)}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{\tau\in
M_{\theta}^{(m)}}{C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m})\prod_{i=1}^{m}{T_{q,x_{i}}^{(i-1)\theta+\tau_{i}^{+}-\tau_{i}^{-}}}}.$
(5.8)

Above we used the notation of Section 4: $M_{\theta}^{(m)}$ is the set of all
strictly upper triangular matrices with entries in $\\{0,1,\dots,\theta\\}$,
and for $\tau\in M_{\theta}^{(m)}$, we let
$\tau_{i}^{+}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum\limits_{j>i}{\tau_{i,j}}$,
$\tau_{i}^{-}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum\limits_{k<i}{\tau_{k,i}}$. Also for $\tau\in
M_{\theta}^{(m)}$, let $C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m})$ be the
rational functions defined in (4.3). The function
$\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ is defined by the formula above in the
domain

$\displaystyle\mathcal{U}_{m}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\bigcap_{k\in\mathbb{Z}}{\bigcap_{i<j}\big{\\{}x_{i}\neq
q^{k}x_{j}\big{\\}}}\cap\bigcap_{k\geq 1}{\bigcap_{i=1}^{m}{\big{\\{}x_{i}\neq
q^{-k}t^{1-m}\big{\\}}}}\cap\bigcap_{i=1}^{m}{\\{x_{i}\neq 0\\}},$

and admits an analytic continuation to the domain
$\bigcap\limits_{i=1}^{m}{\\{x_{i}\neq
0\\}}=(\mathbb{C}\setminus\\{0\\})^{m}$. Moreover if $\nu_{1}\geq 0$, the
function $\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ can be analytically continued to
$\mathbb{C}^{m}$.

The convergence (5.7) is uniform on compact subsets of
$(\mathbb{C}\setminus\\{0\\})^{m}$ and if $\nu_{1}\geq 0$, then it is uniform
on compact subsets of $\mathbb{C}^{m}$.

######  Remark 5.6.

For $\theta=1$, our theorem has a different form than [24, Theorem 6.5]. It is
not immediately clear that the two answers are the same.

######  Remark 5.7.

From their definition, it is clear that the rational functions
$\big{\\{}C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m})\colon$ $\tau\in
M_{\theta}^{(m)}\big{\\}}$ are holomorphic in a domain of the form
$\bigcap\limits_{-N_{1}\leq k\leq N_{1}}{\bigcap\limits_{i<j}\\{x_{i}\neq
q^{k}x_{j}\\}}$, for large enough $N_{1}\in\mathbb{N}$. In particular, all
functions $\big{\\{}C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m})\colon\tau\in
M_{\theta}^{(m)}\big{\\}}$ are holomorphic on $\mathcal{U}_{m}$.

###### Proof.

This result is a consequence of Theorem 5.2 and the multiplicative formula for
Macdonald polynomials, Theorem 4.1. Let us give more details. As before we
prove first the uniform limit on compact subsets of $\mathcal{U}_{m}$. Begin
by applying Theorem 4.1 for the signature $\lambda(N)\in\mathbb{GT}_{N}$ and
$t^{N-1}x_{i}$ instead of $x_{i}$, for $i=1,\dots,m$. Since $P_{\lambda(N)}$
is a homogeneous Laurent polynomial of degree $|\lambda(N)|$, the resulting
left-hand side is

$\displaystyle\frac{P_{\lambda(N)}\big{(}x_{1},\dots,x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}.$
(5.9)

As for the right side, the factor

$\displaystyle\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\ 0\leq
k<\theta\end{subarray}}{\big{(}t^{N-1}x_{i}-t^{N-1}x_{j}q^{k}\big{)}^{-1}}\times(q-1)^{-\theta{m\choose
2}}\times\prod_{i=1}^{m}{\frac{[\theta(N-i+1)-1]_{q}!}{[\theta N-1]_{q}!}}$

equals

$\displaystyle(-1)^{\theta{m\choose 2}}\frac{q^{-\theta^{2}(N-1){m\choose
2}}}{\prod_{\begin{subarray}{c}1\leq i<j\leq m\\\ 0\leq
k<\theta\end{subarray}}{\big{(}x_{i}-q^{k}x_{j}\big{)}}}\prod_{i=1}^{m}{\frac{1}{\big{(}t^{N-i+1};q\big{)}_{\theta(i-1)}}}.$
(5.10)

We can also obtain easily the polynomial equality

$\displaystyle\prod_{j=1}^{k}{\big{(}t^{N-1}x-q^{j-\theta}\big{)}}=(-1)^{k}q^{{k+1\choose
2}-\theta k}\prod_{i=1}^{k}{\big{(}1-xq^{\theta N-i}\big{)}},\qquad\text{for
any}\quad k\in\mathbb{N}.$

Therefore for any $k\in\mathbb{N}$, we have

$\displaystyle\prod_{i=1}^{m}\prod_{j=1}^{k-1}{\big{(}x_{i}t^{N-1}-q^{j-\theta}\big{)}}=(-1)^{m(k-1)}q^{m\left({k\choose
2}-\theta(k-1)\right)}\prod_{i=1}^{m}{\big{(}x_{i}q^{\theta
N-k+1};q\big{)}_{k-1}}.$ (5.11)

Observe also that the rational functions are invariant under the simultaneous
transformations $x_{i}\mapsto t^{N-1}x_{i}$ $\forall\,i=1,2,\dots,m$, that is

$\displaystyle
C_{\tau}^{(q,q^{\theta})}\big{(}t^{N-1}x_{1},\dots,t^{N-1}x_{m}\big{)}=C_{\tau}^{(q,q^{\theta})}(x_{1},\dots,x_{m}).$
(5.12)

By combining (5.9), (5.10), (5.11) for $k=\theta(N-m+1),\theta N$, and (5.12),
we obtain

$\displaystyle\frac{P_{\lambda(N)}\big{(}x_{1},\dots,x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}=(-1)^{\theta{m\choose
2}}q^{-2\theta^{2}{m\choose 3}-{\theta+1\choose 2}{m\choose
2}}\times\frac{1}{\prod\limits_{i=1}^{m}{\big{(}t^{N-i+1};q\big{)}_{\theta(i-1)}}}$
$\displaystyle\qquad{}\times\frac{1}{\prod\limits_{\begin{subarray}{c}1\leq
i<j\leq m\\\ 0\leq
k<\theta\end{subarray}}{\big{(}x_{i}-q^{k}x_{j}\big{)}}}\frac{1}{\prod\limits_{i=1}^{m}{\big{(}x_{i}t^{m-1}q;q\big{)}_{\theta(N-m+1)-1}}}$
$\displaystyle\qquad{}\times\widetilde{D}_{q,\theta}^{(m)}\left\\{\prod\limits_{i=1}^{m}{\frac{P_{\lambda(N)}\big{(}x_{i},t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}(x_{i}q;q)_{\theta
N-1}}\right\\}.$

The following limits hold uniformly for $(x_{1},\dots,x_{m})$ belonging to
compact subsets of $\mathcal{U}_{m}$

$\displaystyle\lim_{N\rightarrow\infty}{\prod_{i=1}^{m}{\big{(}t^{N-1+1};q\big{)}}_{\theta(i-1)}}=1,\qquad\lim_{N\rightarrow\infty}{\frac{1}{\prod\limits_{i=1}^{m}{\big{(}x_{i}t^{m-1}q;q\big{)}_{\theta(N-m+1)-1}}}}=\frac{1}{\prod\limits_{i=1}^{m}{\big{(}x_{i}t^{m-1}q;q\big{)}_{\infty}}}.$

We have moreover the following limit holds uniformly for $(x_{1},\dots,x_{m})$
belonging to compact subsets of $(\mathbb{C}\setminus\\{0\\})^{m}$, because of
Theorem 5.2,

$\displaystyle\lim_{N\rightarrow\infty}{\prod_{i=1}^{m}{\frac{P_{\lambda(N)}\big{(}x_{i},t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}(x_{i}q;q)_{\theta
N-1}}}=\prod_{i=1}^{m}{\Phi^{\nu}(x_{i};q,t)(x_{i}q;q)_{\infty}}.$

It is not difficult to observe that if $U\subset\mathbb{C}^{m}$ is a domain
preserved by the map of multiplication by $q$, and $\\{f_{n}\\}_{n\geq 1}$,
$f$ are sequences of holomorphic functions on $U$ for which
$\lim\limits_{n\rightarrow\infty}{f_{n}(x)}=f(x)$ uniformly on compact subsets
of $U$, then

$\displaystyle\lim_{n\rightarrow\infty}{T_{q,x}^{s}f_{n}(x)}=\lim_{n\rightarrow\infty}{f_{n}(q^{s}x)}=f(q^{s}x)=T_{q,x}^{s}f(x)$

uniformly for $x$ belonging to compact subsets of $U$. As an implication, the
order of the limit as $N\rightarrow\infty$ and the $q$-difference operator
$\widetilde{D}_{q,\theta}^{(m)}$ can be interchanged. All the considerations
above immediately imply the desired uniform limit for $(x_{1},\dots,x_{m})$
belonging to compact subsets of $\mathcal{U}_{m}$.

As in the proof of Theorem 5.2, the limit in the statement will hold also
uniformly for compact subsets of $(\mathbb{C}\setminus\\{0\\})^{m}$ if we show
that $\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ admits an analytic continuation to
this larger domain. The extension of Riemann’s theorem for removable
singularities for several complex variables, [45, Theorem 8], shows that
$\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ admits an analytic continuation to all
$\left(\\{z\in\mathbb{C}\setminus\\{0\\}\colon|z|\leq R\\}\right)^{m}$ if we
showed only that $\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ is bounded on
$(\\{z\in\mathbb{C}\setminus\\{0\\}\colon|z|\leq
R\\})^{m}\cap\mathcal{U}_{m}$. The latter can be proved by repeating the
argument in the proof of Theorem 5.2 above.

Finally, if $\nu_{1}\geq 0$ then
$\lim\limits_{N\rightarrow\infty}{\lambda_{N}(N)}=\nu_{1}$ implies
$\lambda(N)_{N}\geq 0$ for large enough $N$. It follows that
$P_{\lambda(N)}\big{(}x_{1},\dots,x_{m},t^{-m},\dots,t^{1-N}\big{)}/P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}$
is a polynomial in $x_{1},\dots,x_{m}$ for large enough $N$, and therefore
holomorphic on $\mathbb{C}^{m}$. The same argument as in the proof of Theorem
5.2 again shows $\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ admits an analytic
continuation to $\mathbb{C}^{m}$ and the limit holds uniformly for compact
subsets of $\mathbb{C}^{m}$. ∎

##  6 Preliminaries on the $\boldsymbol{(q,t)}$-Gelfand–Tsetlin graph

In this section, assume $q,t\in(0,1)$. We use the notation
$\mathbb{P}-\lim\limits_{k\rightarrow\infty}M_{k}=M$ to indicate that a
sequence of probability measures $\\{M_{k}\\}_{k\geq 1}$ converges weakly to
$M$.

###  6.1 The $\boldsymbol{(q,t)}$-Gelfand–Tsetlin graph

The $(q,t)$-Gelfand–Tsetlin graph is an undirected, $\mathbb{Z}_{\geq
0}$-graded graph with countable vertices, together with a sequence of
cotransition probabilities between the levels of the graph (considered as
discrete spaces).

Begin by defining the set of vertices of the graph as the set of all
signatures

$\displaystyle\mathbb{GT}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\bigsqcup_{N\geq 0}{\mathbb{GT}_{N}},$

where, for convenience, we have also included the singleton
$\mathbb{GT}_{0}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\\{\varnothing\\}$.

The edges are determined by the interlacing constraints: edges only join
vertices associated to signatures whose lengths differ by $1$ and
$\mu\in\mathbb{GT}_{N}$ is joined to $\lambda\in\mathbb{GT}_{N+1}$ if and only
if $\mu\prec\lambda$, i.e.,
$\lambda_{N+1}\leq\mu_{N}\leq\cdots\leq\lambda_{2}\leq\mu_{1}\leq\lambda_{1}$.
We also assume $\varnothing\prec(k)$, $\forall\,(k)\in\mathbb{GT}_{1}$. We
call the graph with vertices and edges just described the Gelfand–Tsetlin
graph222GT graph, for short.. Next we introduce a $(q,t)$-deformation of the
GT graph by considering certain cotransition probabilities.

######  Definition 6.1.

Consider the numbers $\Lambda^{N+1}_{N}(\lambda,\mu)$, $\mu\prec\lambda$,
defined by the expression

$\displaystyle\Lambda^{N+1}_{N}(\lambda,\mu)=t^{|\mu|}\psi_{\lambda/\mu}(q,t)\frac{P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N}\big{)}}=\psi_{\lambda/\mu}(q,t)\frac{P_{\mu}\big{(}t,t^{2},\dots,t^{N}\big{)}}{P_{\lambda}(1,t,\dots,t^{N}\big{)}},$
(6.1)

for all $N\in\mathbb{Z}_{\geq 0}$, $\lambda\in\mathbb{GT}_{N+1}$,
$\mu\in\mathbb{GT}_{N}$, $\mu\prec\lambda$, and where the branching
coefficients $\psi_{\lambda/\mu}(q,t)$ are defined in Theorem 2.5. For
convenience, also let $\Lambda^{N+1}_{N}(\lambda,\mu)=0$ if
$\mu\not\prec\lambda$, i.e., if $\mu$ is not adjacent to $\lambda$.

The $(q,t)$-Gelfand–Tsetlin graph333$(q,t)$-GT graph, for short. is the
sequence $\big{\\{}\mathbb{GT}_{N},\Lambda^{N+1}_{N}\colon
N=0,1,2,\dots\big{\\}}$ of data consisting of the GT graph and the
$\mathbb{GT}_{N+1}\times\mathbb{GT}_{N}$ matrices
$\big{[}\Lambda^{N+1}_{N}(\lambda,\mu)\big{]}$, $N\geq 0$, defined above.

In general, the numbers $\Lambda^{N+1}_{N}(\lambda,\mu)$ depend on the values
$q$, $t$, but for simplicity we omit that dependence from the notation. By
virtue of the evaluation identity, Theorem 2.3, and the assumption
$q,t\in(0,1)$, we have

$\displaystyle\Lambda^{N+1}_{N}(\lambda,\mu)\geq
0,\qquad\forall\,\lambda\in\mathbb{GT}_{N+1},\quad\mu\in\mathbb{GT}_{N}.$
(6.2)

Moreover, the branching rule for Macdonald polynomials, Theorem 2.5, shows

$\displaystyle
P_{\lambda}\big{(}1,t,t^{2},\dots,t^{N}\big{)}=\sum_{\mu\colon\mu\prec\lambda}{\psi_{\lambda/\mu}(q,t)P_{\mu}\big{(}t,t^{2},\dots,t^{N}\big{)}}$

and then

$\displaystyle
1=\sum_{\mu\colon\mu\prec\lambda}{\Lambda^{N+1}_{N}(\lambda,\mu)}.$ (6.3)

Equations (6.2) and (6.3) show that
$\big{[}\Lambda^{N+1}_{N}(\lambda,\mu)\big{]}$ is a stochastic matrix of
format $\mathbb{GT}_{N+1}\times\mathbb{GT}_{N}$, for each $N\geq 0$. Thus
$\Lambda^{N+1}_{N}$ determines a Markov kernel
$\mathbb{GT}_{N+1}\dashrightarrow\mathbb{GT}_{N}$. For this reason, we call
$\Lambda^{N+1}_{N}(\lambda,\mu)$ the cotransition probabilities. Let us also
define the more general Markov kernels
$\Lambda^{M}_{N}\colon\mathbb{GT}_{M}\dashrightarrow\mathbb{GT}_{N}$, $M>N$,
by

$\displaystyle\Lambda^{M}_{N}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\Lambda^{M}_{M-1}\Lambda^{M-1}_{M-2}\cdots\Lambda^{N+1}_{N},$

or more explicitly

$\displaystyle\Lambda^{M}_{N}(\lambda,\mu)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{\lambda\succ\lambda^{(M-1)}\succ\cdots\succ\lambda^{(N+1)}\succ\mu}{\Lambda^{M}_{M-1}\big{(}\lambda,\lambda^{(M-1)}\big{)}\cdots\Lambda^{N+1}_{N}\big{(}\lambda^{(N+1)},\mu\big{)}}.$

By duality, the kernel $\Lambda^{M}_{N}$ also determines a map
$\mathcal{M}_{\rm prob}(\mathbb{GT}_{M})\rightarrow\mathcal{M}_{\rm
prob}(\mathbb{GT}_{N})$ between the spaces of probability measures on
$\mathbb{GT}_{M}$ and $\mathbb{GT}_{N}$, that we denote by the same symbol
$\Lambda^{M}_{N}$. For example, if $\lambda\in\mathbb{GT}_{M}$ and
$\delta_{\lambda}$ is the delta mass at $\lambda$, then
$\Lambda^{M}_{N}\delta_{\lambda}$ is the probability measure on
$\mathbb{GT}_{N}$ given by

$\displaystyle\Lambda^{M}_{N}\delta_{\lambda}(\mu)=\Lambda^{M}_{N}(\lambda,\mu).$
(6.4)

######  Definition 6.2.

A sequence $\\{M_{N}\\}_{N\geq 0}$, such that each $M_{N}$ is a probability
measure on $\mathbb{GT}_{N}$, is called a $(q,t)$-coherent sequence if the
following relations are satisfied

$\displaystyle
M_{N}(\mu)=\sum_{\lambda\in\mathbb{GT}_{N+1}}{M_{N+1}(\lambda)\Lambda^{N+1}_{N}(\lambda,\mu)},\qquad\forall\,N\geq
0,\quad\forall\,\mu\in\mathbb{GT}_{N}.$ (6.5)

Similarly, a finite sequence $\\{M_{N}\\}_{N=0,1,\dots,k}$ is said to be a
$(q,t)$-coherent sequence if the relations (6.5) hold for $N=0,1,\dots,k-1$
and all $\mu\in\mathbb{GT}_{N}$.

It is clear that for any probability measure $M_{N}$ on $\mathbb{GT}_{N}$,
there exist probability measures $M_{0},M_{1},\dots,M_{N-1}$ on
$\mathbb{GT}_{0},\mathbb{GT}_{1},\dots,\mathbb{GT}_{N-1}$ such that
$\\{M_{m}\\}_{m=0,1,\dots,N}$ is a $(q,t)$-coherent sequence, and moreover
$M_{0},\dots,M_{N-1}$ are uniquely determined by this condition: in fact,
$M_{m}=\Lambda^{N}_{m}M_{N}$ $\forall\,0\leq m\leq N-1$.

The set of (infinite) $(q,t)$-coherent sequences $\\{M_{N}\\}_{N\geq 0}$ is a
convex set. Theorem 1.3 is, in different terms, a characterization of the
extreme points of the set of $(q,t)$-coherent sequences.

###  6.2 The path-space $\boldsymbol{\mathcal{T}}$ and
$\boldsymbol{(q,t)}$-central measures

The set of $(q,t)$-coherent sequences defined before is in bijection with a
class of probability measures in the path-space of the GT graph $\mathcal{T}$
that we define next.

######  Definition 6.3.

The path-space $\mathcal{T}$ is the set of (infinite) paths in the GT graph
that begin at $\varnothing\in\mathbb{GT}_{0}$:

$\displaystyle\mathcal{T}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\big{\\{}\tau=\big{(}\varnothing=\tau^{(0)}\prec\tau^{(1)}\prec\tau^{(2)}\prec\cdots\big{)}\colon\tau^{(n)}\in\mathbb{GT}_{n}\
\forall n\geq 0\big{\\}}.$

For any finite path
$\phi=(\phi^{(0)}\prec\phi^{(1)}\prec\dots\prec\phi^{(n)})$, define the
cylinder set $S_{\phi}$ (or simply $S(\phi)$) by

$\displaystyle
S_{\phi}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\big{\\{}\tau\in\mathcal{T}\colon\tau^{(i)}=\phi^{(i)}\
\forall\,i=0,1,\dots,n\big{\\}}\subset\mathcal{T}.$

The set $\mathcal{T}$ is equipped with the $\sigma$-algebra generated by the
cylinder sets $S_{\phi}$, where $\phi$ varies over all finite paths in the GT
graph. We always consider $\mathcal{T}$ as a measurable space.

An interesting class of probability measures on $\mathcal{T}$ consists of the
ones that are coherent with the sequence of stochastic matrices
$\big{\\{}\Lambda^{N+1}_{N}\big{\\}}_{N\geq 0}$. To clarify what such
coherence is, define the natural projection maps

$\displaystyle\operatorname{Proj}_{N}\colon\ \mathcal{T}\subset\prod_{n\geq
0}{\mathbb{GT}_{n}}\longrightarrow\mathbb{GT}_{N},$
$\displaystyle\hphantom{\operatorname{Proj}_{N}\colon}{}\
\tau=\big{(}\tau^{(0)}\prec\tau^{(1)}\prec\tau^{(2)}\prec\cdots\big{)}\mapsto\tau^{(N)}.$

It is a standard exercise to show that the $\sigma$-algebra of $\mathcal{T}$
is the smallest one for which all the maps $\operatorname{Proj}_{N}$ are
measurable. Consequently, for any probability measure $M$ on $\mathcal{T}$, we
can associate to it the sequence of its pushforwards under the maps
$\operatorname{Proj}_{N}$, namely the sequence $\\{M_{N}\\}_{N\geq 0}$,
$M_{N}=(\operatorname{Proj}_{N})_{*}M$.

######  Definition 6.4.

A probability measure $M$ on $\mathcal{T}$ is said to be a $(q,t)$-central
measure if the following relations hold

$\displaystyle
M\big{(}S\big{(}\phi^{(0)}\prec\phi^{(1)}\dots\prec\phi^{(N-1)}\prec\phi^{(N)}\big{)}\big{)}=\Lambda^{N}_{N-1}\big{(}\phi^{(N)},\phi^{(N-1)}\big{)}\cdots\Lambda^{1}_{0}\big{(}\phi^{(1)},\phi^{(0)}\big{)}M_{N}\big{(}\phi^{(N)}\big{)}$
$\displaystyle\qquad{}=t^{|\phi^{(N-1)}|+\cdots+|\phi^{(0)}|}\frac{\psi_{\phi^{(N)}/\phi^{(N-1)}}(q,t)\cdots\psi_{\phi^{(1)}/\phi^{(0)}}(q,t)}{P_{\phi^{(N)}}\big{(}1,t,\dots,t^{N-1}\big{)}}M_{N}\big{(}\phi^{(N)}\big{)},$

for all $N\geq 0$, all finite paths
$\phi=\big{(}\phi^{(0)}\prec\dots\prec\phi^{(N)}\big{)}$, and for some
probability measures $M_{N}$ on $\mathbb{GT}_{N}$. The branching coefficients
$\psi_{\mu/\nu}(q,t)$ are explicit in the statement of Theorem 2.5.

One can verify easily that, if the relations above hold, then the measure
$M_{N}$ is the pushforward $(\operatorname{Proj}_{N})_{*}M$, for all $N\geq
0$. Moreover, $\\{M_{N}\\}_{N\geq 0}$ is automatically a $(q,t)$-coherent
sequence.

We denote by $M_{\rm prob}(\mathcal{T})$ the set of $(q,t)$-central
(probability) measures. The set of $(q,t)$-central measures is a convex set.
The set of extreme points of $M_{\rm prob}(\mathcal{T})$, equipped with its
inherited topology, is called the boundary of the $(q,t)$-GT graph444We may
also call it the minimal boundary of the $(q,t)$-GT graph, to differentiate it
from the Martin boundary of the $(q,t)$-GT graph, see Section 7.1 below. and
is denoted by $\Omega_{q,t}$.

The following proposition implies that the correspondence between the set of
$(q,t)$-central measures and the set of $(q,t)$-coherent sequences is a
bijection.

######  Proposition 6.5.

Any probability measure $M$ on $\mathcal{T}$ has an associated sequence
$\\{M_{N}\\}_{N\geq 0}$ of probability measures on
$\\{\mathbb{GT}_{N}\\}_{N\geq 0}$, as shown above. The map
$M\mapsto\\{M_{N}\\}_{N\geq 0}$ is a bijection between the set $M_{\rm
prob}(\mathcal{T})$ of $(q,t)$-central measures on $\mathcal{T}$, and the set
of $(q,t)$-coherent sequences. The bijection is an isomorphism of convex sets.

Let $\big{\\{}M^{(i)}\big{\\}}_{i\geq 1},M$ be elements on $M_{\rm
prob}(\mathcal{T})$ and
$\big{\\{}M^{(i)}_{m}\big{\\}}_{\begin{subarray}{c}i\geq 1,m\geq
0\end{subarray}}$, $\\{M_{m}\\}_{m\geq 0}$ be their corresponding
$(q,t)$-coherent sequences. Then the weak limit
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{(i)}}=M$ holds if and only if
the weak limits
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{(i)}_{m}}=M_{m}$ hold for all
$m\in\mathbb{N}$.

###### Proof.

Similar statements are known for other branching graphs, e.g., the case $t=q$
of our proposition is given in [21, Propositions 4.4 and 4.9], and the case
$t=q\rightarrow 1$ is in [40, Proposition 10.3]. In the case $t=q$, this
proposition is given in [21, Propositions 4.4 and 4.9]. The proof at our level
of generality can be easily adapted from the proofs in [21]; details are left
to the reader. ∎

###  6.3 Macdonald generating functions

We introduce Macdonald generating functions, which will be very helpful in our
study of $(q,t)$-coherent sequences.

######  Definition 6.6.

Let $M_{N}$ be a probability measure on $\mathbb{GT}_{N}$, then its Macdonald
generating function is the formal sum

$\displaystyle\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{\lambda\in\mathbb{GT}_{N}}{M_{N}(\lambda)\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{N}t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}.$

Note that $\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})$ depends on the values $q$,
$t$, but we omit such dependence for simplicity.

The sum above is absolutely convergent on the torus
$(x_{1},\dots,x_{m})\in\mathbb{T}^{m}$. In fact, Theorem 2.5 and the fact that
all the branching coefficients $\psi_{\mu/\nu}(q,t)$ are nonnegative imply
$\big{|}P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{N}t^{N-1}\big{)}\big{|}\leq
P_{\lambda}\big{(}|x_{1}|,|x_{2}t|,\dots,\big{|}x_{N}t^{N-1}\big{|}\big{)}=P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}$.
Thus not only is $\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})$ well-defined as a
function on $\mathbb{T}^{N}$, but also
$\sup\limits_{x_{1},\dots,x_{N}\in\mathbb{T}}{|\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})|}\leq
1$. Therefore $\mathcal{P}_{M_{N}}\in L^{\infty}(\mathbb{T}^{m})$.

If $M_{N}$ is supported on the set of positive signatures
$\mathbb{GT}^{+}_{N}$, then each
$P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{N}t^{N-1}\big{)}$ is a polynomial in
$x_{1},\dots,x_{N}$ and therefore the sum defining $\mathcal{P}_{M_{N}}$ is
absolutely convergent on the closed unit disk
$(x_{1},\dots,x_{m})\in\mathbb{D}^{m}$. Moreover $\mathcal{P}_{M_{N}}\in
L^{\infty}(\mathbb{D}^{m})$ if $M_{N}$ is supported on $\mathbb{GT}^{+}_{N}$.

In general, $\mathcal{P}_{M_{N}}\in L^{\infty}(\mathbb{T}^{m})\subset
L^{2}(\mathbb{T}^{m})$. The Fourier series expansion of $\mathcal{P}_{M_{N}}$
can be obtained by using Corollary 2.6. In fact,

$\displaystyle\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})=\sum_{\lambda\in\mathbb{GT}_{N}}{M_{N}(\lambda)\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{N}t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}$
$\displaystyle\hphantom{\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})}{}=\sum_{\lambda\in\mathbb{GT}_{N}}{M_{N}(\lambda)\sum_{\mu\in\mathbb{GT}_{N}}{\frac{c_{\lambda,\mu}m_{\mu}\big{(}x_{1},\dots,x_{N}t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}}$
$\displaystyle\hphantom{\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})}{}=\sum_{\mu\in\mathbb{GT}_{N}}{m_{\mu}\big{(}x_{1},\dots,x_{N}t^{N-1}\big{)}\sum_{\lambda\in\mathbb{GT}_{N}}{\frac{c_{\lambda,\mu}M_{N}(\lambda)}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}},$

where the interchange in the order of summation follows from the absolute
convergence of all the sums involved. (For the absolute convergence, the
nonnegativity of all coefficients $c_{\lambda,\mu}$ is needed.) From the
expansion above, we can extract the coefficient of $x_{1}^{\kappa_{1}}\cdots
x_{N}^{\kappa_{N}}$ in the Fourier series, for any
$\kappa=(\kappa_{1}\geq\dots\geq\kappa_{N})\in\mathbb{GT}_{N}$. In fact, such
term appears only in $m_{\kappa}\big{(}x_{1},\dots,x_{N}t^{N-1}\big{)}$ with
coefficient $t^{n(\kappa)}$, where
$n(\kappa)=\kappa_{2}+2\kappa_{3}+\cdots+(N-1)\kappa_{N}$. Thus the Fourier
coefficient of $x_{1}^{\kappa_{1}}\cdots x_{N}^{\kappa_{N}}$ in the expansion
of $\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})$ is

$\displaystyle
f_{\kappa_{1},\dots,\kappa_{N}}=t^{n(\kappa)}\sum_{\lambda\in\mathbb{GT}_{N}}{\frac{c_{\lambda,\kappa}M_{N}(\lambda)}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}.$
(6.6)

If $M_{N}$ is supported on $\mathbb{GT}^{+}_{N}$, then the sum defining
$f_{\kappa_{1},\dots,\kappa_{N}}$ above is finite. Indeed the only signatures
with a nonzero contribution are $\lambda\in\mathbb{GT}^{+}_{N}$ with
$|\lambda|=|\kappa|$. But then $|\kappa|=|\lambda|\geq\lambda_{1}$, and there
are finitely many signatures $\lambda\in\mathbb{GT}_{N}$ with
$|\kappa|\geq\lambda_{1}\geq\dots\geq\lambda_{N}\geq 0$. This observation will
be put to use several times.

######  Lemma 6.7.

Let $M_{N}$, $M_{N}^{\prime}$ be probability measures on $\mathbb{GT}_{N}$
that are supported on $\mathbb{GT}^{+}_{N}$. If

$\displaystyle\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})=\mathcal{P}_{M_{N}^{\prime}}(x_{1},\dots,x_{N})\qquad\forall\,(x_{1},\dots,x_{N})\in\mathbb{T}^{N}$

then $M_{N}=M_{N}^{\prime}$.

###### Proof.

Both $\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})$ and
$\mathcal{P}_{M_{N}^{\prime}}(x_{1},\dots,x_{N})$ belong to
$L^{2}\big{(}\mathbb{T}^{N}\big{)}$. The equality of these functions implies
that their Fourier coefficients agree. From (6.6), this means

$\displaystyle\sum_{\mu\in\mathbb{GT}^{+}_{N}}{\frac{c_{\mu,\kappa}M_{N}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}}}=\sum_{\mu\in\mathbb{GT}^{+}_{N}}{\frac{c_{\mu,\kappa}M_{N}^{\prime}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}}}\qquad\forall\,\kappa\in\mathbb{GT}_{N}.$
(6.7)

Observe that we have restricted the sum above to $\mu\in\mathbb{GT}^{+}_{N}$,
because $M_{N}$, $M_{N}^{\prime}$ are supported on positive signatures. Let
$n\in\mathbb{Z}_{\geq 0}$ be arbitrary. We show that
$M_{N}(\kappa)=M_{N}^{\prime}(\kappa)$ for all $\kappa\in\mathbb{GT}^{+}_{N}$
with $|\kappa|=n$.

Let $C$ be the matrix whose rows and columns are parametrized by
$\lambda\in\mathbb{GT}^{+}_{N}$ with $|\lambda|=n$, and such that its entry
$C(\kappa,\mu)$ is $c_{\mu,\kappa}/P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}$.
Observe that $C$ is a finite and square matrix. Also let $M$ (resp.
$M^{\prime}$) be the column vector whose entries are parametrized by
$\lambda\in\mathbb{GT}^{+}_{N}$ with $|\lambda|=n$ and whose entry $\mu$ is
$M_{N}(\mu)$ (resp. $M_{N}^{\prime}(\mu)$). Then (6.7) yields
$CM=CM^{\prime}$. The matrix $C$ is upper-triangular with respect to the order
$\geq$ on signatures because $c_{\mu,\kappa}=0$ unless $\mu\geq\kappa$.
Moreover the diagonal entries are
$c_{\mu,\mu}/P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}=1/P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}\neq
0$. It follows that $C$ has an inverse and
$M=C^{-1}CM=C^{-1}CM^{\prime}=M^{\prime}$. Therefore
$M_{N}(\kappa)=M_{N}^{\prime}(\kappa)$ $\forall\,\kappa\in\mathbb{GT}^{+}_{N}$
with $|\kappa|=n$. Since $n\in\mathbb{Z}_{\geq 0}$ was arbitrary and both
$M_{N}$, $M_{N}^{\prime}$ are supported on $\mathbb{GT}^{+}_{N}$, we conclude
$M_{N}=M_{N}^{\prime}$. ∎

######  Proposition 6.8.

If the sequence $\\{M_{N}\\}_{N\geq 0}$ $($resp. finite sequence
$\\{M_{N}\\}_{N=0,1,\dots,k})$ is a $(q,t)$-coherent sequence, then

$\displaystyle\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})=\mathcal{P}_{M_{N+1}}(1,x_{1},\dots,x_{N}),\qquad\forall\,(x_{1},\dots,x_{N})\in\mathbb{T}^{N}$

for all $N\geq 0$ $($resp. for all $N=0,1,\dots,k-1)$. The converse statement
holds if, for each $N\geq 0$ $($resp. $N=0,1,\dots,k-1)$, $M_{N}$ is supported
on $\mathbb{GT}^{+}_{N}$.

###### Proof.

Let us prove the first part. Let $\\{M_{N}\\}_{N\geq 0}$ be a $(q,t)$-coherent
sequence. By making use of the branching rule, Theorem 2.5, the fact that
$P_{\mu}$ is homogeneous of degree $|\mu|$, and making a change in the order
of summation, we obtain

$\displaystyle\mathcal{P}_{M_{N+1}}(1,x_{1},\dots,x_{N})=\sum_{\lambda\in\mathbb{GT}_{N+1}}{M_{N+1}(\lambda)\frac{P_{\lambda}\big{(}1,x_{1}t,\dots,x_{N}t^{N}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N}\big{)}}}$
$\displaystyle\qquad{}=\sum_{\lambda\in\mathbb{GT}_{N+1}}{\frac{M_{N+1}(\lambda)}{P_{\lambda}\big{(}1,t,\dots,t^{N}\big{)}}\sum_{\mu\in\mathbb{GT}_{N}}{\psi_{\lambda/\mu}(q,t)P_{\mu}\big{(}x_{1}t,\dots,x_{N}t^{N}\big{)}}}$
$\displaystyle\qquad{}=\sum_{\lambda\in\mathbb{GT}_{N+1}}{\frac{M_{N+1}(\lambda)}{P_{\lambda}\big{(}1,t,\dots,t^{N}\big{)}}\sum_{\mu\in\mathbb{GT}_{N}}{t^{|\mu|}\psi_{\lambda/\mu}(q,t)P_{\mu}\big{(}x_{1},\dots,x_{N}t^{N-1}\big{)}}}$
$\displaystyle\qquad{}=\sum_{\mu\in\mathbb{GT}_{N}}{\frac{P_{\mu}(x_{1},\dots,x_{N}t^{N-1})}{P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}}\sum_{\lambda\in\mathbb{GT}_{N+1}}{t^{|\mu|}\psi_{\lambda/\mu}(q,t)\frac{P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N}\big{)}}M_{N+1}(\lambda)}}$
$\displaystyle\qquad{}=\sum_{\mu\in\mathbb{GT}_{N}}{\frac{P_{\mu}\big{(}x_{1},\dots,x_{N}t^{N-1}\big{)}}{P_{\mu}\big{(}1,t,\dots,t^{N-1}\big{)}}M_{N}(\mu)}=\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N}).$

We can easily show that all sums above are absolutely convergent, so the
change in the order of summations can be justified.

Next we prove the converse statement. Assume that $M_{N}$, $M_{N+1}$ are
probability measures on $\mathbb{GT}_{N}$, $\mathbb{GT}_{N+1}$. Assume that
they are supported on $\mathbb{GT}^{+}_{N}$, $\mathbb{GT}^{+}_{N+1}$,
respectively, and also that
$\mathcal{P}_{M_{N}}(x_{1},\dots,x_{N})=\mathcal{P}_{M_{N+1}}(1,x_{1},\dots,x_{N})$
on $\mathbb{T}^{N}$. Let $M_{N}^{\prime}$ be the measure on $\mathbb{GT}_{N}$
defined by

$\displaystyle
M_{N}^{\prime}(\mu)=\sum_{\lambda\in\mathbb{GT}_{N+1}}{M_{N+1}(\lambda)\Lambda^{N+1}_{N}(\lambda,\mu)}\qquad\forall\,\mu\in\mathbb{GT}_{N}.$

Since $\big{[}\Lambda^{N+1}_{N}(\lambda,\mu)\big{]}$ is a stochastic matrix
and $M_{N+1}$ is a probability measure on $\mathbb{GT}_{N+1}$, then
$M_{N}^{\prime}$ is a probability measure on $\mathbb{GT}_{N}$. Moreover since
$M_{N+1}$ is supported on $\mathbb{GT}^{+}_{N+1}$, it follows that
$M_{N}^{\prime}$ is supported on $\mathbb{GT}^{+}_{N}$. In fact, if
$\mu\notin\mathbb{GT}_{N}\setminus\mathbb{GT}^{+}_{N}$ (or equivalently
$\mu_{N}<0$), then for any $\lambda\in\mathbb{GT}_{N+1}$, either
$\lambda_{N+1}<0$ in which case $M_{N+1}(\lambda)=0$, or $\lambda_{N+1}\geq 0$
in which case $\Lambda^{N+1}_{N}(\lambda,\mu)=0$.

We will be done if we showed $M_{N}=M_{N}^{\prime}$. From what we have proved
in the first part of the argument, we have
$\mathcal{P}_{M_{N}^{\prime}}(x_{1},\dots,x_{N})=\mathcal{P}_{M_{N+1}}(1,x_{1},\dots,x_{N})$
on $\mathbb{T}^{N}$. It follows that
$\mathcal{P}_{M_{N}}=\mathcal{P}_{M_{N}^{\prime}}$ on $\mathbb{T}^{N}$. An
application of Lemma 6.7 concludes the proof. ∎

######  Proposition 6.9.

Let $N\in\mathbb{N}$ be arbitrary. If $\\{M^{m}\\}_{m\geq 1}$, $M$ are all
probability measures on $\mathbb{GT}_{N}$ such that the weak convergence holds
$\mathbb{P}-\lim\limits_{m\rightarrow\infty}{M^{m}}=M$, then

$\displaystyle\lim_{m\rightarrow\infty}{\mathcal{P}_{M^{m}}(x_{1},\dots,x_{N})}=\mathcal{P}_{M}(x_{1},\dots,x_{N})\qquad\forall\,(x_{1},\dots,x_{N})\in\mathbb{T}^{N}.$
(6.8)

The convergence above is uniform on $\mathbb{T}^{N}$.

###### Proof.

Let $\epsilon>0$ be a very small real number. Since $M$ is a probability
measure, there exists $c>0$ large enough so that

$\displaystyle M\left(\\{\lambda\in\mathbb{GT}_{N}\colon
c\geq\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N}\geq-c\\}\right)>1-\epsilon.$
(6.9)

From the weak convergence
$\mathbb{P}-\lim\limits_{m\rightarrow\infty}{M^{m}}=M$, there exists
$N_{1}\in\mathbb{N}$ so that $m>N_{1}$ implies

$\displaystyle M^{m}\left(\\{\lambda\in\mathbb{GT}_{N}\colon
c\geq\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{N}\geq-c\\}\right)>1-2\epsilon.$
(6.10)

Consider the set
$\mathbb{GT}_{N}^{[-c,c]}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\\{\lambda\in\mathbb{GT}_{N}\colon
c\geq\lambda_{1}\geq\dots\geq\lambda_{N}\geq-c\\}\subset\mathbb{GT}_{N}$. It
is clear that $\mathbb{GT}_{N}^{[-c,c]}$ is finite and has cardinality no
greater than $(2c+1)^{N}$. Also, since $\mathbb{GT}_{N}^{[-c,,c]}$ is a finite
set, there exists $N_{2}\in\mathbb{N}$ such that $m>N_{2}$ implies

$\displaystyle\left|M(\lambda)-M^{m}(\lambda)\right|<\frac{\epsilon}{(2c+1)^{N}}\qquad\forall\,\lambda\in\mathbb{GT}_{N}^{[-c,c]}.$
(6.11)

We are ready to make the desired estimate. Use (6.9), (6.10), (6.11) and the
triangle inequality to argue that for any $m>\max\\{N_{1},N_{2}\\}$, we have

$\displaystyle\sup_{(x_{1},\dots,x_{N})\in\mathbb{T}^{N}}{\left|\mathcal{P}_{M^{m}}(x_{1},\dots,x_{N})-\mathcal{P}_{M}(x_{1},\dots,x_{N})\right|}$
$\displaystyle\qquad{}\leq\sum_{\lambda\in\mathbb{GT}_{N}^{[-c,c]}}{|M^{m}(\lambda)-M(\lambda)|\sup_{(x_{1},\dots,x_{N})\in\mathbb{T}^{N}}{\left|\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{N}t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}\right|}}$
$\displaystyle\qquad\quad{}+\sum_{\lambda\in\mathbb{GT}_{N}\setminus\mathbb{GT}_{N}^{[-c,c]}}{\left(M^{m}(\lambda)+M(\lambda)\right)\sup_{(x_{1},\dots,x_{N})\in\mathbb{T}^{N}}{\left|\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{N}t^{N-1}\big{)}}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}\right|}}$
$\displaystyle\qquad{}\leq\sum_{\lambda\in\mathbb{GT}_{N}^{[-c,c]}}{|M^{m}(\lambda)-M(\lambda)|}+\sum_{\lambda\in\mathbb{GT}_{N}\setminus\mathbb{GT}_{N}^{[-c,c]}}{\left(M^{m}(\lambda)+M(\lambda)\right)}$
$\displaystyle\qquad{}\leq\frac{\epsilon}{(2c+1)^{N}}\big{|}\mathbb{GT}_{N}^{[-c,c]}\big{|}+M^{m}\big{(}\mathbb{GT}_{N}\setminus\mathbb{GT}_{N}^{[-c,c]}\big{)}+M\big{(}\mathbb{GT}_{N}\setminus\mathbb{GT}_{N}^{[-c,c]}\big{)}$
(6.12) $\displaystyle\qquad{}\leq\epsilon+2\epsilon+\epsilon=4\epsilon.$ ∎

A partial converse to the previous proposition is Proposition 6.10 below.

######  Proposition 6.10.

Let $\\{M^{m}\\}_{m\geq 1}$, $M$, be probability measures on
$\mathbb{GT}_{N}$, supported on $\mathbb{GT}^{+}_{N}$. Assume the following
convergence

$\displaystyle\lim_{m\rightarrow\infty}{\mathcal{P}_{M^{m}}(x_{1},\dots,x_{N})}=\mathcal{P}_{M}(x_{1},\dots,x_{N})$

holds uniformly on $\mathbb{T}^{N}$, then we have the weak convergence
$\mathbb{P}-\lim\limits_{m\rightarrow\infty}{M^{m}}=M$.

###### Proof.

All functions $\\{\mathcal{P}_{M^{m}}\\}_{m\geq 1}$, $\mathcal{P}_{M}$ belong
to $L^{2}(\mathbb{T}^{m})$. Therefore the limit
$\lim\limits_{m\rightarrow\infty}\mathcal{P}_{M^{m}}(x_{1},\dots,$
$x_{N})=\mathcal{P}_{M}(x_{1},\dots,x_{N})$ implies the convergence of the
Fourier coefficients. Due to (6.6), this implies that, for any
$\kappa\in\mathbb{GT}_{N}$, we have

$\displaystyle\lim_{m\rightarrow\infty}{\sum_{\lambda\in\mathbb{GT}^{+}_{N}}{\frac{c_{\lambda,\kappa}M^{m}(\lambda)}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}}=\sum_{\lambda\in\mathbb{GT}^{+}_{N}}{\frac{c_{\lambda,\kappa}M(\lambda)}{P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}}}.$

We show that $\lim\limits_{m\rightarrow\infty}{M^{m}(\lambda)}=M(\lambda)$ for
any $\lambda\in\mathbb{GT}^{+}_{N}$. In fact, let $n\in\mathbb{Z}_{\geq 0}$ be
arbitrary and let us prove
$\lim\limits_{m\rightarrow\infty}{M^{m}(\lambda)}=M(\lambda)$ for any
$\lambda\in\mathbb{GT}^{+}_{N}$ with $|\lambda|=n$. Consider the finite,
square matrix $C$ whose rows and columns are parametrized by
$\lambda\in\mathbb{GT}^{+}_{N}$, $|\lambda|=n$, and whose entries are
$C(\kappa,\lambda)=c_{\lambda,\kappa}/P_{\lambda}\big{(}1,t,\dots,t^{N-1}\big{)}$.
Also let $\\{M^{m}\\}_{m\geq 0},M$ be column vectors whose entries are
parametrized by $\lambda\in\mathbb{GT}^{+}_{N}$ with $|\lambda|=n$, and whose
entries, at $\lambda\in\mathbb{GT}^{+}_{N}$, are $\\{M^{m}(\lambda)\\}_{m\geq
0},M(\lambda)$. From the limit relation above, we have the entrywise limit of
column vectors $\lim\limits_{m\rightarrow\infty}{CM^{m}}=CM$. The matrix $C$
is upper triangular (with respect to the order on signatures given in
Corollary 2.6) and has nonzero diagonal entries, thus it has an inverse
$C^{-1}$. Each entry of the column vectors
$M^{m}=\big{(}C^{-1}C\big{)}M^{m}=C^{-1}(CM^{m})$ is a finite linear
combination of entries of $CM^{m}$, and the same can be said about the entries
of $C^{-1}CM=M$. Thus the entry-wise limit of column vectors
$\lim\limits_{m\rightarrow\infty}{M^{m}}=M$ follows.

By assumption, $M^{m}(\lambda)=M(\lambda)=0$ for any
$\lambda\notin\mathbb{GT}^{+}_{N}$, so also
$\lim\limits_{m\rightarrow\infty}{M^{m}(\lambda)}=M(\lambda)$ in this case.
Hence the weak convergence
$\mathbb{P}-\lim\limits_{m\rightarrow\infty}{M^{m}}=M$ is proved. ∎

###  6.4 Automorphisms $\boldsymbol{A_{k}}$

Recall the set $\mathcal{N}$ of nonincreasing integer sequences, given in
Definition 5.1. Equip $\mathcal{N}$ with the topology of pointwise
convergence. We denote a generic element of $\mathcal{N}$ by
$\nu=(\nu_{1}\leq\nu_{2}\leq\cdots)$. For each $k\in\mathbb{Z}$, we can define
the continuous map $A_{k}\colon\mathcal{N}\rightarrow\mathcal{N}$ by
$\nu\mapsto A_{k}\nu=(\nu_{1}+k\leq\nu_{2}+k\leq\cdots)$. It is clear that
$A_{k}$ has inverse $A_{-k}$, so each $A_{k}$ is a homeomorphism.

Similar automorphisms can be constructed for $\mathbb{GT}$ and $\mathcal{T}$.
In detail, we can define the map
$A_{k}\colon\mathbb{GT}\rightarrow\mathbb{GT}$ by $\lambda\mapsto
A_{k}\lambda=(\lambda_{1}+k\geq\lambda_{2}+k\geq\cdots)$,
$A_{k}\varnothing=\varnothing$, whose inverse is $A_{-k}$, and moreover it
restricts to automorphisms $\mathbb{GT}_{N}\rightarrow\mathbb{GT}_{N}$ for
each $N\in\mathbb{Z}_{\geq 0}$. It is clear that $\mu\prec\lambda$ implies
$A_{k}\mu\prec A_{k}\lambda$, so the automorphism $A_{k}$ of $\mathbb{GT}$
induces the automorphism of measurable spaces
$A_{k}\colon\mathcal{T}\rightarrow\mathcal{T}$,
$\tau=\big{(}\tau^{(0)}\prec\tau^{(1)}\prec\tau^{(2)}\prec\cdots\big{)}\mapsto\big{(}A_{k}\tau^{(0)}\prec
A_{k}\tau^{(1)}\prec A_{k}\tau^{(2)}\prec\cdots\big{)}$.

The same notation $A_{k}$ is used to define automorphisms of the spaces
$\mathcal{N}$, $\mathbb{GT}$ and $\mathcal{T}$, but there should be no
confusion each time it is used in the future.

We have introduced the automorphisms $A_{k}$ because, in Lemma 6.12 below, we
will relate the extreme central probability measures associated to $\nu$ and
$A_{k}\nu$. The starting point is the following simple statement, which has
nothing to do with probability.

######  Lemma 6.11.

Recall the functions $\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$, defined in the
statement of Theorem 5.5. Let $\nu\in\mathcal{N}$ and $k\in\mathbb{Z}$ be
arbitrary. The following equality holds

$\displaystyle\Phi^{A_{k}\nu}(x_{1},\dots,x_{m};q,t)=t^{k{m\choose
2}}(x_{1}\cdots x_{m})^{k}\Phi^{\nu}(x_{1},\dots,x_{m};q,t),$
$\displaystyle\forall\,(x_{1},\dots,x_{m})\in(\mathbb{C}\setminus\\{0\\})^{m}.$
(6.13)

###### Proof.

As both sides of the identity (6.13) are analytic functions on
$(\mathbb{C}\setminus\\{0\\})^{m}$, we only need to prove the equality for
$(x_{1},\dots,x_{m})\in\mathcal{U}_{m}$, where the domain $\mathcal{U}_{m}$
was defined in the statement of Theorem 5.5. We can now make use of formula
(5.8) for $\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$. Observe that the only place
where $\nu$ appears in the right-hand side is inside the univariate functions
$\Phi^{\nu}(x_{i};q,t)$. The operator $\widetilde{D}_{q,\theta}^{(m)}$
satisfies that for any Laurent polynomial $f$ on variables
$x_{1},\dots,x_{m}$, the following identity holds

$\widetilde{D}_{q,\theta}^{(m)}\big{\\{}(x_{1}\cdots
x_{m})^{k}f\big{\\}}=t^{k{m\choose 2}}(x_{1}\cdots
x_{m})^{k}\widetilde{D}_{q,\theta}^{(m)}\\{f\\}.$

We deduce that the lemma will be proved for all $m\in\mathbb{N}$ once we prove
it for $m=1$, that is, we need $\Phi^{A_{k}\nu}(x;q,t)=x^{k}\Phi^{\nu}(x;q,t)$
$\forall\,x\in\mathcal{U}=\bigcap\limits_{k\geq 1}{\\{x\neq
q^{-k}\\}}\cap\\{x\neq 0\\}$. The latter statement easily follows from the
integral definition of $\Phi^{\nu}(x;q,t)$ in Theorem 5.2 (to be precise, our
desired statement follows after a change of variables $z\mapsto z+k$ in the
integral). ∎

Next we introduce new maps $\\{A_{k}\\}_{k\in\mathbb{Z}}$ on spaces of
probability measures.

For a probability measure $M_{m}$ on $\mathbb{GT}_{m}$, define $A_{k}M_{m}$ as
the pushforward of $M_{m}$ under the automorphism $A_{k}$ of
$\mathbb{GT}_{m}$, i.e.,
$A_{k}M_{m}(\mu)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}M_{m}(A_{-k}\mu)$ for all $\mu\in\mathbb{GT}_{m}$. Observe that
if we let $\delta_{\lambda}$ be the probability measure on $\mathbb{GT}_{m}$
given by the delta mass at $\lambda\in\mathbb{GT}_{m}$, then
$A_{k}\delta_{\lambda}=\delta_{A_{k}\lambda}$ for any $k\in\mathbb{Z}$.

Similarly if $M$ is a probability measure on $\mathcal{T}$, define $A_{k}M$ as
the pushforward of $M$ under the automorphism $A_{k}$ of $\mathcal{T}$. This
can be described concretely as follows. The automorphism $A_{k}$ of
$\mathcal{T}$ induces automorphisms $A_{k}$ on the set of paths of length $m$
in the GT graph, for any $m\in\mathbb{N}$:

$\displaystyle\phi=\big{(}\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(m)}\big{)}\mapsto
A_{k}\phi\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\big{(}A_{k}\phi^{(0)}\prec A_{k}\phi^{(1)}\prec\cdots\prec
A_{k}\phi^{(m)}\big{)}.$

It is therefore natural to define also an automorphism on the family of
cylinder sets by
$A_{k}S_{\phi}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}S_{A_{k}\phi}$. Then $A_{k}M$ is given by
$A_{k}M(S_{\phi})\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}M(A_{-k}S_{\phi})=M(S_{A_{-k}\phi})$ for all finite paths
$\phi=\big{(}\phi^{(0)}\prec\phi^{(1)}\prec\dots\prec\phi^{(m)}\big{)}$.

######  Lemma 6.12.

Let $M\in M_{\rm prob}(\mathcal{T})$ be a $(q,t)$-central probability measure
and $\\{M_{m}\\}_{m\geq 0}$ be its associated $(q,t)$-coherent sequence. Then
also $A_{k}M\in M_{\rm prob}(\mathcal{T})$ and its associated $(q,t)$-coherent
sequence is $\\{A_{k}M_{m}\\}_{m\geq 0}$.

###### Proof.

Clearly $A_{k}M\in M_{\rm prob}(\mathcal{T})$ is a consequence of the
definition of $A_{k}M$ as a pushforward of the probability measure $M$. The
second claim can be restated as

$(\operatorname{Proj}_{m})_{*}A_{k}M=A_{k}M_{m}.$

This equality follows from the definitions of $A_{k}M$ and $A_{k}M_{m}$ as
pushforwards of $M$ and $M_{m}$, the fact that
$(\operatorname{Proj}_{m})_{*}M=M_{m}$, and the evident commutativity of the
diagram

${\mathcal{T}}$${\mathcal{T}}$${\mathbb{GT}_{m}}$${\mathbb{GT}_{m}.}$$\scriptstyle{A_{k}}$$\scriptstyle{\operatorname{Proj}_{m}}$$\scriptstyle{\operatorname{Proj}_{m}}$$\scriptstyle{A_{k}}$
∎

##  7 The boundary of the $\boldsymbol{(q,t)}$-Gelfand–Tsetlin graph

In this section we prove Theorem 1.3, which characterizes the (minimal)
boundary of the $(q,t)$-GT graph. Along the way, we also define and
characterize the Martin boundary of the $(q,t)$-GT graph.

Assume throughout this section that $q\in(0,1)$, $\theta\in\mathbb{N}$ and set
$t=q^{\theta}$. Recall the notation
$\mathbb{P}-\lim\limits_{k\rightarrow\infty}M_{k}=M$ indicates that a sequence
of probability measures $\\{M_{k}\\}_{k\geq 1}$ converges weakly to $M$.

###  7.1 The Martin boundary: definition and preliminaries

For any $\lambda\in\mathbb{GT}_{N}$, let $\delta_{\lambda}$ be the delta mass
at $\lambda$. As remarked in Section 6.1, there exists a unique
$(q,t)$-coherent sequence $\\{M_{m}^{\lambda}\\}_{m=0,1,\dots,N}$ such that
each $M_{m}^{\lambda}$ is a probability measure on $\mathbb{GT}_{m}$ and
$M_{N}^{\lambda}=\delta_{\lambda}$. Such a sequence is given explicitly by
$M^{\lambda}_{N}=\delta_{\lambda}$ and
$M^{\lambda}_{m}=\Lambda^{N}_{m}\delta_{\lambda}$ $\forall\,0\leq m\leq N-1$,
where the probability measures $\Lambda^{N}_{m}\delta_{\lambda}$ on
$\mathbb{GT}_{m}$ are given explicitly in (6.4). Moreover recall that for a
$(q,t)$-central probability measure $M$ on $\mathcal{T}$, we can associate a
$(q,t)$-coherent sequence $\\{M_{m}\\}_{m\geq 0}$ as in Definition 6.4.

######  Definition 7.1.

The Martin boundary of the $(q,t)$-Gelfand–Tsetlin graph is the subset of
$(q,t)$-central probability measures $M\in M_{\rm prob}(\mathcal{T})$ for
which there exists a sequence $\\{\lambda(N)\\}_{N\geq 1}$,
$\lambda(N)\in\mathbb{GT}_{N}$, such that the following weak limits hold

$\displaystyle\mathbb{P}-\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}=M_{m}\qquad\forall\,m=0,1,2,\dots.$

Let us denote the Martin boundary by $\Omega_{q,t}^{\rm Martin}\subset M_{\rm
prob}(\mathcal{T})$ and equip it with its subspace topology, namely with the
topology of weak convergence.

To characterize the Martin boundary $\Omega_{q,t}^{\rm Martin}$, we begin by
giving necessary conditions on sequences of signatures
$\\{\lambda(N)\\}_{N\geq 1}$ which yield a weak convergence as above. In this
section, we sometimes denote
$M^{\lambda}_{N}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\delta_{\lambda}$ the delta mass at $\lambda\in\mathbb{GT}_{N}$,
and $\\{M_{m}^{\lambda}=\Lambda^{N}_{m}\delta_{\lambda}\\}_{m=0,1,\dots,N}$
the corresponding $(q,t)$-coherent sequence.

######  Lemma 7.2.

Assume that $\\{\lambda(N)\\}_{N\geq 1}$, $\lambda(N)\in\mathbb{GT}_{N}$, is a
sequence of signatures such that
$\\{\Lambda^{N}_{1}\delta_{\lambda(N)}\\}_{N\geq 1}$ converges weakly, as
$N\rightarrow\infty$, to some probability measure $\mathfrak{m}$ on
$\mathbb{GT}_{1}=\mathbb{Z}$. Then the sequence $\\{\lambda(N)_{N}\\}_{N\geq
1}$ is bounded.

###### Proof.

We use the Macdonald generating functions of Section 6.3 above. By Proposition
6.8, we have

$\displaystyle\mathcal{P}_{\delta_{\lambda(N)}}\big{(}1^{N-1},z\big{)}=\mathcal{P}_{M^{\lambda(N)}_{1}}(z)$

and then

$\displaystyle\frac{P_{\lambda(N)}\big{(}1,t,\dots,t^{N-2},t^{N-1}z\big{)}}{P_{\lambda(N)}\big{(}1,t,\dots,t^{N-2},t^{N-1}\big{)}}=\frac{P_{\lambda(N)}\big{(}z,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}=\sum_{n\in\mathbb{Z}}{M^{\lambda(N)}_{1}(n)z^{n}}.$
(7.1)

In the sum of the right-hand side above, $n$ ranges from $\lambda(N)_{N}$ to
$\lambda(N)_{1}$ because of the branching rule for Macdonald polynomials. We
multiply the equality by $z^{-\lambda(N)_{N}}$ and then set $z=0$, so in the
right-hand side one clearly picks up the coefficient of $z^{\lambda(N)_{N}}$,
namely $M^{\lambda(N)}_{1}(\lambda(N)_{N})$. By the index stability of
Macdonald polynomials, 2.2), the left-hand side is

$\displaystyle\left.z^{-\lambda(N)_{N}}\frac{P_{\lambda(N)}\big{(}z,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|_{z=0}$
$\displaystyle\qquad{}=\left.\frac{\big{(}z\cdot t^{-1}\cdots
t^{1-N}\big{)}^{-\lambda(N)_{N}}}{\big{(}1\cdot t^{-1}\cdots
t^{1-N}\big{)}^{-\lambda(N)_{N}}}\frac{P_{\lambda(N)}\big{(}z,t^{-1},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|_{z=0}$
$\displaystyle\qquad{}=\left.\frac{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N},0)}\big{(}z,t^{-1},\dots,t^{1-N}\big{)}}{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N},0)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}\right|_{z=0}$
$\displaystyle\qquad{}=\frac{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N})}\big{(}t^{-1},\dots,t^{1-N}\big{)}}{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N},0)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}.$

Thanks to Theorem 2.3, we can then obtain a lower bound for
$M^{\lambda(N)}_{1}(\lambda(N)_{N})$ as follows:

$\displaystyle
M^{\lambda(N)}_{1}(\lambda(N)_{N})=\frac{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N})}\big{(}t^{-1},\dots,t^{1-N}\big{)}}{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N},0)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}$
$\displaystyle\qquad{}=\frac{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N})}\big{(}t^{N-2},\dots,t,1\big{)}}{P_{(\lambda(N)_{1}-\lambda(N)_{N},\dots,\lambda(N)_{N-1}-\lambda(N)_{N},0)}\big{(}t^{N-1},\dots,t,1\big{)}}$
$\displaystyle\qquad{}=\prod_{1\leq i<j\leq
N-1}{\frac{\big{(}q^{\lambda(N)_{i}-\lambda(N)_{j}}t^{j-i};q\big{)}_{\infty}(t^{j-i+1};q)_{\infty}}{\big{(}q^{\lambda(N)_{i}-\lambda(N)_{j}}t^{j-i+1};q\big{)}_{\infty}(t^{j-i};q)_{\infty}}}$
$\displaystyle\qquad\quad{}\times\prod_{1\leq i<j\leq
N}{\frac{\big{(}q^{\lambda(N)_{i}-\lambda(N)_{j}}t^{j-i+1};q\big{)}_{\infty}(t^{j-i};q)_{\infty}}{\big{(}q^{\lambda(N)_{i}-\lambda(N)_{j}}t^{j-i};q\big{)}_{\infty}(t^{j-i+1};q)_{\infty}}}$
$\displaystyle\qquad{}=\prod_{i=1}^{N-1}{\frac{\big{(}q^{\lambda(N)_{i}-\lambda(N)_{N}}t^{N-i+1};q\big{)}_{\infty}(t^{N-i};q)_{\infty}}{\big{(}q^{\lambda(N)_{i}-\lambda(N)_{N}}t^{N-i};q\big{)}_{\infty}(t^{N-i+1};q)_{\infty}}}$
$\displaystyle\qquad{}=\prod_{i=1}^{N-1}{\frac{\big{(}1-t^{N-i}\big{)}\big{(}1-qt^{N-i}\big{)}\cdots\big{(}1-q^{\theta-1}t^{N-i}\big{)}}{\big{(}1-q^{\lambda(N)_{i}-\lambda(N)_{N}}t^{N-i}\big{)}\cdots\big{(}1-q^{\lambda(N)_{i}-\lambda(N)_{N}+\theta-1}t^{N-i}\big{)}}}$
$\displaystyle\qquad{}\geq\prod_{i=1}^{N-1}{\big{(}1-t^{N-i}\big{)}\big{(}1-qt^{N-i}\big{)}\cdots\big{(}1-q^{\theta-1}t^{N-i}\big{)}}\geq\prod_{i=1}^{N-1}{\big{(}1-t^{N-i}\big{)}^{\theta}}\geq((t;t)_{\infty})^{\theta}.$

On the other hand, since $(t;t)_{\infty}\in(0,1)$ and
$\mathbb{P}-\lim\limits_{N\rightarrow\infty}{M^{\lambda(N)}_{1}}=\mathfrak{m}$,
then there exist $N_{1},N_{2}\in\mathbb{N}$ large enough such that

$\displaystyle M^{\lambda(N)}_{1}\left(\\{(n)\in\mathbb{GT}_{1}\colon-
N_{1}\leq n\leq
N_{1}\\}\right)>1-((t;t)_{\infty})^{\theta}\qquad\forall\,N>N_{2}.$

We conclude that $-N_{1}\leq\lambda(N)_{N}\leq N_{1}$, for all $N>N_{2}$.
Therefore the sequence $\\{\lambda(N)_{N}\\}_{N\geq 1}$ is bounded. ∎

One actually has the following more general statement.

######  Lemma 7.3.

Let $k\in\mathbb{N}$. Assume that $\\{\lambda(N)\\}_{N\geq 1}$,
$\lambda(N)\in\mathbb{GT}_{N}$, is a sequence of signatures such that the
sequence $\Lambda^{N}_{k}\delta_{\lambda(N)}$ converges weakly, as
$N\rightarrow\infty$, to some probability measure $\mathfrak{m}$ on
$\mathbb{GT}_{k}$. Then for any $i=1,\dots,k$, the sequence
$\\{\lambda(N)_{N-i+1}\\}_{N\geq 1}$, is bounded.

###### Proof.

The argument here is very similar to that of the previous proof for $k=1$. As
before, by making use of Macdonald generating functions, we can derive a more
general equation than (7.1), which is

$\displaystyle\frac{P_{\lambda(N)}\big{(}z_{1}t^{1-k},\dots,z_{k-1}t^{-1},z_{k},t^{-k},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}$
$\displaystyle\qquad{}=\sum_{\mu\in\mathbb{GT}_{k}}{M_{k}^{\lambda(N)}(\mu)\frac{P_{\mu}\big{(}z_{1}t^{1-k},\dots,z_{k-1}t^{-1},z_{k}\big{)}}{P_{\mu}\big{(}1,t^{-1},\dots,t^{1-k}\big{)}}}.$
(7.2)

In the sum of the right-hand side above, note that $\mu$ ranges over
signatures in $\mathbb{GT}_{k}$ such that

$\displaystyle\mu_{k}\geq\lambda(N)_{N},\quad\mu_{k-1}\geq\lambda(N)_{N-1},\quad\dots,\quad\mu_{1}\geq\lambda(N)_{N-k+1}.$
(7.3)

This is a consequence of the branching rule for Macdonald polynomials. Another
relevant observation is that for any $\mu\in\mathbb{GT}_{k}$ satisfying (7.3),
any monomial $c_{m_{1},\dots,m_{k}}z_{1}^{m_{1}}\cdots z_{k}^{m_{k}}$ with
$c_{m_{1},\dots,m_{k}}\neq 0$ in the expansion of
$P_{\mu}(z_{1}t^{1-k},\dots,z_{k-1}t^{-1},z_{k})$ satisfies:
$m_{k}\geq\lambda(N)_{N}$; if $m_{k}=\lambda(N)_{N}$ then
$m_{k-1}\geq\lambda(N)_{N-1}$; and so on until, if
$m_{2}=\lambda(N)_{N-k+2},\dots,m_{k}\geq\lambda(N)_{N}$, then
$m_{1}\geq\lambda(N)_{N-k+1}$. This is a consequence of the triangularity
property of the Macdonald polynomials, see Definition/Proposition 2.1.

In equation (7.2) above, multiply both sides by $z_{k}^{-\lambda(N)_{N}}$ and
then set $z_{k}=0$. From the branching rule for Macdonald polynomials and the
fact that the branching coefficients satisfy

$\psi_{\lambda(N)/(\lambda(N)_{1},\dots,\lambda(N)_{N-1})}(q,t)=1,$

the resulting left-hand side is

$\frac{P_{(\lambda(N)_{1},\dots,\lambda(N)_{N-1})}\big{(}z_{1}t^{1-k},\dots,z_{k-1}t^{-1},t^{-k},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}.$

(Note that the argument $z_{k}$ is no longer present in the numerator.)
Similarly, from the property

$\psi_{\mu/(\mu_{1},\dots,\mu_{k-1})}(q,t)=1,\qquad\textrm{for
any}\quad\mu\in\mathbb{GT}_{k},$

the resulting right-hand side is

$\sum_{\mu\in\mathbb{GT}_{k}\colon\mu_{k}=\lambda(N)_{N}}{M_{k}^{\lambda(N)}(\mu)\frac{P_{(\mu_{1},\dots,\mu_{k-1})}\big{(}z_{1}t^{1-k},\dots,z_{k-1}t^{-1}\big{)}}{P_{\mu}\big{(}1,t^{-1},\dots,t^{1-k}\big{)}}}.$

After that, multiply both sides by $(z_{k-1}t^{-1})^{-\lambda(N)_{N-1}}$ and
then set $z_{k-1}=0$; this gives

$\displaystyle\frac{P_{(\lambda(N)_{1},\dots,\lambda(N)_{N-2})}\big{(}z_{1}t^{1-k},\dots,z_{k-2}t^{-2},t^{-k},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}$
$\displaystyle\qquad{}=\sum_{\begin{subarray}{c}\mu\in\mathbb{GT}_{k}\colon\\\
\mu_{k-1}=\lambda(N)_{N-1},\
\mu_{k}=\lambda(N)_{N}\end{subarray}}{M_{k}^{\lambda(N)}(\mu)\frac{P_{(\mu_{1},\dots,\mu_{k-2})}\big{(}z_{1}t^{1-k},\dots,z_{k-2}t^{-2}\big{)}}{P_{\mu}\big{(}1,t^{-1},\dots,t^{1-k}\big{)}}}.$

Repeat the same procedure $k$ times, until we have multiplied both sides by
$\big{(}z_{1}t^{1-k}\big{)}^{-\lambda(N)_{N-k+1}}$ and set $z_{1}=0$. The end
result is

$\displaystyle
M_{k}^{\lambda(N)}\left(\lambda(N)_{N-k+1},\dots,\lambda(N)_{N}\right)$
$\displaystyle=\frac{P_{(\lambda(N)_{1},\dots,\lambda(N)_{N-k})}\big{(}t^{-k},\dots,t^{1-N}\big{)}P_{(\lambda(N)_{N-k+1},\dots,\lambda(N)_{N})}\big{(}1,t^{-1},\dots,t^{1-k}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},\dots,t^{1-N}\big{)}}$
$\displaystyle=t^{(N-k)(\lambda(N)_{N-k+1}+\lambda(N)_{N-k+2}+\dots+\lambda(N)_{N})}$
$\displaystyle\times\frac{P_{(\lambda(N)_{1},\dots,\lambda(N)_{N-k})}\big{(}1,t,\dots,t^{N-k-1}\big{)}P_{(\lambda(N)_{N-k+1},\dots,\lambda(N)_{N})}\big{(}1,t,\dots,t^{k-1}\big{)}}{P_{\lambda(N)}\big{(}1,t,t^{2},\dots,t^{N-1}\big{)}}$
$\displaystyle=\prod_{i=1}^{N-k}\prod_{j=N-k+1}^{N}\\!{\frac{\big{(}1-t^{j-i}\big{)}\big{(}1-qt^{j-i}\big{)}\cdots\big{(}1-q^{\theta-1}t^{j-i}\big{)}}{\big{(}1\\!-\\!q^{\lambda(N)_{i}-\lambda(N)_{j}}t^{j-i}\big{)}\big{(}1\\!-\\!q^{\lambda(N)_{i}-\lambda(N)_{j}+1}t^{j-i}\big{)}\cdots\big{(}1\\!-\\!q^{\lambda(N)_{i}-\lambda(N)_{j}+\theta-1}t^{j-i}\big{)}}}$
$\displaystyle\geq\prod_{i=1}^{N-k}\prod_{j=N-k+1}^{N}{\big{(}1-t^{j-i}\big{)}\big{(}1-qt^{j-i}\big{)}\cdots\big{(}1-q^{\theta-1}t^{j-i}\big{)}}\geq((t;t)_{\infty})^{\theta
k},$

where we used the homogeneity of Macdonald polynomials for the second
equality, as well as three applications of Theorem 2.3 for the third equality.
Then $M_{k}^{\lambda(N)}(\lambda(N)_{N-k+1},\dots,\lambda(N)_{N})\geq
c_{k}=((t;t)_{\infty})^{\theta k}>0$. On the other hand, since
$M_{k}^{\lambda(N)}$ converges weakly to $\mathfrak{m}$, there exist
$N_{1},N_{2}\in\mathbb{N}$ large enough such that

$\displaystyle
M_{k}^{\lambda(N)}(\\{(n_{1},\dots,n_{k})\in\mathbb{GT}_{k}\colon-N_{1}\leq
n_{1},\dots,n_{k}\leq N_{1}\\})>1-c_{k}\qquad\forall\,N>N_{2}.$

Therefore we conclude that
$-N_{1}\leq\lambda(N)_{N-k+1},\dots,\lambda(N)_{N}\leq N_{1}$ for all
$N>N_{2}$. We conclude that each sequence $\\{\lambda(N)_{N-i+1}\\}_{N\geq
1}$, $i=1,\dots,k$, is uniformly bounded by a constant. ∎

Next we show that $\Omega_{q,t}^{\rm Martin}$ is in bijection with the set
$\mathcal{N}$. We do so by first constructing a map $\Omega_{q,t}^{\rm
Martin}\rightarrow\mathcal{N}$ that will later be shown to be bijective.

######  Proposition 7.4.

Let $M$ be a $(q,t)$-central probability measure on $\mathcal{T}$ and
$\\{M_{m}\\}_{m=0,1,2,\dots}$ be its associated $(q,t)$-coherent system. If
$M$ belongs to $\Omega_{q,t}^{{\rm Martin}}$, then there exists a unique
$\nu\in\mathcal{N}$ such that

$\displaystyle\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t\big{)}=\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m}),$
$\displaystyle\forall\,(x_{1},\dots,x_{m})\in\mathbb{T}^{m},\quad\forall\,m\geq
1.$ (7.4)

The function $\Phi^{\nu}(z_{1},\dots,z_{m};q,t)$ was defined in Theorem 5.2
for $m=1$, and in Theorem 5.5 for general $m$.

###### Proof.

Let $M\in\Omega_{q,t}^{{\rm Martin}}$. By definition, there exists a sequence
$\\{\lambda(N)\\}_{N\geq 1}$ such that $\lambda(N)\in\mathbb{GT}_{N}$, $N\geq
1$, and
$\mathbb{P}-\lim\limits_{m\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}=M_{m}$
holds weakly, for any $m\in\mathbb{N}$.

By Lemma 7.3, it follows that for any $i=1,2,\dots$, the sequence
$\\{\lambda(N)_{N-i+1}\\}_{N\geq 1}$ is uniformly bounded. In particular,
since $\\{\lambda(N)_{N}\\}_{N\geq 1}$ is uniformly bounded, there exists a
subsequence
$\big{\\{}N^{1}_{1}<N^{1}_{2}<N^{1}_{3}<\cdots\big{\\}}\subset\mathbb{N}$ such
that
$\lambda\big{(}N^{1}_{1}\big{)}_{N^{1}_{1}}=\lambda\big{(}N^{1}_{2}\big{)}_{N^{1}_{2}}=\cdots$.
Analogously, using that $\\{\lambda(N)_{N-1}\\}_{N\geq 1}$ is bounded, there
exists a subsequence
$\big{\\{}N^{2}_{1}<N^{2}_{2}<N^{2}_{3}<\cdots\big{\\}}\subset\big{\\{}N^{1}_{1}<N^{1}_{2}<N^{1}_{3}<\cdots\big{\\}}$
such that
$\lambda\big{(}N^{2}_{1}\big{)}_{N^{2}_{1}-1}=\lambda\big{(}N^{2}_{2}\big{)}_{N^{2}_{2}-1}=\cdots$.
In a similar fashion, we can define subsequences
$\big{\\{}N^{k}_{1}<N^{k}_{2}<\cdots\big{\\}}$ inductively.

Now consider the subsequence
$\big{\\{}N_{1}=N^{1}_{1}<N_{2}=N^{2}_{2}<N_{3}=N_{3}^{3}<\cdots\big{\\}}\subset\mathbb{N}$.
By construction, the sequence $\\{\lambda(N_{k})\\}_{k\geq 1}$ is such that
the limits $\lim\limits_{k\rightarrow\infty}{\lambda(N_{k})_{N_{k}-i+1}}$
exist for all $i=1,2,\dots$. Consequently, there exists $\nu\in\mathcal{N}$
such that $\\{\lambda(N_{k})\\}_{k\geq 1}$ stabilizes to $\nu$ in the sense of
Definition 5.1. From Theorem 5.5, the following limit

$\displaystyle\lim_{k\rightarrow\infty}{\frac{P_{\lambda(N_{k})}\big{(}x_{1}t^{1-m},x_{2}t^{2-m},\dots,x_{m},t^{-m},\dots,t^{1-N_{k}}\big{)}}{P_{\lambda(N_{k})}\big{(}1,t^{-1},\dots,t^{1-N_{k}}\big{)}}}$
$\displaystyle\qquad{}=\Phi^{\nu}\big{(}x_{1}t^{1-m},x_{2}t^{2-m},\dots,x_{m};q,t\big{)}$
(7.5)

holds uniformly for $(x_{1},\dots,x_{m})$ in compact subsets of the domain
$(\mathbb{C}\setminus\\{0\\})^{m}$, in particular the convergence is uniform
on $\mathbb{T}^{m}$.

On the other hand, since we also have the weak convergence
$\mathbb{P}-\lim\limits_{k\rightarrow\infty}{\Lambda^{N_{k}}_{m}\delta_{\lambda(N_{k})}}=M_{m}$,
then by Proposition 6.9:

$\displaystyle\lim_{k\rightarrow\infty}{\mathcal{P}_{\Lambda^{N_{k}}_{m}\delta_{\lambda(N_{k})}}(x_{1},\dots,x_{m})}=\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})$
(7.6)

uniformly on $\mathbb{T}^{m}$. By Proposition 6.8 and the homogeneity of the
Macdonald polynomials, the Macdonald generating function of
$\Lambda^{N_{k}}_{m}\delta_{\lambda(N_{k})}$ equals

$\displaystyle\mathcal{P}_{\Lambda^{N_{k}}_{m}\delta_{\lambda(N_{k})}}(x_{1},\dots,x_{m})=\mathcal{P}_{\delta_{\lambda(N_{k})}}\big{(}1^{N_{k}-m},x_{1},\dots,x_{m}\big{)}$
$\displaystyle\qquad{}=\frac{P_{\lambda(N_{k})}\big{(}1,t,\dots,t^{N_{k}-m-1},t^{N_{k}-m}x_{1},\dots,t^{N_{k}-1}x_{m}\big{)}}{P_{\lambda(N_{k})}\big{(}1,t,t^{2},\dots,t^{N_{k}-1}\big{)}}$
$\displaystyle\qquad{}=\frac{P_{\lambda(N_{k})}\big{(}x_{1}t^{1-m},x_{2}t^{2-m},\dots,x_{m},t^{-m},\dots,t^{1-N_{k}}\big{)}}{P_{\lambda(N_{k})}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N_{k}}\big{)}}.$
(7.7)

By combining (7.5), (7.6) and (7.7), we conclude

$\displaystyle\Phi^{\nu}\big{(}x_{1}t^{1-m},x_{2}t^{2-m},\dots,x_{m};q,t\big{)}=\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})\qquad\text{for
all}\quad(x_{1},\dots,x_{m})\in\mathbb{T}^{m}$

and all $m\geq 1$, as desired. The statement about the uniqueness of
$\nu\in\mathcal{N}$ follows from Lemma 5.4. ∎

Because of Proposition 7.4, for each $M\in\Omega_{q,t}^{\rm Martin}$, there
exists a unique $\nu\in\mathcal{N}$ such that equation (7.4) is satisfied for
all $m\in\mathbb{N}$. Thus there is a well-defined map of sets

$\displaystyle\mathbf{N}\colon\ \Omega_{q,t}^{{\rm
Martin}}\rightarrow\mathcal{N},$ $\displaystyle\hphantom{\mathbf{N}\colon}{}\
M\mapsto\nu,$ (7.8)

which is determined by setting $\nu=\mathbf{N}(M)$ be the unique element of
$\mathcal{N}$ such that (7.4) is satisfied. We prove below that $\mathbf{N}$
is bijective, but first we show the convenient fact that $\mathbf{N}$ commutes
with the automorphisms $A_{k}$, $k\in\mathbb{Z}$, of Section 6.4.

######  Lemma 7.5.

Let $M\in\Omega_{q,t}^{\rm Martin}$, $k\in\mathbb{Z}$. Then
$A_{k}M\in\Omega_{q,t}^{\rm Martin}$ and
$\mathbf{N}(A_{k}M)=A_{k}\mathbf{N}(M)$.

###### Proof.

As $M\in\Omega_{q,t}^{\rm Martin}\subset M_{\textrm{Prob}}(\mathcal{T})$,
Lemma 6.12 shows $A_{k}M\in M_{\textrm{Prob}}(\mathcal{T})$ and
$\\{A_{k}M_{m}\\}_{m\geq 0}$ is its corresponding $(q,t)$-coherent sequence.
By definition, there exists a sequence $\\{\lambda(N)\\}_{N\geq 1}$,
$\lambda(N)\in\mathbb{GT}_{N}$, such that the weak convergence holds
$\mathbb{P}-\lim\limits_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}=M_{m}$
$\forall\,m\geq 0$. We claim that
$\mathbb{P}-\lim\limits_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{A_{k}\lambda(N)}}=A_{k}M_{m}$
$\forall\,m\geq 0$, which would show $A_{k}M\in\Omega_{q,t}^{\rm Martin}$
indeed.

Take any $m\in\mathbb{Z}_{\geq 0}$, $\mu\in\mathbb{GT}_{m}$, then

$\displaystyle\big{(}\Lambda^{N}_{m}\delta_{A_{k}\lambda(N)}\big{)}(\mu)=\Lambda^{N}_{m}(A_{k}\lambda(N),\mu)=\Lambda^{N}_{m}(\lambda(N),A_{-k}\mu)$
$\displaystyle\hphantom{\big{(}\Lambda^{N}_{m}\delta_{A_{k}\lambda(N)}\big{)}(\mu)}{}=\big{(}\Lambda^{N}_{m}\delta_{\lambda(N)}\big{)}(A_{-k}\mu)\xrightarrow{N\rightarrow\infty}M_{m}(A_{-k}\mu)=A_{k}M_{m}(\mu),$

where we have used
$\Lambda^{N}_{m}(A_{k}\nu,A_{k}\kappa)=\Lambda^{N}_{m}(\nu,\kappa)$,
previously stated in the proof of Lemma 6.12.

Let us move on to the second part of the lemma. Let $\mathbf{N}(M)=\nu$; to
show $\mathbf{N}(A_{k}M)=A_{k}\nu$, we need

$\displaystyle\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})=\Phi^{A_{k}\nu}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t\big{)}$
$\displaystyle\forall\,(x_{1},\dots,x_{m})\in\mathbb{T}^{m},\quad\forall\,m\in\mathbb{N}.$

For any $m\in\mathbb{N}$, $(x_{1},\dots,x_{m})\in\mathbb{T}^{m}$, we have

$\displaystyle\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})=\sum_{\lambda\in\mathbb{GT}_{m}}{A_{k}M_{m}(\lambda)\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{m-1}\big{)}}}$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=\sum_{\lambda\in\mathbb{GT}_{m}}{M_{m}(A_{-k}\lambda)\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{m-1}\big{)}}}$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=\sum_{\lambda\in\mathbb{GT}_{m}}{M_{m}(\lambda)\frac{P_{A_{k}\lambda}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}}{P_{A_{k}\lambda}\big{(}1,t,t^{2},\dots,t^{m-1}\big{)}}}$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=(x_{1}x_{2}\cdots
x_{m})^{k}\cdot\sum_{\lambda\in\mathbb{GT}_{m}}{M_{m}(\lambda)\frac{P_{\lambda}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}}{P_{\lambda}\big{(}1,t,t^{2},\dots,t^{m-1}\big{)}}}$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=(x_{1}x_{2}\cdots
x_{m})^{k}\cdot\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=(x_{1}\cdots
x_{m})^{k}\cdot\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m};q,t\big{)}$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=t^{k{m\choose
2}}\big{(}\big{(}x_{1}t^{1-m}\big{)}\cdots(x_{m})\big{)}^{k}\cdot\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m};q,t\big{)}$
$\displaystyle\hphantom{\mathcal{P}_{A_{k}M_{m}}(x_{1},\dots,x_{m})}{}=\Phi^{A_{k}\nu}\big{(}x_{1}t^{1-m},\dots,x_{m};q,t\big{)},$

where the last equality follows from Lemma 6.11. ∎

######  Proposition 7.6.

The Martin boundary $\Omega_{q,t}^{{\rm Martin}}$ of the $(q,t)$-GT graph is
bijective to $\mathcal{N}$ under the map $\mathbf{N}$ defined in (7.8) above.

###### Proof.

Step 1. We prove $\mathbf{N}$ is surjective.

Let $\nu\in\mathcal{N}$ be arbitrary; we want to show it belongs to the range
of $\mathbf{N}$. From Lemma 7.5, we may assume $\nu_{1}=0$ without any loss of
generality. Consider the sequence
$\\{\lambda(N)=(\nu_{N}\geq\nu_{N-1}\geq\dots\geq\nu_{1})\\}_{N\geq 1}$ of
signatures that stabilizes to $\nu$. Let $m\in\mathbb{N}$ be arbitrary. The
first claim is that the limit
$\lim\limits_{N\rightarrow\infty}{M^{\lambda(N)}_{m}(\lambda)}$ exists for any
$\lambda\in\mathbb{GT}_{m}$.

By definition (6.4),
$M^{\lambda(N)}_{m}(\lambda)=\Lambda^{N}_{m}(\lambda(N),\lambda)=0$ unless
$\lambda_{m}\geq\lambda(N)_{N}=\nu_{1}=0$. Thus the previous claim is clear if
$\lambda_{m}<\nu_{1}=0$, in which case
$\lim\limits_{N\rightarrow\infty}{M^{\lambda(N)}_{m}(\lambda)}=0$. Assume now
$\lambda_{m}\geq\nu_{1}=0$, i.e., $\lambda\in\mathbb{GT}^{+}_{m}$.

From Theorem 5.5, we have the uniform limit

$\displaystyle\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}}$
$\displaystyle\qquad{}=\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t\big{)}$
(7.9)

on $\mathbb{T}^{m}$. Correspondingly, the Fourier coefficients of the
normalized Macdonald characters converge to those of
$\Phi^{\nu}(x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t)$. Proposition
refprop:coherentsequences and the expansion of Corollary 2.6 give

$\displaystyle\frac{P_{\lambda(N)}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}$
$\displaystyle\qquad{}=\sum_{\mu\in\mathbb{GT}_{m}}{M_{m}^{\lambda(N)}(\mu)\frac{P_{\mu}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}$
$\displaystyle\qquad{}=\sum_{\kappa\in\mathbb{GT}_{m}}{m_{\kappa}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}\sum_{\mu\in\mathbb{GT}_{m}}{\frac{c_{\mu,\kappa}M_{m}^{\lambda(N)}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}}.$
(7.10)

Let $\kappa\in\mathbb{GT}_{m}$ be arbitrary, and denote
$n(\kappa)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\kappa_{2}+2\kappa_{3}+\dots+(m-1)\kappa_{m}$. Observe that
$x_{1}^{\kappa_{1}}\cdots x_{m}^{\kappa}$ appears only in the monomial
symmetric polynomial $m_{\kappa}(x_{1},x_{2}t,\dots,x_{m}t^{m-1})$ and the
corresponding term is $t^{n(\kappa)}x_{1}^{\kappa_{1}}\cdots
x_{m}^{\kappa_{m}}$, so (7.10) is essentially the Fourier expansion of the
prelimit functions in (7.9). Then we have that, for any
$\kappa\in\mathbb{GT}_{m}$, the following limit

$\displaystyle\lim_{N\rightarrow\infty}{t^{n(\kappa)}\sum_{\mu\in\mathbb{GT}_{m}}{\frac{c_{\mu,\kappa}M_{m}^{\lambda(N)}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}}$
(7.11)

exists and equals the Fourier coefficient of $x_{1}^{\kappa_{1}}\cdots
x_{m}^{\kappa_{m}}$ in the function
$\Phi^{\nu}(x_{1}t^{m-1},\dots,x_{m};q,t)$. As mentioned before,
$M^{\lambda(N)}_{m}(\mu)=0$ unless $\mu_{m}\geq\nu_{1}\geq 0$. Thus we can
restrict the sum in (7.11) to $\mu\in\mathbb{GT}^{+}_{m}$. From the same
analysis as in Proposition 6.10, we can obtain that the limit
$\lim\limits_{N\rightarrow\infty}{M^{\lambda(N)}_{m}(\mu)}$ exists for any
$\mu\in\mathbb{GT}^{+}_{m}$ with $|\lambda|=n$; let us denote
$M_{m}(\mu)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\lim\limits_{N\rightarrow\infty}{M_{m}^{\lambda(N)}(\mu)}$.
Immediately from the definition (and Fatou’s lemma) it follows that

$\displaystyle M_{m}(\lambda)\geq 0\qquad\forall\,\lambda\in\mathbb{GT}_{m},$
$\displaystyle\sum_{\lambda\in\mathbb{GT}_{m}}{M_{m}(\lambda)}\leq 1.$

Recall that we observed $M^{\lambda(N)}_{m}(\lambda)=0$ if
$\lambda\notin\mathbb{GT}^{+}_{m}$, therefore $M_{m}(\lambda)=0$ if
$\lambda\notin\mathbb{GT}^{+}_{m}$.

Next consider the following function

$\displaystyle
F(x_{1},\dots,x_{m})=\sum_{\mu\in\mathbb{GT}_{m}}{M_{m}(\mu)\frac{P_{\mu}\big{(}x_{1},x_{2}t,\dots,x_{m}t^{m-1}\big{)}}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}.$

Clearly $F$ defines a function on $\mathbb{T}^{m}$, as it is defined by an
absolutely convergent series on the $m$-dimensional torus. Moreover its
absolute value is upper bounded by $1$, therefore $F\in
L^{\infty}(\mathbb{T}^{m})\subset L^{2}(\mathbb{T}^{m})$. By the same argument
preceding (6.6), one shows, for any $\kappa\in\mathbb{GT}_{m}$, that the
Fourier coefficient of $x_{1}^{\kappa_{1}}\cdots x_{m}^{\kappa_{m}}$ in $F$ is

$\displaystyle
t^{n(\kappa)}\sum_{\mu\in\mathbb{GT}_{m}}{\frac{c_{\mu,\kappa}M_{m}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}.$
(7.12)

Observe that, if $\kappa\in\mathbb{GT}_{m}$ is fixed, the sums in both (7.11)
and (7.12) are finite because $c_{\mu,\kappa}M_{m}(\mu)=0$ unless
$\mu\geq\kappa$ and $\mu\in\mathbb{GT}^{+}_{m}$. Therefore
$\lim\limits_{N\rightarrow\infty}{M_{m}^{\lambda(N)}(\mu)}=M_{m}(\mu)$ for all
$\mu\in\mathbb{GT}^{+}_{m}$ implies the equality between (7.11) and (7.12). In
other words, the following convergence holds

$\displaystyle\frac{P_{\lambda(N)}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}\xrightarrow{\textrm{Fourier}}F(x_{1},\dots,x_{m})$
(7.13)

in the sense that all Fourier coefficients of the left side of (7.13) converge
to the corresponding Fourier coefficients of $F(x_{1},\dots,x_{m})$, as $N$
tends to infinity. But we already knew that the Fourier coefficients of the
left side of (7.13) converge to the corresponding Fourier coefficients of
$\Phi^{\nu}(x_{1}t^{1-m},\dots,x_{m};q,t)$. Therefore all the Fourier
coefficients of the difference
$F(x_{1},\dots,x_{m})-\Phi^{\nu}(x_{1}t^{1-m},\dots,x_{m};q,t)$ of square-
integrable functions on $\mathbb{T}^{m}$ must be zero. It follows that
$F(x_{1},\dots,x_{m})=\Phi^{\nu}(x_{1}t^{1-m},\dots,x_{m};q,t)$. In
particular, the equality holds for $x_{1}=\dots=x_{m}=1$, resulting in

$\displaystyle\sum_{\mu\in\mathbb{GT}_{m}}{M_{m}(\mu)}=F(1,\dots,1)=\Phi^{\nu}\big{(}t^{1-m},t^{2-m},\dots,1;q,t\big{)}$
$\displaystyle\qquad{}=\lim_{N\rightarrow\infty}{\left.\frac{P_{\lambda(N)}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m},t^{-m},\dots,t^{1-N}\big{)}}{P_{\lambda(N)}\big{(}1,t^{-1},t^{-2},\dots,t^{1-N}\big{)}}\right|_{x_{1}=\dots=x_{m}=1}}=\lim_{N\rightarrow\infty}{1}=1.$

Moreover $\Phi^{\nu}(x_{1}t^{1-m},\dots,x_{m};q,t)=F(x_{1},\dots,x_{m})$ is
the Macdonald generating function of $M_{m}$.

We are almost done. For each $m\in\mathbb{N}$, we have constructed probability
measures $M_{m}$ as limits of $M^{\lambda(N)}_{m}$ and shown that
$\Phi^{\nu}(x_{1}t^{1-m},\dots,x_{m};q,t)$ is the generating function of
$M_{m}$. Complete the sequence with $M_{0}=\delta_{\varnothing}$, the delta
mass at $\varnothing$. We claim that $\\{M_{m}\\}_{m\geq 0}$ is a
$(q,t)$-coherent sequence. In fact, since
$\\{M_{m}^{\lambda(N)}\\}_{m=0,1,\dots,N}$ is a $(q,t)$-coherent sequence,
then

$\displaystyle
M_{m}^{\lambda(N)}(\mu)=\sum_{\lambda\in\mathbb{GT}_{m+1}}{M_{m+1}^{\lambda(N)}(\lambda)\Lambda^{m+1}_{m}(\lambda,\mu)}$
(7.14)

for any $0\leq m<N$, $\mu\in\mathbb{GT}_{m}$. As $N$ goes to infinity, the
left side of (7.14) tends to $M_{m}(\mu)$. By an argument similar to that in
the proof of Proposition 6.9, one shows that the right side of (7.14)
converges to
$\sum\limits_{\lambda\in\mathbb{GT}_{m+1}}M_{m+1}(\lambda)\Lambda^{m+1}_{m}(\lambda,\mu)$.
Indeed, the argument simply relies on the weak convergence
$M_{m+1}^{\lambda(N)}\rightarrow M_{m+1}$ and the uniform (on $\lambda$) bound
$|\Lambda^{m+1}_{m}(\lambda,\mu)|=\Lambda^{m+1}_{m}(\lambda,\mu)\leq 1$.
Therefore the limit of (7.14) as $N\rightarrow\infty$ is

$\displaystyle
M_{m}(\mu)=\sum_{\lambda\in\mathbb{GT}_{m}}{M_{m+1}(\lambda)\Lambda^{m+1}_{m}(\lambda,\mu)},$

for any $m\in\mathbb{Z}_{\geq 0}$, $\mu\in\mathbb{GT}_{m}$. Thus
$\\{M_{m}\\}_{m\geq 0}$ is a $(q,t)$-coherent sequence and has an associated
probability measure $M$ on $\mathcal{T}$, as given by Proposition 6.5. By the
definition of $\mathbf{N}$, we conclude $\mathbf{N}(M)=\nu$.

Step 2. Next we show that $\mathbf{N}$ is injective.

Let $M,M^{\prime}\in\Omega_{q,t}^{\rm Martin}$ have the same image $\nu$ under
the map $\mathbf{N}$. The goal is to prove $M=M^{\prime}$. From Lemma 7.5, we
may assume $\nu_{1}=0$ without any loss of generality. Furthermore, we can
assume that $M$ is the element of $\Omega_{q,t}^{\rm Martin}$ such that
$\mathbf{N}(M)=\nu$ and that was contructed in the first step.

Let $\\{M_{m}\\}_{m\geq 0}$ and $\\{M^{\prime}_{m}\\}_{m\geq 0}$ be the
$(q,t)$-coherent sequences associated to $M,M^{\prime}$, then

$\displaystyle\mathcal{P}_{M_{1}}(x)=\Phi^{\nu}(x;q,t)=\mathcal{P}_{M^{\prime}_{1}}(x)\qquad\forall\,x\in\mathbb{T}.$
(7.15)

As it was mentioned in Section 6.3, Macdonald generating functions are
uniformly bounded on the torus, in particular, $\mathcal{P}_{\mathfrak{m}}\in
L^{\infty}(\mathbb{T})\subset L^{2}(\mathbb{T})$ for any probability measure
$\mathfrak{m}$ on $\mathbb{GT}_{1}=\mathbb{Z}$. Write the first equality of
(7.15) for $x=e^{i\theta}$ as follows:

$\displaystyle\sum_{n\in\mathbb{Z}}{M_{1}(n)e^{in\theta}}=\sum_{n\in\mathbb{Z}}{M_{1}^{\prime}(n)e^{in\theta}}.$

Both sums above are expansions of a square integrable function on $\mathbb{T}$
in terms of the basis $\\{e^{in\theta}\\}\subset L^{2}(\mathbb{T})$. Thus the
(Fourier) coefficients in both sums must agree, i.e.,
$M_{1}(n)=M^{\prime}_{1}(n)$ $\forall\,n\in\mathbb{Z}$, and so
$M_{1}=M^{\prime}_{1}$.

We aim to apply a similar argument to show that $M_{m}=M^{\prime}_{m}$ for any
$m\in\mathbb{N}$. We will be done once this is proved, as Proposition 6.5
would then show $M=M^{\prime}$. For general $m$, we make use of the fact that
$M\in\Omega_{q,t}^{\rm Martin}$ is the probability measure constructed in step
1: we use that each $M_{m}$ is supported on $\mathbb{GT}^{+}_{m}$.

As above, the definition of the map $\mathbf{N}$ implies

$\displaystyle\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})=\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t\big{)}$
$\displaystyle\hphantom{\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})}{}=\mathcal{P}_{M^{\prime}_{m}}(x_{1},\dots,x_{m})\qquad\forall\,(x_{1},\dots,x_{m})\in\mathbb{T}^{m}.$

The equality of the functions above implies the equality of corresponding
Fourier coefficients. It follows that for any $\kappa\in\mathbb{GT}_{m}$ we
have, see (6.6),

$\displaystyle\sum_{\mu\in\mathbb{GT}_{m}}{\frac{c_{\mu,\kappa}M_{m}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}=\sum_{\mu\in\mathbb{GT}_{m}}{\frac{c_{\mu,\kappa}M_{m}^{\prime}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}.$
(7.16)

When $\kappa\notin\mathbb{GT}^{+}_{m}$ (or equivalently $\kappa_{m}<0$), we
claim that the left side of (7.16) is zero. In fact, for any
$\mu\in\mathbb{GT}_{m}$, either $c_{\mu,\kappa}=0$ when $\mu_{m}\geq 0$ or
$M_{m}(\mu)=0$ when $\mu_{m}<0$, because $M_{m}$ is supported on
$\mathbb{GT}^{+}_{m}$. Then also the right side of (7.16) is zero if
$\kappa\notin\mathbb{GT}^{+}_{m}$. By using also the properties of the
coefficients $c_{\mu,\kappa}$ stated in Corollary 2.6, we have

$\displaystyle
0=\sum_{\mu\in\mathbb{GT}_{m}}{\frac{c_{\mu,\kappa}M_{m}^{\prime}(\mu)}{P_{\mu}\big{(}1,t,\dots,t^{m-1}\big{)}}}\geq\frac{c_{\kappa,\kappa}M_{m}^{\prime}(\kappa)}{P_{\kappa}\big{(}1,t,\dots,t^{m-1}\big{)}}=\frac{M_{m}^{\prime}(\kappa)}{P_{\kappa}\big{(}1,t,\dots,t^{m-1}\big{)}}\geq
0.$

Then we must have $M_{m}^{\prime}(\kappa)=0$ if
$\kappa\notin\mathbb{GT}^{+}_{m}$, i.e., $M_{m}^{\prime}$ is supported on
$\mathbb{GT}^{+}_{m}$. Finally an application of Lemma 6.7 says that if
$\mathcal{P}_{M_{m}}=\mathcal{P}_{M_{m}^{\prime}}$ on $\mathbb{T}^{m}$, and
$M_{m}$, $M_{m}^{\prime}$ are both supported on $\mathbb{GT}^{+}_{m}$, implies
$M_{m}=M^{\prime}_{m}$, thus finishing the proof. ∎

###  7.2 Characterization of the Martin boundary

Our next goal is to characterize the topological space $\Omega_{q,t}^{\rm
Martin}$ completely. Recall the map $\mathbf{N}\colon\Omega_{q,t}^{\rm
Martin}\rightarrow\mathcal{N}$, defined above in (7.8), by letting
$\mathbf{N}(M)=\nu$ be the unique element of $\mathcal{N}$ such that

$\displaystyle\Phi^{\nu}\big{(}x_{1}t^{1-m},\dots,x_{m-1}t^{-1},x_{m};q,t\big{)}=\mathcal{P}_{M_{m}}(x_{1},\dots,x_{m})\\!\qquad\forall\,(x_{1},\dots,x_{m})\in\mathbb{T}^{m},\\!\quad\forall\,m\geq
1.$

Proposition 7.6 shows that $\mathbf{N}$ is a bijection, so the inverse map
$\mathbf{N}^{-1}$ is well-defined.

######  Definition 7.7.

For any $\nu\in\mathcal{N}$, we denote $\mathbf{N}^{-1}(\nu)$ by $M^{\nu}$ and
the corresponding $(q,t)$-coherent sequence by $\\{M^{\nu}_{m}\\}_{m\geq 0}$.

From step 1 of the proof of Proposition 7.6, and Lemma 7.5, we have that for
the sequence of signatures
$\\{\lambda(N)=(\nu_{N}\geq\cdots\geq\nu_{1})\\}_{N\geq 1}$ which stabilizes
to $\nu$, the following weak convergence holds

$\displaystyle\mathbb{P}-\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}=M^{\nu}_{m}\qquad\forall\,m\in\mathbb{N},$

i.e.,

$\displaystyle\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}(\mu)=\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}(\lambda(N),\mu)}=M^{\nu}_{m}(\mu)\qquad\forall\,m\in\mathbb{N},\quad\forall\,\mu\in\mathbb{GT}_{m}.$

In fact, we note the same analysis as in step 1 of the proof of Proposition
7.6 shows that the weak convergence above holds for any sequence of signatures
$\\{\lambda(N)\\}_{N\geq 1}$ stabilizing to $\nu$.

We need the following lemma to prove that $\mathbf{N}$ is a homeomorphism.

######  Lemma 7.8.

  1. $1.$

For any $\nu\in\mathcal{N}$, $m\in\mathbb{N}$, $M_{m}^{\nu}$ is supported on
$\\{\mu=(\mu_{1}\geq\dots\geq\mu_{m-1}\geq\mu_{m})\in\mathbb{GT}_{m}\colon\mu_{m}\geq\nu_{1},\,\mu_{m-1}\geq\nu_{2},\,\dots,\,\mu_{1}\geq\nu_{m}\\}$.

  2. $2.$

For any $m\in\mathbb{N}$, there exists $c_{m}\in(0,1)$ such that for any
integers $n_{1}\leq n_{2}\leq\dots\leq n_{m}$ and $\nu\in\mathcal{N}$ with
$\nu_{1}=n_{1},\,\nu_{2}=n_{2},\,\dots,\,\nu_{m}=n_{m}$, we have
$M^{\nu}_{m}(n_{m}\geq\dots\geq n_{2}\geq n_{1})\geq c_{m}$.

  3. $3.$

Let $m\in\mathbb{N}$ and let $\widetilde{\nu},\nu\in\mathcal{N}$ be such that
$\widetilde{\nu}_{i}\geq\nu_{i}$ $\forall\,i>m$ and
$\widetilde{\nu}_{i}=\nu_{i}$ $\forall\,1\leq i\leq m$, then
$M^{\widetilde{\nu}}_{m}(\nu_{m}\geq\dots\geq\nu_{2}\geq\nu_{1})\leq
M^{\nu}_{m}(\nu_{m}\geq\cdots\geq\nu_{2}\geq\nu_{1})$.

###### Proof.

Let us prove (1). We use the weak convergence of probabilities
$\mathbb{P}-\lim\limits_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}=M^{\nu}_{m}$
$\forall\,m\in\mathbb{N}$, for the sequence of signatures
$\\{\lambda(N)=(\nu_{N}\geq\dots\geq\nu_{2}\geq\nu_{1})\\}_{N\geq 1}$. Because
of the definition (6.4) for the maps $\Lambda^{N}_{m}$, it follows that
$\Lambda^{N}_{m}\delta_{\lambda(N)}(\mu)=\Lambda^{N}_{m}(\lambda(N),\mu)=0$,
unless
$\mu_{m}\geq\lambda(N)_{N}=\nu_{1},\dots,\mu_{1}\geq\lambda(N)_{N-m+1}=\nu_{m}$.
Thus also $M^{\nu}_{m}(\mu)=0$ unless
$\mu_{m}\geq\nu_{1},\dots,\mu_{1}\geq\nu_{m}$.

Next we show (2); to get started fix $n_{1}\in\mathbb{N}$ and let us show that
$M^{\nu}_{1}(n_{1})\geq((t;t)_{\infty})^{\theta}$ for any $\nu\in\mathcal{N}$
with $\nu_{1}=n_{1}$ (so we can set $c_{1}=((t;t)_{\infty})^{\theta}$). The
weak convergence mentioned above gives
$M_{1}^{\nu}(n_{1})=\lim\limits_{N\rightarrow\infty}{\Lambda^{N}_{1}\delta_{\lambda(N)}(n_{1})}=\lim\limits_{N\rightarrow\infty}{\Lambda^{N}_{1}\delta_{\lambda(N)}(\lambda(N)_{N})}$,
for any $\nu\in\mathcal{N}$ with $\nu_{1}=n_{1}$ and $\\{\lambda(N)\\}_{N\geq
1}$ as constructed above. The calculations in the proof of Lemma 7.2 show
$\Lambda^{N}_{1}\delta_{\lambda(N)}(\lambda(N)_{N})\geq((t;t)_{\infty})^{\theta}$
for all $N\in\mathbb{N}$, and thus also $M^{\nu}_{1}(n_{1})\geq
c_{1}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}((t;t)_{\infty})^{\theta}>0$, for any $\nu\in\mathcal{N}$ with
$\nu_{1}=n_{1}$.

For a general $m\in\mathbb{N}$, we have

$M_{m}^{\nu}(n_{m}\geq\dots\geq n_{2}\geq
n_{1})=\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}(\lambda(N)_{N-m+1}\geq\dots\geq\lambda(N)_{N})},$

for $\\{\lambda(N)\\}_{N\geq 1}$ as constructed above. In the proof of Lemma
7.3, we showed $\Lambda^{N}_{m}\delta_{\lambda(N)}(\lambda(N)_{N-m+1}$
$\geq\cdots\geq\lambda(N)_{N})\geq
c_{m}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}((t;t)_{\infty})^{\theta m}$ holds for all $N\geq 1$. Therefore
$M_{m}^{\nu}(n_{m}\geq\dots\geq n_{2}\geq n_{1})\geq c_{m}>0$, for any
$\nu\in\mathcal{N}$ with $\nu_{1}=n_{1}$, $\nu_{2}=n_{2}$, $\dots$,
$\nu_{m}=n_{m}$.

Finally we move on to (3). To get started, we prove it for $m=1$, so let
$\widetilde{\nu},\nu\in\mathcal{N}$ be such that
$\widetilde{\nu}_{i}\geq\nu_{i}$ $\forall\,i\geq 2$ and
$\widetilde{\nu}_{1}=\nu_{1}$. Consider the following pair of sequences of
signatures
$\\{\widetilde{\lambda}(N)=(\widetilde{\nu}_{N}\geq\dots\geq\widetilde{\nu}_{2}\geq\widetilde{\nu}_{1})\\}_{N\geq
1}$ and $\\{\lambda(N)=(\nu_{N}\geq\dots\geq\nu_{2}\geq\nu_{1})\\}_{N\geq 1}$.
Then we have the weak limits

$\displaystyle\mathbb{P}-\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\widetilde{\lambda}(N)}}=M^{\widetilde{\nu}}_{m},\qquad\mathbb{P}-\lim_{N\rightarrow\infty}{\Lambda^{N}_{m}\delta_{\lambda(N)}}=M^{\nu}_{m}\hskip
7.22743pt\forall m\in\mathbb{N}.$ (7.17)

By the calculations in Lemma 7.2, we have

$\displaystyle\Lambda^{N}_{1}\delta_{\lambda(N)}(\nu_{1})=\Lambda^{N}_{1}\delta_{\lambda(N)}(\lambda(N)_{N})$
$\displaystyle\hphantom{\Lambda^{N}_{1}\delta_{\lambda(N)}(\nu_{1})}{}=\prod_{i=1}^{N-1}{\frac{(1-t^{N-i})(1-qt^{N-i})\cdots(1-q^{\theta-1}t^{N-i})}{(1-q^{\lambda(N)_{i}-\lambda(N)_{N}}t^{N-i})\cdots(1-q^{\lambda(N)_{i}-\lambda(N)_{N}+\theta-1}t^{N-i})}}$
$\displaystyle\hphantom{\Lambda^{N}_{1}\delta_{\lambda(N)}(\nu_{1})}{}=\prod_{i=1}^{N-1}{\frac{(1-t^{N-i})(1-qt^{N-i})\cdots(1-q^{\theta-1}t^{N-i})}{(1-q^{\nu_{N-i+1}-\nu_{1}}t^{N-i})\cdots(1-q^{\nu_{N-i+1}-\nu_{1}+\theta-1}t^{N-i})}}$
$\displaystyle\hphantom{\Lambda^{N}_{1}\delta_{\lambda(N)}(\nu_{1})}{}\geq\prod_{i=1}^{N-1}{\frac{(1-t^{N-i})(1-qt^{N-i})\cdots(1-q^{\theta-1}t^{N-i})}{(1-q^{\widetilde{\nu}_{N-i+1}-\widetilde{\nu}_{1}}t^{N-i})\cdots(1-q^{\widetilde{\nu}_{N-i+1}-\widetilde{\nu}_{1}+\theta-1}t^{N-i})}}$
$\displaystyle\hphantom{\Lambda^{N}_{1}\delta_{\lambda(N)}(\nu_{1})}{}=\prod_{i=1}^{N-1}{\frac{(1-t^{N-i})(1-qt^{N-i})\cdots(1-q^{\theta-1}t^{N-i})}{(1-q^{\widetilde{\lambda}(N)_{i}-\widetilde{\lambda}(N)_{N}-\theta}t^{N-i+1})\cdots(1-q^{\widetilde{\lambda}(N)_{i}-\widetilde{\lambda}(N)_{N}-1}t^{N-i+1})}}$
$\displaystyle\hphantom{\Lambda^{N}_{1}\delta_{\lambda(N)}(\nu_{1})}{}=\Lambda^{N}_{1}\delta_{\widetilde{\lambda}(N)}(\widetilde{\lambda}(N)_{N})=\Lambda^{N}_{1}\delta_{\widetilde{\lambda}(N)}(\nu_{1}).$

Thus taking into account the limits (7.17) for $m=1$, we deduce
$M^{\nu}_{1}(\nu_{1})\geq M^{\widetilde{\nu}}_{1}(\nu_{1})$.

The proof of the third item for a general $m\in\mathbb{N}$ follows from
similar calculations that are used to prove Lemma 7.3. We leave the details to
the reader. ∎

######  Theorem 7.9.

The bijective map $\mathbf{N}\colon\Omega_{q,t}^{\rm
Martin}\rightarrow\mathcal{N}$ is a homeomorphism.

###### Proof.

The first step shows that $\mathbf{N}^{-1}$ is continuous and the second one
shows that $\mathbf{N}$ is continuous.

Step 1. Let $\big{\\{}M^{\nu^{(i)}}\big{\\}}_{i\geq
1}\subset\Omega_{q,t}^{{\rm Martin}}$ and $\big{\\{}\nu^{(i)}\big{\\}}_{i\geq
1}$ be the corresponding images under the map $\mathbf{N}$. If
$\lim\limits_{i\rightarrow\infty}{\nu^{(i)}}=\nu\in\mathcal{N}$ pointwise,
then we prove the weak limit
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{\nu^{(i)}}}=M^{\nu}$.

By applying some automorphism $A_{k}$ with large $k$, if necessary, and
invoking Lemma 7.5, we can assume $\nu_{1}\geq 0$. Observe that
$\lim\limits_{i\rightarrow\infty}{\nu^{(i)}_{1}}=\nu_{1}\geq 0$ implies
$\nu^{(i)}_{1}\geq 0$ for large enough $i$. Thus let us also assume
$\nu^{(i)}_{1}\geq 0$ $\forall\,i\geq 1$, for simplicity.

We reduce our desired statement to simpler ones. We claim that
$\lim\limits_{i\rightarrow\infty}{\nu^{(i)}}=\nu$ implies

$\displaystyle\lim_{i\rightarrow\infty}{\Phi^{\nu^{(i)}}(x_{1},\dots,x_{m};q,t)}=\Phi^{\nu}(x_{1},\dots,x_{m};q,t)\qquad\forall\,m\in\mathbb{N}$
(7.18)

uniformly on $\mathbb{T}^{m}$.

First let us deduce our desired weak convergence from the limit above. Recall
that, due to Theorem 5.5, all functions
$\big{\\{}\Phi^{\nu^{(i)}}(x_{1},\dots,x_{m};q,t)\big{\\}}_{i\geq 1}$ and
$\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ are entire functions and, in particular,
they are continuous on the torus $\mathbb{T}^{m}$. So the convergence (7.18)
implies the convergence of Fourier coefficients. Thus for any
$\kappa\in\mathbb{GT}_{m}$ we obtain, see (6.6),

$\displaystyle\lim_{i\rightarrow\infty}{\sum_{\mu\in\mathbb{GT}^{+}_{m}}{\frac{c_{\mu,\kappa}M^{\nu^{(i)}}_{m}(\mu)}{P_{\mu}(1,t,\dots,t^{m-1})}}}=\sum_{\mu\in\mathbb{GT}^{+}_{m}}{\frac{c_{\mu,\kappa}M^{\nu}_{m}(\mu)}{P_{\mu}(1,t,\dots,t^{m-1})}}.$
(7.19)

Note that in both sides of (7.19), the sums have been restricted to
$\mathbb{GT}^{+}_{m}$. In fact, since we are assuming $\nu^{(i)}_{1}$,
$\nu_{1}\geq 0$ $\forall\,i\geq 1$, Lemma 7.8(1) implies that all probability
measures $\big{\\{}M^{\nu^{(i)}}_{m}\big{\\}}_{i\geq 1},M^{\nu}_{m}$ are
supported on $\mathbb{GT}^{+}_{m}$.

From Proposition 6.10, the limit (7.19) yields
$\lim\limits_{i\rightarrow\infty}{M_{m}^{\nu^{(i)}}(\mu)}=M_{m}^{\nu}(\mu)$
$\forall\,\mu\in\mathbb{GT}^{+}_{m}$. But also
$M^{\nu^{(i)}}_{m}(\mu)=M_{m}^{\nu}(\mu)=0$ for any $i\geq 1$ and any
$\mu\notin\mathbb{GT}^{+}_{m}$. Therefore
$\lim\limits_{i\rightarrow\infty}{M_{m}^{\nu^{(i)}}(\mu)}=M_{m}^{\nu}(\mu)$
holds for all $\mu\in\mathbb{GT}_{m}$, and so
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{\nu^{(i)}}_{m}}=M_{m}$. Note
that we proved the weak convergence for any $m\in\mathbb{N}$. By Proposition
6.5, we conclude
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{\nu^{(i)}}}=M$.

We are left with the task of proving (7.18). We show the uniform convergence
in a neighborhood of the torus $\mathbb{T}^{m}$. But it suffices to prove the
limit on compact subsets of $\mathcal{U}_{m}$. From the definition of
$\Phi^{\nu}(x_{1},\dots,x_{m};q,t)$ on $\mathcal{U}_{m}$, see (5.8), and
observing that $\mathbb{T}^{m}\subset\mathcal{U}_{m}$, it follows that the
result holds for any $m\in\mathbb{N}$ provided it holds for $m=1$. We prove
(7.18) for $m=1$ on an open neighborhood of $\mathbb{T}$.

Clearly a small enough open neighborhood of $\mathbb{T}$ is a subset of
$\mathcal{U}$. By the definition of $\Phi^{\nu}(x;q,t)$ on $\mathcal{U}$, see
(5.2), the desired uniform convergence will hold if we verify

$\displaystyle\lim_{i\rightarrow\infty}{\int_{\mathcal{C}^{+}}{x^{z}\prod_{j=1}^{\infty}{\frac{(q^{-z+\nu_{j}^{(i)}}t^{j};q)_{\infty}}{(q^{-z+\nu_{j}^{(i)}}t^{j-1};q)_{\infty}}}}}=\int_{\mathcal{C}^{+}}{x^{z}\prod_{j=1}^{\infty}{\frac{(q^{-z+\nu_{j}}t^{j};q)_{\infty}}{(q^{-z+\nu_{j}}t^{j-1};q)_{\infty}}}}$
(7.20)

uniformly on compact subsets of $x\in\mathbb{C}\setminus\\{0\\}$. Note that,
since all $\big{\\{}\nu^{(i)}_{1}\big{\\}}_{i\geq 1}$, $\nu_{1}$, are
nonnegative, we can take the same contour $\mathcal{C}^{+}$ for both of the
integrals in (7.20).

The pointwise convergence of integrands in (7.20) is clear. We still need some
uniform estimates for the contribution of the tails of the left side in
(7.20). This is similar to the proof of Theorem 5.2.

Let $K\subset\mathbb{C}\setminus\\{0\\}$ be any compact set. Parametrize the
tails of $\mathcal{C}^{+}$ as $z=r\pm\frac{\pi\sqrt{-1}}{\ln{q}}$; for $r$
ranging from some large $R>0$ to $+\infty$, we want to show that the
contribution of each of these lines is small. We have

$\displaystyle\sup_{x\in
K}\left|x^{z}\cdot\prod_{j=1}^{\infty}{\frac{\big{(}q^{-z+\nu_{j}^{(i)}}t^{j};q\big{)}_{\infty}}{\big{(}q^{-z+\nu_{j}^{(i)}}t^{j-1};q\big{)}_{\infty}}}\right|$
$\displaystyle\qquad{}\leq{\rm
const}\times|x|^{r}\times\prod_{j=1}^{\infty}{\frac{1}{\big{(}1+q^{-r+\nu_{j}^{(i)}+\theta(j-1)}\big{)}\cdots\big{(}1+q^{-r+\nu_{j}^{(i)}+\theta
j-1}\big{)}}}$ $\displaystyle\qquad{}\leq{\rm
const}\cdot\frac{|x|^{r}}{q^{-kr}q^{\nu^{(i)}_{1}+\nu^{(i)}_{2}+\dots+\nu^{(i)}_{k}}}={\rm
const}\cdot\frac{\big{(}|x|q^{k}\big{)}^{r}}{q^{\nu_{1}^{(i)}+\dots+\nu_{k}^{(i)}}},$

for any $z=r\pm\frac{\pi\sqrt{-1}}{\ln{q}}$, and any $k\in\mathbb{N}$. The
constant above depends on $K$ but not on $i$. Choose $k\in\mathbb{N}$ large
enough so that $a\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sup\limits_{x\in K}{|x|}\cdot q^{k}\in(0,1)$. Since
$\lim\limits_{i\rightarrow\infty}{\nu_{j}^{(i)}}=\nu_{j}$ for all
$j=1,2,\dots,k$, then $\sup\limits_{i\geq
1}{q^{-\nu_{1}^{(i)}-\nu_{2}^{(i)}-\dots-\nu_{k}^{(i)}}}\leq{\rm
const}<\infty$. Then there exists a constant $c=c_{K}>0$, independent of $i$,
such that

$\displaystyle\sup_{i\geq 1}\sup_{x\in
K}\left|x^{z}\cdot\prod_{j=1}^{\infty}{\frac{\big{(}q^{-z+\nu_{j}^{(i)}}t^{j};q\big{)}_{\infty}}{\big{(}q^{-z+\nu_{j}^{(i)}}t^{j-1};q\big{)}_{\infty}}}\right|\leq
c\cdot a^{r},\qquad\textrm{if}\quad z=r\pm\frac{\pi\sqrt{-1}}{\ln{q}}.$

Since $\int_{R}^{\infty}{a^{r}{\rm
d}r}=-e^{R\ln{a}}/\ln{a}\xrightarrow{R\rightarrow\infty}0$, we have just shown
that the contribution of the tails of $\mathcal{C}^{+}$ is uniformly small. We
can then apply dominated convergence theorem to conclude (7.20), as desired.

Step 2. As in the step above, let $\big{\\{}M^{\nu^{(i)}}\big{\\}}_{i\geq 1}$
be a sequence in $\Omega_{q,t}^{{\rm Martin}}$, whose images under
$\mathbf{N}$ are $\big{\\{}\nu^{(i)}\big{\\}}_{i\geq 1}$. If the weak limit
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{\nu^{(i)}}}=M$ holds, then we
show that $M\in\Omega_{q,t}^{{\rm Martin}}$, that there a pointwise limit
$\lim\limits_{i\rightarrow\infty}{\nu^{(i)}}=\nu\in\mathcal{N}$ and moreover
$M=M^{\nu}$.

Let $m\in\mathbb{N}$ be arbitrary. Thanks to Proposition 6.5, the limit
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M^{\nu^{(i)}}}=M$ implies
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M_{m}^{\nu^{(i)}}}=M_{m}$. By
Lemma 7.8(2), there exists $c_{m}\in(0,1)$ such that
$M_{m}^{\nu^{(i)}}\big{(}\nu_{m}^{(i)}\geq\cdots\geq\nu_{1}^{(i)}\big{)}\geq
c_{m}>0$ for all $i\geq 1$. Since $M_{m}$ is a probability measure, there
exists $N_{0}\in\mathbb{N}$ large enough such that
$M_{m}((\lambda_{1},\dots,\lambda_{m})\in\mathbb{GT}_{m}\colon
N_{0}\geq\lambda_{1},\dots,\lambda_{m}\geq-N_{0})>1-\frac{c_{m}}{2}$. Then
$\mathbb{P}-\lim\limits_{i\rightarrow\infty}{M_{m}^{\nu^{(i)}}}=M_{m}$ implies
$M_{m}^{\nu^{(i)}}((\lambda_{1},\dots,\lambda_{m})\in\mathbb{GT}_{m}\colon
N_{0}\geq\lambda_{1},\dots,\lambda_{m}\geq-N_{0})>1-c_{m}$ for all $i$ large
enough. It follows that $N_{0}\geq\nu_{m}^{(i)},\dots,\nu_{1}^{(i)}\geq-N_{0}$
for all $i$ large enough. In particular, the sequence
$\big{\\{}\nu_{m}^{(i)}\big{\\}}_{i\geq 1}$ is bounded.

By using the boundedness of the sequences
$\big{\\{}\nu^{(i)}_{m}\big{\\}}_{i\geq 1}$ and the “diagonal argument” of
Proposition 7.4, there exists a subsequence $1\leq i_{1}<i_{2}<i_{3}<\cdots$
such that the limits $\lim\limits_{k\rightarrow\infty}{\nu^{(i_{k})}_{m}}$
exist, for all $m\in\mathbb{N}$. In other words, we have the pointwise limit
$\lim\limits_{k\rightarrow\infty}{\nu^{(i_{k})}}=\nu$, for some
$\nu\in\mathcal{N}$.

We have the obvious implication

$\displaystyle\mathbb{P}-\lim_{i\rightarrow\infty}{M^{\nu^{(i)}}}=M\Longrightarrow\mathbb{P}-\lim_{k\rightarrow\infty}{M^{\nu^{(i_{k})}}}=M,$

But step 1 in this proof shows that
$\lim\limits_{k\rightarrow\infty}{\nu^{(i_{k})}}=\nu$ implies
$\mathbb{P}-\lim\limits_{k\rightarrow\infty}{M^{\nu^{(i_{k})}}}=M^{\nu}$. By
uniqueness of weak limits, we have $M=M^{\nu}$.

We are left to show that $\lim\limits_{i\rightarrow\infty}{\nu^{(i)}}=\nu$.
Above we only showed the existence of a subsequence $\\{i_{k}\\}_{k\geq 1}$
such that $\lim\limits_{k\rightarrow\infty}{\nu^{(i_{k})}}=\nu$. Nevertheless
the same argument gives us more. In fact, it shows that any subsequence
$\big{\\{}\nu^{(i_{r})}\big{\\}}_{r\geq
1}\subset\big{\\{}\nu^{(i)}\big{\\}}_{i\geq 1}$ must have a subsubsequence
$\big{\\{}\nu^{(i_{r(s)})}\big{\\}}_{s\geq
1}\subset\big{\\{}\nu^{(i_{r})}\big{\\}}_{r\geq 1}$ such that a pointwise
limit $\lim\limits_{s\rightarrow\infty}{\nu^{(i_{r(s)})}}=\nu^{\prime}$
exists. Then $M=M^{\nu^{\prime}}=M^{\nu}$, but since the map $\mathbf{N}$ is
bijective, we have $\nu=\nu^{\prime}$. We therefore conclude that the sequence
$\big{\\{}\nu^{(i)}\big{\\}}_{i\geq 1}$ itself must converge to $\nu$,
finishing the proof. ∎

###  7.3 Relation between the Martin boundary and the (minimal) boundary

The basic relation between the Martin and minimal boundary of the $(q,t)$-GT
graph is the following statement, which actually holds in a much greater
generality.

######  Proposition 7.10 (consequence of [37, Theorem 6.1]).

The following inclusion holds $\Omega_{q,t}\subseteq\Omega_{q,t}^{\rm
Martin}$. In other words, for any $M\in\Omega_{q,t}$, there exists a sequence
$\\{\lambda(N)\\}_{N\geq 1}$, $\lambda(N)\in\mathbb{GT}_{N}$, such that

$\displaystyle\mathbb{P}-\lim_{N\rightarrow\infty}{M^{\lambda(N)}_{m}}=M_{m}\qquad\forall\,m=0,1,2,\dots.$

In many examples, especially in the context of asymptotic representation
theory, it is known that the Martin boundary of a branching graph is equal to
its minimal boundary, e.g., [21, 37]. In our case, we will also prove that
this is the case by following the ideas in [21, 41]. The following statement,
which also holds in greater generality, will be useful.

######  Proposition 7.11 (consequence of [37, Theorem 9.2]).

Let $M^{\prime}$ be any probability measure on $M_{\rm prob}(\mathcal{T})$.
There exists a unique Borel probability measure $\pi$ belonging to
$\Omega_{q,t}$ such that

$\displaystyle
M^{\prime}(S_{\phi})=\int_{M\in\Omega_{q,t}}{M(S_{\phi})\pi({\rm d}M)},$

for any finite path
$\phi=\big{(}\phi^{(0)}\prec\phi^{(1)}\prec\cdots\prec\phi^{(n)}\big{)}$ in
the $(q,t)$-GT graph.

###  7.4 Characterization of the boundary of the
$\boldsymbol{(q,t)}$-Gelfand–Tsetlin graph

###### Proof of Theorem 1.3.

Let us now prove the items (1), (2) stated in the theorem.

(1) Thanks to Proposition 7.10, we have
$\Omega_{q,t}\subseteq\Omega_{q,t}^{\rm Martin}\subseteq M_{\rm
prob}(\mathcal{T})$. We claim that $\Omega_{q,t}^{\rm
Martin}\subseteq\Omega_{q,t}$. Because of Theorem 7.9, the Martin boundary
$\Omega_{q,t}^{\rm Martin}$ (with its induced topology from $M_{\rm
prob}(\mathcal{T})$) is homeomorphic to $\mathcal{N}$, under the map
$\mathbf{N}$. The claim would then show that the minimal boundary
$\Omega_{q,t}$ of the $(q,t)$-GT graph is equal to the Martin boundary
$\Omega_{q,t}^{\rm Martin}$ and therefore homeomorphic to $\mathcal{N}$, under
the map $\mathbf{N}$.

Let us prove the claim above. Let $\nu\in\mathcal{N}$ be arbitrary and let
$M^{\nu}\in\Omega_{q,t}^{\rm Martin}$ be the corresponding element of
$\Omega_{q,t}^{\rm Martin}$. Let
$\mathfrak{i}\colon\Omega_{q,t}\hookrightarrow\Omega_{q,t}^{\rm Martin}$ be
the natural inclusion, considered as a measurable map. By Proposition 7.11,
there exists a unique probability measure $\pi$ on $\Omega_{q,t}$ such that

$\displaystyle M^{\nu}=\int_{M\in\Omega_{q,t}}{M\pi({\rm
d}M)}=\int_{M\in\Omega_{q,t}^{\rm Martin}}{M(\mathfrak{i}_{*}\pi)({\rm d}M)}.$
(7.21)

Note that $i_{*}\pi$ is a probability measure on $\Omega_{q,t}^{\rm Martin}$
with $(i_{*}\pi)(\Omega_{q,t})=1$. Let $\widetilde{\pi}$ be the pushforward of
$\mathfrak{i}_{*}\pi$ under the homeomorphism
$\mathbf{N}\colon\Omega_{q,t}^{\rm Martin}\rightarrow\mathcal{N}$, so
$\widetilde{\pi}$ is a Borel probability measure on $\mathcal{N}$. Equation
(7.21) can be rewritten as

$\displaystyle
M^{\nu}=\int_{\widetilde{\nu}\in\mathcal{N}}{M^{\widetilde{\nu}}\widetilde{\pi}({\rm
d}\widetilde{\nu})}.$ (7.22)

We make a subclaim: $\widetilde{\pi}$ is the delta mass at
$\nu\in\mathcal{N}$. Let us first deduce $\Omega_{q,t}^{\rm
Martin}\subseteq\Omega_{q,t}$ from this latter claim. In fact, if
$\widetilde{\pi}$ is the delta mass at $\nu\in\mathcal{N}$, then
$\mathfrak{i}_{*}\pi$ is the delta mass at $M^{\nu}$. But since we had
$(i_{*}\pi)(\Omega_{q,t})=1$, then $M^{\nu}\in\Omega_{q,t}$. Since $M^{\nu}$
was an arbitrary element of $\Omega_{q,t}^{\rm Martin}$, then we conclude
$\Omega_{q,t}^{\rm Martin}\subseteq\Omega_{q,t}$.

Let us now prove the subclaim that the probability measure $\widetilde{\pi}$
on $\mathcal{N}$ satisfying (7.22) must be the delta mass at
$\nu\in\mathcal{N}$. We show first that $\widetilde{\pi}$ is supported on
$\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}\geq\nu\\}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}_{1}\geq\nu_{1},\,\widetilde{\nu}_{2}\geq\nu_{2},\,\widetilde{\nu}_{3}\geq\nu_{3},\dots\\}$.
Since $\widetilde{\pi}$ is a Borel measure, the opposite would mean the
existence of $m\in\mathbb{N}$ and
$\kappa_{1}\leq\kappa_{2}\leq\cdots\leq\kappa_{m}$, such that
$\kappa_{i}<\nu_{i}$ for some $1\leq i\leq m$, and

$\displaystyle\widetilde{\pi}\big{(}\big{\\{}\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}_{1}=\kappa_{1},\dots,\widetilde{\nu}_{m}=\kappa_{m}\big{\\}}\big{)}>0.$

As a consequence of (7.22) we have, for all $m\in\mathbb{N}$,

$\displaystyle
M_{m}^{\nu}=\int_{\widetilde{\nu}\in\mathcal{N}}{M_{m}^{\widetilde{\nu}}\widetilde{\pi}({\rm
d}\widetilde{\nu})}.$ (7.23)

We can now apply (7.23) to
$\kappa=(\kappa_{m}\geq\dots\geq\kappa_{1})\in\mathbb{GT}_{m}$:

$\displaystyle
M_{m}^{\nu}(\kappa)=\int_{\widetilde{\nu}\in\mathcal{N}}{M_{m}^{\widetilde{\nu}}(\kappa)\widetilde{\pi}({\rm
d}\widetilde{\nu})}.$

From Lemma 7.8(1), the left-hand side of the equality above vanishes, while
Lemma 7.8(2) shows that the right-hand side is at least
$c_{m}\cdot\widetilde{\pi}(\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}_{1}=\kappa_{1},\dots,\widetilde{\nu}_{m}=\kappa_{m}\\})>0$,
thus there is a contradiction.

From the fact that $\widetilde{\pi}$ is supported on
$\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}\geq\nu\\}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}_{1}\geq\nu_{1},\widetilde{\nu}_{2}\geq\nu_{2},\dots\\}$,
and Lemma 7.8, parts (1) (3), we have that (7.23) evaluated at
$(\nu_{m}\geq\cdots\geq\nu_{1})$ is

$\displaystyle
M^{\nu}_{m}(\nu_{m}\geq\dots\geq\nu_{1})=\int_{\widetilde{\nu}\in\mathcal{N}}{{M_{m}^{\widetilde{\nu}}(\nu_{m}\geq\dots\geq\nu_{1})\widetilde{\pi}({\rm
d}\widetilde{\nu})}}=\int_{\begin{subarray}{c}\widetilde{\nu}\in\mathcal{N}\\\
\widetilde{\nu}\geq\nu\end{subarray}}{{M_{m}^{\widetilde{\nu}}(\nu_{m}\geq\dots\geq\nu_{1})\widetilde{\pi}({\rm
d}\widetilde{\nu})}}$
$\displaystyle\qquad{}=\int_{\begin{subarray}{c}\widetilde{\nu}\in\mathcal{N},\,\widetilde{\nu}\geq\nu\\\
\widetilde{\nu}_{i}=\nu_{i}\ \forall
i=1,\dots,m\end{subarray}}{{M_{m}^{\widetilde{\nu}}(\nu_{m}\geq\dots\geq\nu_{1})\widetilde{\pi}({\rm
d}\widetilde{\nu})}}$
$\displaystyle\qquad\quad{}+\int_{\begin{subarray}{c}\widetilde{\nu}\in\mathcal{N},\,\widetilde{\nu}\geq\nu\\\
\widetilde{\nu}_{i}\neq\nu_{i}\textrm{ for some
}i\in\\{1,\dots,m\\}\end{subarray}}{{M_{m}^{\widetilde{\nu}}(\nu_{m}\geq\dots\geq\nu_{1})\widetilde{\pi}({\rm
d}\widetilde{\nu})}}$
$\displaystyle\qquad{}=\int_{\begin{subarray}{c}\widetilde{\nu}\in\mathcal{N},\widetilde{\nu}\geq\nu\\\
\widetilde{\nu}_{i}=\nu_{i}\
\forall\,i=1,\dots,m\end{subarray}}{{M_{m}^{\widetilde{\nu}}(\nu_{m}\geq\dots\geq\nu_{1})\widetilde{\pi}({\rm
d}\widetilde{\nu})}}$ $\displaystyle\qquad\leq
M_{m}^{\nu}(\nu_{m}\geq\dots\geq\nu_{1})\cdot\widetilde{\pi}(\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}_{1}=\nu_{1},\dots,\widetilde{\nu}_{m}=\nu_{m}\\}).$

Next Lemma 7.8(2) says that $M_{m}^{\nu}(\nu_{m}\geq\dots\geq\nu_{1})\geq
c_{m}>0$, so we must have
$\widetilde{\pi}(\\{\widetilde{\nu}\in\mathcal{N}\colon\widetilde{\nu}_{1}=\nu_{1},\dots,\widetilde{\nu}_{m}=\nu_{m}\\})=1$.
Since this is true for any $m\in\mathbb{N}$, it follows that $\widetilde{\pi}$
must be the delta mass at $\nu$, thus proving our second claim and the full
characterization of $\Omega_{q,t}$.

Let us return to the statement of item (1) in Theorem 1.3. By definition of
the map $\mathbf{N}$, the relations (1.2) hold. We have already observed that
the Macdonald generating function defining
$\mathcal{P}_{M_{m}^{\nu}}(x_{1},\dots,x_{m})$ is absolutely convergent on
$\mathbb{T}^{m}$. The last statement that says $M^{\nu}$ is determined by the
relations (1.2) follows from the uniqueness statement in Proposition 7.4.

(2) Let $\\{M_{m}^{\nu}\\}_{m\geq 0}$, $\\{M_{m}^{A_{k}\nu}\\}_{m\geq 0}$, be
the $(q,t)$-coherent sequences associated to $M^{\nu}$ and $M^{A_{k}\nu}$,
respectively. By Lemma 7.5, we have the first statement
$M^{A_{k}\nu}(S_{A_{k}\phi})=M^{\nu}(S_{\phi})$, for any finite path $\phi$.
Next by virtue of Lemma 6.12, $M^{A_{k}\nu}_{m}=A_{k}M^{\nu}_{m}$ for all
$m\geq 0$. Thus by following the definitions,
$M^{A_{k}\nu}_{m}(A_{k}\lambda)=A_{k}M^{\nu}_{m}(A_{k}\lambda)=M^{\nu}_{m}(A_{-k}A_{k}\lambda)=M^{\nu}_{m}(\lambda)$,
for any $\lambda\in\mathbb{GT}_{m}$ and $m\in\mathbb{Z}_{\geq 0}$. ∎

##  Appendix A Basics on $\boldsymbol{q}$-analysis

A good reference for the material on $q$-analysis is [2, Chapter 10]. Assume
$|q|<1$ is an arbitrary complex number. Most statements work if $q$ is an
indeterminate too.

The $q$-numbers and the $q$-factorial are defined by

$\displaystyle[n]_{q}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\frac{1-q^{n}}{1-q},\qquad n\in\mathbb{N},$
$\displaystyle[n]_{q}!\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}[n]_{q}\cdots[2]_{q}[1]_{q},\qquad
n\in\mathbb{N},\qquad[0]_{q}!\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}1.$

It is evident that $[n]_{q}\rightarrow n$ and $[n]_{q}!\rightarrow n!$, as
$q\rightarrow 1$, for any $n\in\mathbb{Z}_{\geq 0}$. Observe that we can also
define $[x]_{q}$ for any $x\in\mathbb{C}$, as before, and it also holds that
$[x]_{q}\rightarrow x$ as $q\rightarrow 1$. The $q$-Gamma function is defined
by

$\displaystyle\Gamma_{q}(z)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}(1-q)^{1-z}\frac{(q;q)_{\infty}}{(q^{z};q)_{\infty}}.$

From the definition, the $q$-functional equation

$\displaystyle\Gamma_{q}(z+1)=[z]_{q}\Gamma_{q}(z),\qquad
z\notin\\{\dots,-2,-1,0\\},$

is evident. The $q$-Gamma function is a meromorphic function with simple poles
at $z=0,-1,-2,\dots$ and all their shifts by an integral multiple of
$2\pi\sqrt{-1}/\ln{q}$. The $q$-Gamma function has no zeroes in $\mathbb{C}$.
Moreover, we have the following convergence to the Gamma function.

######  Theorem A.1 ([2, Corollary 10.3.4]).

For any $z\in\mathbb{C}\setminus\\{\dots,-2,-1,0\\}$, we have

$\displaystyle\lim_{q\rightarrow 1}{\Gamma_{q}(z)}=\Gamma(z).$

######  Remark A.2.

As a consequence of Stieltjes–Vitali theorem, the convergence
$\lim\limits_{q\rightarrow 1}{\Gamma_{q}(z)}=\Gamma(z)$ holds uniformly on
compact subsets of $\mathbb{C}\setminus\\{\dots,-2,-1,0\\}$.

Other important identities we use in our paper are the $q$-binomial theorems.
To state them, we need to define the $q$-Pochhammer symbols $(x;q)_{n}$ and
$(x;q)_{\infty}$, for any $x\in\mathbb{C}$ and $n\in\mathbb{Z}_{\geq 0}$ by

$\displaystyle(x;q)_{n}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\begin{cases}1&\textrm{if }n=0,\\\
\displaystyle\prod\limits_{i=1}^{n}{\big{(}1-xq^{i-1}\big{)}}&\textrm{if
}n\geq 1,\\\
\displaystyle\prod\limits_{i=1}^{\infty}{\big{(}1-xq^{i-1}\big{)}}&\textrm{if
}n=\infty.\end{cases}$

Note that $|q|<1$ implies that the product defining $(x;q)_{\infty}$ is
uniformly convergent for $x\in\mathbb{C}$, and thus $(x;q)_{\infty}$ is an
entire function.

The $q$-binomial formula is the following

######  Theorem A.3 ([2, Theorem 10.2.1]).

For $|z|<1$,

$\displaystyle\sum_{n=0}^{\infty}{\frac{(a;q)_{n}}{(q;q)_{n}}z^{n}}=\frac{(az;q)_{\infty}}{(z;q)_{\infty}}$

######  Corollary A.4.

For $z\in\mathbb{C}$, $m\in\mathbb{N}$,

$\displaystyle\sum_{n=0}^{M}{\frac{(q^{-1};q^{-1})_{M}}{(q^{-1};q^{-1})_{n}(q^{-1};q^{-1})_{M-n}}(-1)^{n}q^{-{n\choose
2}}z^{n}}=\big{(}z;q^{-1}\big{)}_{M}.$

###### Proof.

Let $a=q^{-M}$ in Theorem A.3, and use $(q;q)_{n}=(-1)^{n}q^{{n+1\choose
2}}(q^{-1};q^{-1})_{n}$. The statement is then proved for any $|z|<1$;
therefore it also holds for any $z\in\mathbb{C}$ because both sides are
polynomials on $z$. ∎

Another application of the $q$-binomial theorem is the following limit.

######  Theorem A.5 ([2, Theorem 10.2.4]).

For any $a,b\in\mathbb{R}$ such that $b-a\notin\mathbb{Z}$, the following
limit

$\displaystyle\lim_{q\rightarrow
1}{\frac{(xq^{a};q)_{\infty}}{(xq^{b};q)_{\infty}}}=(1-x)^{b-a}$

holds uniformly on compact subsets of $\\{x\in\mathbb{C}\colon|x|\leq 1,x\neq
1\\}$.

######  Remark A.6.

If $b-a\in\mathbb{Z}$, the limit in Theorem A.5 holds uniformly on compact
subsets of $\mathbb{C}\setminus\\{1\\}$.

##  Appendix B Some properties of the rational functions
$\boldsymbol{C^{(q,t)}_{\tau_{1},\dots,\tau_{n}}(u_{1},\dots,u_{n})}$

######  Lemma B.1.

Assume $t=q^{\theta}$, for some $\theta\in\mathbb{N}$. Let
$\tau_{1},\dots,\tau_{n}\in\mathbb{Z}_{\geq 0}$ and $u_{1},\dots,u_{n}$ be $n$
variables, then $C^{(q,t)}_{\tau_{1},\dots,\tau_{n}}(u_{1},\dots,u_{n})=0$ if
some of the integers $\tau_{1},\dots,\tau_{n}$ is strictly larger than
$\theta$.

###### Proof.

Let us first rewrite the expression for
$C^{(q,t)}_{\tau_{1},\dots,\tau_{n}}(u_{1},\dots,u_{n})$, as it was done in
[30, Section 6]. Let $v_{i}=q^{\tau_{i}}u_{i}$ for $i=1,\dots,n$. Also, given
$\tau=(\tau_{1},\dots,\tau_{n})$, let
$T=T_{\tau}\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\\{k\in\\{1,2,\dots,n\\}\colon\tau_{k}\neq 0\\}$. Then

$\displaystyle C^{(q,t)}_{\tau}(u_{1},\dots,u_{n})=\prod_{k\in
T}{t^{\tau_{k}-1}\frac{(q/t;q)_{\tau_{k}-1}}{(q;q)_{\tau_{k}-1}}\frac{(qu_{k};q)_{\tau_{k}}}{(qtu_{k};q)_{\tau_{k}}}}$
$\displaystyle\hphantom{C^{(q,t)}_{\tau}(u_{1},\dots,u_{n})=}{}\times\prod_{1\leq
i<j\leq
n}{\frac{(qu_{i}/tu_{j};q)_{\tau_{i}}}{(qu_{i}/u_{j};q)_{\tau_{i}}}\frac{(tu_{i}/v_{j};q)_{\tau_{i}}}{(u_{i}/v_{j};q)_{\tau_{i}}}}F_{\tau}(u;q,t),$
(B.1)

where

$\displaystyle
F_{\tau}(u;q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}\sum_{K\subset T}(-1)^{|K|}(1/t)^{{|K|\choose 2}}\prod_{j\in
T-K}{\frac{t-q^{\tau_{j}}}{1-q^{\tau_{j}}}}$
$\displaystyle\hphantom{F_{\tau}(u;q,t)\mathrel{\stackrel{{\scriptstyle\makebox[0.0pt]{\mbox{\tiny
def}}}}{{=}}}}{}\times\prod_{\begin{subarray}{c}k\in K\\\ j\in
T-K\end{subarray}}{\frac{v_{j}-v_{k}/t}{v_{j}-v_{k}}}\prod_{k\in
K}{\left(\frac{1-tv_{k}}{1-v_{k}}\prod_{\begin{subarray}{c}i\in T\\\ i\neq
k\end{subarray}}{\frac{u_{i}-v_{k}}{u_{i}-v_{k}/t}}\right)}.$

Due to the factor $\prod\limits_{k\in T}{(q/t;q)_{\tau_{k}-1}}$ in (B.1), it
follows that $\tau_{k}-\theta\in\mathbb{N}=\\{1,2,\dots\\}$, for some $k$,
implies $C^{(q,t)}_{\tau_{1},\dots,\tau_{n}}(u_{1},\dots,u_{n})=0$, as
desired. ∎

######  Lemma B.2 ([30, Lemma 6.1]).

Assume $t=q$. Let $\tau_{1},\dots,\tau_{n}\in\mathbb{Z}_{\geq 0}$ and
$u_{1},\dots,u_{n}$ be $n$ variables, then
$C^{(q,t)}_{\tau_{1},\dots,\tau_{n}}(u_{1},\dots,u_{n})=0$ if some of the
integers $\tau_{1},\dots,\tau_{n}$ is strictly larger than $1$. If all
$\tau_{1},\dots,\tau_{n}\in\\{0,1\\}$, then
$C^{(q,t)}_{\tau_{1},\dots,\tau_{n}}(u_{1},\dots,u_{n})$ does not depend on
the variables $u_{1},\dots,u_{n}$ and

$\displaystyle
C_{\tau_{1},\dots,\tau_{n}}^{(q,t)}=(-1)^{\tau_{1}+\dots+\tau_{n}}.$

######  Lemma B.3.

Assume $\theta\in\mathbb{N}$. If we let
$a_{n}^{(\theta)}=\frac{C_{n}^{(q,q^{\theta})}(x_{2}^{-1}x_{1}q^{-\theta})}{\prod\limits_{i=0}^{\theta-1}{(x_{1}-q^{i}x_{2})}}$,
for all $0\leq n\leq\theta$, then these expressions satisfy the relations

$\displaystyle
a_{n}^{(\theta)}=\frac{1}{x_{1}-x_{2}}\big{(}T_{q,x_{2}}a_{n}^{(\theta-1)}-T_{q,x_{1}}a_{n-1}^{(\theta-1)}\big{)},\qquad
1\leq n\leq\theta-1,$ $\displaystyle
a_{0}^{(\theta)}=\frac{1}{\prod\limits_{i=0}^{\theta-1}{(x_{1}-q^{i}x_{2})}},\qquad
a_{\theta}^{(\theta)}=\frac{1}{\prod\limits_{i=0}^{\theta-1}{(x_{2}-q^{i}x_{1})}}.$

###### Proof.

The expression $C_{n}^{(q,t)}(u)$, for $n\in\mathbb{Z}_{\geq 0}$, is much
simpler than the general expression (2.5). It was first found by Jing and
Joźefiak in [26] and it reads
$C_{n}^{(q,t)}(u)=t^{n}\frac{(1/t;q)_{n}}{(q;q)_{n}}\frac{(u;q)_{n}}{(qtu;q)_{n}}\frac{1-q^{2n}u}{1-u}$.
Then, for $0\leq n\leq\theta$:

$\displaystyle
C_{n}^{(q,q^{\theta})}\big{(}x_{1}/\big{(}q^{\theta}x_{2}\big{)}\big{)}=\frac{q^{\theta}x_{2}-q^{2n}x_{1}}{q^{\theta}x_{2}-x_{1}}\frac{1}{t^{n}}\prod_{i=1}^{n}{\left\\{\frac{q^{\theta}-q^{i-1}}{1-q^{i}}\frac{q^{\theta}x_{2}-q^{i-1}x_{1}}{x_{2}-q^{i}x_{1}}\right\\}}.$
(B.2)

From (B.2), it is only a matter of tedious computation to check the three
identities given in the lemma. ∎

### Acknowledgements

It is my pleasure to thank Alexei Borodin for his generous sharing of time and
ideas. I am equally indebted to Vadim Gorin, for his interest in my work, many
helpful discussions and for sharing some of his notes on the $q$-GT graph.
This work would not exist without them. I would also like to thank Jiaoyang
Huang, for being an excellent sounding board at the beginning stage of this
project, Konstantin Matveev for his expert help with the software Mathematica,
and Grigori Olshanski for comments in a previous draft of this paper and for
asking a question that led to my proof of Theorem 1.3. The suggestions of the
referees helped improved this text greatly; many thanks are due to them.

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