entirely lie in the frozen region, as well as the horizontal lines
$[\eta\,a_{0},\eta\,a_{m}]\times\\{0\\}$ and
$[\eta\,a_{0},\eta\,a_{m}]\times\\{1\\}$ (recall the discussion above
Proposition 25).

####  3.3.1. The shape of the liquid region

We conclude this section proving Proposition 19 from the introduction, and
giving an alternative description of the liquid region (see Proposition 27
below). These results have all rather standard proofs, which we include for
the sake of completeness.

###### Proof of Proposition 19.

Recall that $U_{c}(x,t)$ is the unique solution with positive imaginary part
of the critical equation $\frac{\partial S}{\partial U}(U)=0$, where $S(U)$ is
as in (3.2). Setting999This substitution is not strictly needed here but makes
the computations nicer and also useful for the proof of Proposition 27.
$\widetilde{U}_{c}(x,t):=U_{c}(x,t)+x$, we get that $\widetilde{U}_{c}(x,t)$
solves the equation

$\log(\widetilde{U}_{c}-x)-\log(1-t)-\sum_{i=0}^{m}\log(\widetilde{U}_{c}-\eta\,a_{i})+\sum_{i=1}^{m}\log(\widetilde{U}_{c}-\eta\,b_{i})=0.$
(54)

Differentiating in $x$ and $t$ the above equation, we obtain that

$\begin{cases}\frac{-1}{\widetilde{U}_{c}-x}=(\widetilde{U}_{c})_{x}\cdot\big{(}\Sigma(\widetilde{U}_{c})-\tfrac{1}{\widetilde{U}_{c}-x}\big{)},\\\
\frac{1}{1-t}=(\widetilde{U}_{c})_{t}\cdot\big{(}\Sigma(\widetilde{U}_{c})-\tfrac{1}{\widetilde{U}_{c}-x}\big{)}.\end{cases}$

where
$\Sigma(s)=\sum_{i=0}^{m}\frac{1}{s-\eta\,a_{i}}-\sum_{i=1}^{m}\frac{1}{s-\eta\,b_{i}}$.
The quantity $\Sigma(\widetilde{U}_{c})-\tfrac{1}{\widetilde{U}_{c}-x}$
corresponds to $\frac{\partial^{2}S}{\partial U^{2}}$, and is thus nonzero in
the liquid region; otherwise $U_{c}$ would be a double root of the critical
equation, which is impossible since the latter has at most 2 non-real
conjugate solutions, counted with multiplicities (Lemma 24). Consequently,
$\widetilde{U}_{c}(x,t)$ solves the equation

$-\frac{(\widetilde{U}_{c})_{t}}{\widetilde{U}_{c}-x}=\frac{(\widetilde{U}_{c})_{x}}{1-t}.$

Substituting $U_{c}(x,t)=\widetilde{U}_{c}(x,t)-x$ in the equation above, we
get (23). ∎

######  Proposition 27.

We have the following equivalent description of the liquid region $L$:

$L=\left\\{(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]\,\middle|\,\operatorname{Disc}_{U}(P_{x,t})<0\right\\},$

where $\operatorname{Disc}_{U}(P_{x,t})$ denotes the discriminant101010Since
$L$ is defined in terms of the sign of the discriminant, let us recall the
standard convention of normalisation of the discriminant:
$\operatorname{Disc}_{U}(P_{x,t})=(-1)^{\frac{m(m+1)}{2}}t^{-1}\,\mathrm{Res}_{U}(P_{x,t},P_{x,t}^{\prime})$,
where $\mathrm{Res}$ is the resultant. of the polynomial $P_{x,t}(U)$
appearing in the critical equation (8). As a consequence, $L$ is an open
subset of $[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$ and the boundary of $L$,
called _frozen boundary curve_ , is given by

$\partial
L=\left\\{(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]\,\middle|\,\operatorname{Disc}_{U}(P_{x,t})=0\right\\}.$
(55)

Moreover, the frozen boundary curve $\partial L$ can be parametrized by
$-\infty<s<\infty$, as follows:

$\begin{cases}x(s)=s-\frac{1}{\Sigma(s)},\\\
t(s)=1-\frac{G(s)}{\Sigma(s)},\end{cases}$

where
$\Sigma(s):=\sum_{i=0}^{m}\frac{1}{s-\eta\,a_{i}}-\sum_{i=1}^{m}\frac{1}{s-\eta\,b_{i}}$
and
$G(s):=\frac{\prod_{i=1}^{m}(s-\eta\,b_{i})}{\prod_{i=0}^{m}(s-\eta\,a_{i})}$.
The tangent vector $(\dot{x}(s),\dot{t}(s))$ to the frozen boundary curve is
parametrized by:

$\begin{cases}\dot{x}(s)=1+\frac{\dot{\Sigma}(s)}{\Sigma(s)^{2}},\\\
\dot{t}(s)=G(s)\left(1+\frac{\dot{\Sigma}(s)}{\Sigma(s)^{2}}\right).\end{cases}$

In particular, it has slope $\frac{\dot{t}(s)}{\dot{x}(s)}=G(s)$.

###### Proof of Proposition 27.

Recall that for a polynomial with real coefficients, its discriminant is
positive (resp. zero) if and only if the number of non-real roots is a
multiple of 4 (resp. the polynomial has a multiple root). Since Lemma 24 shows
that at least $m-1$ roots of the critical equation (8) are simple and real, we
conclude that the existence of two non-real solutions for the critical
equation (8) is equivalent to $\operatorname{Disc}_{U}(P_{x,t})<0$, where we
recall that $\operatorname{Disc}_{U}(P_{x,t})$ denotes the discriminant of the
polynomial $P_{x,t}(U)$ appearing in the critical equation (8).
$\operatorname{Disc}_{U}(P_{x,t})$ is itself a polynomial in $(x,t)$, which
implies that the liquid region $L$ is an open subset of
$[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$ and its boundary is described by (55).

We now turn to the proof of the claimed parametrization for the frozen
boundary curve. Note that, if a point
$(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$ approaches the frozen boundary
curve $\partial L$, the solution $U_{c}(x,t)$ of the critical equation becomes
real and merges with $\overline{U_{c}(x,t)}$. Equivalently, the frozen
boundary can be characterized as the set of points
$(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$ such that the action $S$ in
(3.2) has a double critical point. For computational reasons, it is convenient
to take $\widetilde{U}_{c}(x,t)=U_{c}(x,t)+x$ as parameter to describe
$\partial L$, and express $x$ and $t$ in terms of $\widetilde{U}_{c}(x,t)$.
Hence, we need to impose that $\widetilde{U}_{c}(x,t)$ solves the equation
$\frac{\partial\widetilde{S}}{\partial U}(U)=0$ (written in (54)) and the
equation $\frac{\partial^{2}\widetilde{S}}{\partial U^{2}}(U)=0$. We obtain:

$\begin{cases}\frac{\widetilde{U}_{c}-x}{1-t}=(G(\widetilde{U}_{c}))^{-1},\\\
\frac{1}{\widetilde{U}_{c}-x}=\Sigma(\widetilde{U}_{c}).\end{cases}$

Solving the above linear system for $x$ and $t$, we get the parametrization
claimed in the proposition statement (with $s=\widetilde{U}_{c}$). The claims
for the tangent vector follow from standard computations, noting that
$\dot{G}(s)=-\Sigma(s)\cdot G(s)$. ∎

###  3.4. The imaginary and real part of the action on the real line

The imaginary and real part of the action $S$ (introduced in (3.2)) on the
real line play a key-role in our analysis in the next sections. Hence we
analyze them here.

In what follows, we keep working with the principal determination of the
logarithm, defined on $\mathbb{C}\setminus\\{0\\}$ and continuous on
$\mathbb{C}\setminus\mathbb{R}_{-}$; with the convention that, for negative
real $x$, we have $\log(x)=\log(-x)+\mathrm{i}\pi$.

For $u\in\mathbb{R}$, we have

$\mathfrak{I}S(u)=-\pi
u^{-}+\sum_{i=0}^{m}\pi(u+x_{0}-\eta\,a_{i})^{-}-\sum_{i=1}^{m}\pi(u+x_{0}-\eta\,b_{i})^{-},$
(56)

where $x^{-}=\max(0,-x)$. In particular, $\mathfrak{I}S(u)$ is piecewise
affine, has value $-\pi x_{0}$ for $u<\eta\,a_{0}-x_{0}$ (recall from (2) that
$\sum_{i=0}^{m}a_{i}-\sum_{i=1}^{m}b_{i}=0$), has slope alternatively $+\pi$
and $0$ for $u<0$, then alternatively $0$ or $-\pi$ for $u>0$, and finally
takes value $0$ for $u>\eta\,a_{m}-x_{0}$. See Fig. 11 for an example.

On the other hand, for $u\in\mathbb{R}$, we have

$\mathfrak{R}S(u)=\widetilde{g}(u)-u\cdot\ln(|1-t_{0}|)-\sum_{i=0}^{m}\widetilde{g}(x_{0}-\eta\,a_{i}+u)+\sum_{i=1}^{m}\widetilde{g}(x_{0}-\eta\,b_{i}+u),$
(57)

where $\widetilde{g}(u)=u\cdot\ln(|u|)$ and we use the convention that
$\widetilde{g}(0)=0$. See again Fig. 11 for an example. In particular, the map
$u\to\mathfrak{R}S(u)$ is well-defined and continuous on the real line.
Moreover, since $\widetilde{g}^{\prime}(u)=\ln(|u|)+1$, the map
$u\to\mathfrak{R}S(u)$ is differentiable as a function
$\mathbb{R}\to\mathbb{R}$ except at the points $\eta\,a_{i}-x_{0}$ (where it
has a positive infinite slope) and at the points $0$ and $\eta\,b_{i}-x_{0}$
(where it has a negative infinite slope). Its derivative vanishes exactly when
$u$ satisfies

$|u|\prod_{i=1}^{m}|u+x_{0}-\eta\,b_{i}|=(1-t_{0})\prod_{i=0}^{m}|u+x_{0}-\eta\,a_{i}|,$

i.e. when $u$ satisfies the critical equation (8) or the companion equation

$u\prod_{i=1}^{m}(u+x_{0}-\eta\,b_{i})=-(1-t_{0})\prod_{i=0}^{m}(u+x_{0}-\eta\,a_{i}).$
(58)

Figure 11. Top: Two plots of the function $\mathfrak{I}S(u)$ in (56) in the
specific example of Fig. 10. On the left we set $x_{0}=-0.9$, while on the
right $x_{0}=1$. Bottom: Two plots of the function $\mathfrak{R}S(u)$ in (57)
in the same specific example of Fig. 10. Again, on the left we set
$x_{0}=-0.9$ and $t_{0}=0.5$, while on the right $x_{0}=1$ and $t_{0}=0.5$.

##  4\. Asymptotics for the kernel in the liquid region

Within this section, we assume that $(x_{0},t_{0})$ lies inside the liquid
region, i.e. that the critical equation (8) has two non-real solutions,
denoted by $U_{c}$ and $\overline{U_{c}}$ and with the convention that
$\mathfrak{I}U_{c}>0$.

###  4.1. Landscape of the action

As a preparation for the saddle point analysis, we also need to understand to
some extent the real part $\mathfrak{R}S(U)$ of the action $S$ introduced in
Section 3.2. In particular, we are interested in the shape of the region

$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}:=\Set{U\in\mathbb{C}}{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})}.$

We will use similar notation for various regions below. We invite the reader
to compare the following discussion with the pictures in Fig. 13.

Since $S$ is analytic around $U_{c}$ (and more generally on
$\mathbb{C}\setminus(-\infty,\eta\,a_{m}-x_{0})$) and $U_{c}$ is a simple
critical point of $S$, we have that, locally around $U_{c}$

$S(U)=S(U_{c})+\tfrac{S^{\prime\prime}(U_{c})}{2}\,(U-U_{c})^{2}+O((U-U_{c})^{3}).$

In particular,111111Most of the claims in this paragraph follows by analogy
with the landscape of the complex function $F(U)=U^{2}$ around $0$. For
instance, the four curves corresponding to the level sets
$\\{\mathfrak{R}S(U)=\mathfrak{R}S(U_{c})\\}$ are the analogue of the four
curves corresponding to the level sets
$\\{\mathfrak{R}F(U)=\mathfrak{R}F(0)\\}=\\{U=x+iy\in\mathbb{C}:(x+y)(x-y)=0\\}$.
there are _eight special curves_ leaving $U_{c}$ (see for instance the left-
hand side of Fig. 13): four curves corresponding to the level sets
$\\{\mathfrak{R}S(U)=\mathfrak{R}S(U_{c})\\}$ and four other curves
corresponding to the level sets $\\{\mathfrak{I}S(U)=\mathfrak{I}S(U_{c})\\}$.
We will refer to these curves as _real_ or _imaginary level lines_ ,
respectively. The four real level lines split the neighborhood of $U_{c}$ into
four regions, belonging alternatively to the set
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$ or
$\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$, each of this region containing
one imaginary level line.

Since $S$ is analytic with a non-zero derivative on $\set{V}{V\neq
U_{c},\,\mathfrak{I}(V)>0}$, locally around such $V$ with
$\mathfrak{R}S(V)=\mathfrak{R}S(U_{c})$ (resp. with
$\mathfrak{I}S(V)=\mathfrak{I}S(U_{c})$), the level set
$\\{\mathfrak{R}S(U)=\mathfrak{R}S(U_{c})\\}$ (resp. the level set
$\\{\mathfrak{I}S(U)=\mathfrak{I}S(U_{c})\\}$) looks like a single simple
curve. Therefore the real and imaginary level lines leaving $U_{c}$ go either
to the real line, or to infinity. Moreover, these lines cannot cross in the
closed upper half-plane. Indeed, following a real level line, the analyticity
of $S$ implies that $U\mapsto\mathfrak{I}S(U)$ is strictly monotone: a local
extremum would violate the open mapping theorem. Therefore starting from
$U_{c}$ and following a real level line in the set
$\\{\mathfrak{R}S(u)=\mathfrak{R}S(U_{c})\\}$, we never reach a point $V$ with
$\mathfrak{I}S(V)=\mathfrak{I}S(U_{c})$.

######  Lemma 28.

Exactly three of the real level lines leaving $U_{c}$ go to the real line.

###### Proof.

Since $S(U)\sim|\log(1-t_{0})|\,U$ as $|U|$ tends to infinity (as already
remarked in (48)), we have that $\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})$ when
$\mathfrak{R}U$ goes to $-\infty$ with a fixed imaginary part. Likewise,
$\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})$ when $\mathfrak{R}U$ goes to $+\infty$
with a fixed imaginary part. Therefore, there is an odd number of real level
lines going to the real line.

Assume that there is only one. Then three real level lines are going to
infinity. Because of the asymptotics $S(U)\sim|\log(1-t_{0})|\,U$, real level
lines going to infinity should be asymptotically inside the region
$\Set{y\geq}{x|}:=\Set{x+iy}{y\geq|x|}$ (otherwise we would have
$\lim_{|U|\to+\infty}\mathfrak{R}S(U)=+\infty$ and this would be in
contradiction with the definition of real level line). These three real lines
determine two unbounded regions, each of them containing an imaginary level
line. In particular, these two imaginary level lines go to infinity, inside
the region $\Set{y\geq}{x|}$. But inside this region, we have that
$\lim_{|U|\to+\infty}\mathfrak{I}S(U)=+\infty$ since
$S(U)\sim|\log(1-t_{0})|\,U$. This is in contradiction with the definition of
imaginary level line. We, therefore, conclude that there are exactly three of
the real level lines leaving $U_{c}$ and going to the real line. ∎

We call $A\leq B\leq C$ the intersection points of these three real level
lines and the real axis. Also, let $D$ and $E$ be the intersection points of
the imaginary level lines in between these three level lines and the real
axis. Since real and imaginary level lines do not intersect, we have
$A<D<B<E<C$.

######  Lemma 29.

With the above notation, $D<0<E$. Moreover there is no real $x$ in
$(-\infty,A)\cup(C,+\infty)$ such that
$\mathfrak{I}S(x)=\mathfrak{I}S(U_{c})$.

###### Proof.

Since $D$ and $E$ belong to an imaginary level line leaving $U_{c}$, and since
$\mathfrak{I}S(u)$ is continuous on the closed upper half-plane, we have
$\mathfrak{I}S(D)=\mathfrak{I}S(E)=\mathfrak{I}S(U_{c})$. Furthermore,
$\mathfrak{I}S(U_{c})$ is distinct from the values $\mathfrak{I}S(A)$,
$\mathfrak{I}S(B)$, $\mathfrak{I}S(C)$ since, as explained above,
$\mathfrak{I}S$ is strictly monotone on the real level lines going from
$U_{c}$ to $A$, $B$ or $C$. Looking at the graph of the function
$u\mapsto\mathfrak{I}S(u)$ on the real line (see the computations in Section
3.4), we see that a necessary condition to have
$\mathfrak{I}S(D)=\mathfrak{I}S(E)\neq\mathfrak{I}S(B)$ is that $D<0<E$.

Moreover, looking again at the shape of $\mathfrak{I}S(U)$ and recalling that

$\mathfrak{I}S(A)\neq\mathfrak{I}S(D)=\mathfrak{I}S(U_{c})=\mathfrak{I}S(E)\neq\mathfrak{I}S(C),$

we see that there cannot be an $x$ in either $(-\infty,A)$ or $(C,+\infty)$
such that $\mathfrak{I}S(x)=\mathfrak{I}S(U_{c})$. ∎

As said above, we are interested in describing the regions
$\\{\mathfrak{R}S(u)>\mathfrak{R}S(U_{c})\\}$ or
$\\{\mathfrak{R}S(u)<\mathfrak{R}S(U_{c})\\}$. From the asymptotics
$S(U)\sim|\log(1-t_{0})|\,U$, we know that for $\mathfrak{R}(U)$ negative
(resp. positive) and large in absolute value (in particular larger than
$K|\mathfrak{I}(U)|$ for come constant $K>0$), $U$ belongs to the region
$\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$ (resp.
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$). Thus, the real level lines
leaving $U_{c}$ split the complex plane as shown on the left-hand side of Fig.
13.

Note that we do not exclude the existence of smaller islands inside the main
regions as shown on the right-hand side of Fig. 13. For the sake of
simplicity, we will not draw such potential islands in future figures.

###  4.2. Moving contours and asymptotic of the kernel

We recall from (41) that the renormalized kernel
$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}$ writes as a double contour
integral over contours $\gamma_{Z}$ and $\gamma_{W}$ (where the integrand is
denoted by $\operatorname{Int}_{N}(W,Z)$).

Our strategy for the asymptotic analysis of
$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}$ goes as follows:

  1. (1)

Move the integration contours from $\gamma_{Z}$ and $\gamma_{W}$ to properly
chosen contours $\gamma_{Z}^{\text{new}}$ and $\gamma_{W}^{\text{new}}$ given
in Lemma 30;

  2. (2)

Replace the integrand $\operatorname{Int}_{N}(W,Z)$ by the asymptotically
equivalent expression given in (46):

$(\sqrt{N})^{x_{2}-x_{1}}\mathrm{e}^{\sqrt{N}(S(W)-S(Z))}\,h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z).$

This second step is performed in Proposition 31.

The contours $\gamma_{Z}^{\text{new}}$ and $\gamma_{W}^{\text{new}}$ will be
constructed in such a way that, for almost all
$(W,Z)\in\gamma_{W}^{\text{new}}\times\gamma_{Z}^{\text{new}}$, one has
$S(W)<S(Z)$. In this way, after the second step, the integrand – and thus the
integral – will converge to $0$. The asymptotic expansion of the kernel
$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}$ will then be given by the
potential residue term created by the change of contours.

Recalling the discussion below (41), the integrand has poles for $W=Z$, for
some values of $W$ in the interval $I_{W}=[\eta\,a_{0}-x_{0},o(1)]$ and for
some values of $Z$ in the interval $I_{Z}=[o(1),\eta\,a_{m}-x_{0}-1]$. We also
recall from (42) that we chose the contours $\gamma_{W}=\partial D(0,3L)$ and
$\gamma_{Z}=\partial D(0,4L)$ (both followed in counterclockwise order), with
$L=\eta\max(|a_{0}|,a_{m})$. (Recall that we restricted our analysis to the
case $t_{1}\geq t_{2}$, i.e. when $\gamma_{W}$ is inside $\gamma_{Z}$.)

######  Lemma 30.

There exist two integration contours $\gamma_{W}^{\text{new}}$ and
$\gamma_{Z}^{\text{new}}$ (both followed in counterclockwise direction) such
that

  * •

$\gamma_{W}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$ intersect each other
only at $U_{c}$ and $\overline{U_{c}}$;

  * •

$\gamma_{W}^{\text{new}}$ (resp. $\gamma_{Z}^{\text{new}}$) contains $I_{W}$
(resp. $I_{Z}$) in its interior;

  * •

$\gamma_{W}^{\text{new}}\setminus\\{U_{c},\overline{U_{c}}\\}$ (resp.
$\gamma_{Z}^{\text{new}}\setminus\\{U_{c},\overline{U_{c}}\\}$) lies inside
the region $\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$ (resp.
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$).

We refer the reader to Fig. 13 for an illustration of the original and new
integration contours.

Figure 12. Two possible shapes for the landscape of $\mathfrak{R}S$. The black
fat lines are the real level lines
$\\{\mathfrak{R}S(U)=\mathfrak{R}S(U_{c})\\}$. The yellow regions correspond
to $\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$, while the white regions
correspond to $\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$. On the left-hand
side, we have also represented the imaginary level lines
$\\{\mathfrak{I}S(U)=\mathfrak{I}S(U_{c})\\}$ in dotted lines. The right-hand
side shows another possible landscape of $\mathfrak{R}S$ with a more
complicated configuration (to avoid overloading the picture, we did not draw
imaginary level lines here). For the definition of the points $A,B,C,D,E$, see
the discussion above Lemma 29.

Figure 13.  Left: The picture from Fig. 13 together with the original
integration contours $\gamma_{W}$ (in green) and $\gamma_{Z}$ (in purple)
appearing in the kernel $\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}$. The
green and purple dots are respectively the $W$-poles and $Z$-poles of the
integrand. We also recall that the yellow regions correspond to
$\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$, while the white regions
correspond to $\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$. Right: The same
picture with the new contours $\gamma_{Z}^{\text{new}}$ and
$\gamma_{W}^{\text{new}}$ from Lemma 30.

###### Proof.

We first explain how to construct $\gamma_{Z}^{\text{new}}$, as the
concatenation of two paths going from $U_{c}$ to $\overline{U_{c}}$ (reversing
the right-most one so that we have a counterclockwise contour).

Recall from the previous section (see also Fig. 13) that there is a portion of
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$ between the point $U_{c}$,
$\overline{U_{c}}$, $A$ and $B$ and that this region contains an imaginary
level line of $S$ that intersects the real axis at $D<0$ (Lemma 29). We choose
this imaginary level line as the first path to construct
$\gamma_{Z}^{\text{new}}$.

Around $U_{c}$, we can find another connected area of the region
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$. This part also contains an
imaginary level line of $S$. This imaginary level line cannot go to the real
axis, otherwise the meeting point would be an $x$ in $(C,+\infty)$ with
$\mathfrak{I}S(x)=\mathfrak{I}S(U_{c})$, contradicting Lemma 29. Because of
the asymptotics $S(U)\sim|\log(1-t_{0})|\,U$, this imaginary line should go to
infinity inside the region $\\{y\leq|x|\\}$ and therefore meets the region
$\\{x\geq M\\}$, for any $M>0$. Moreover, we can choose $M>0$ such that
$M\geq\max I_{Z}$ and such that if $|x|\geq M$ and $|y|\leq x$ then
$\mathfrak{R}S(x+iy)>\mathfrak{R}S(U_{c})$. The second path used to construct
$\gamma_{Z}^{\text{new}}$ is then defined as follows: we first follow the
imaginary level line from $U_{c}$ until it meets the region $\\{|x|\geq
M,\,y\leq|x|\\}$, and then join the real axis inside this region. We complete
the path by symmetry until it hits $\overline{U_{c}}$.

By construction, $\gamma_{Z}^{\text{new}}$ goes through $U_{c}$ and
$\overline{U_{c}}$ and
$\gamma_{Z}^{\text{new}}\setminus\\{U_{c},\overline{U_{c}}\\}$ lies in
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$. Also, the intersection points
of $\gamma_{Z}^{\text{new}}$ with the real axis are $D<0$ on the one side and
some number larger than $M>\max I_{Z}$ on the other side, so that
$\gamma_{Z}^{\text{new}}$ encloses $I_{Z}$ as wanted. We construct
$\gamma_{W}^{\text{new}}$ in a symmetric way, completing the proof of the
lemma. ∎

We are now ready to compute the asymptotics of the kernel
$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}$ in (41) when $(x_{0},t_{0})$
lies inside the liquid region. and hence compute the limit of the associated
bead process $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$.

######  Proposition 31.

Assume that $(x_{0},t_{0})$ is in the liquid region. The bead process
$\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ converges in distribution to a
determinantal process with correlation kernel

$K^{(x_{0},t_{0})}_{\infty}\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)}:=\frac{1}{2\mathrm{i}\pi}\int_{\gamma}\frac{1}{1-t_{0}}\,\mathrm{e}^{\frac{W}{1-t_{0}}(t_{1}-t_{2})}\,\left(\frac{W}{1-t_{0}}\right)^{x_{2}-x_{1}}\differential{W},$
(59)

where $\gamma$ is a path going from $\overline{U_{c}}$ to $U_{c}$ and passing
on the left of $0$ for $t_{1}\geq t_{2}$ and on the right of $0$ for
$t_{1}<t_{2}$ (the point 0 is a pole of the integrand in the case
$x_{1}<x_{2}$).

###### Proof.

We first treat the case $t_{1}\geq t_{2}$, so that the original contours in
(41) are such that $\gamma_{W}$ is inside $\gamma_{Z}$. We move the latter
contours to the new contours $\gamma_{W}^{\text{new}}$ and
$\gamma_{Z}^{\text{new}}$ constructed in Lemma 30. Since $I_{W}$ (resp.
$I_{Z}$) is inside both $\gamma_{W}$ and $\gamma_{W}^{\text{new}}$ (resp.
$\gamma_{Z}$ and $\gamma_{Z}^{\text{new}}$), we do not cross any of the
$W$-poles (resp. $Z$-poles). However, since the relative positions of the
contours is not the same ($\gamma_{W}^{\text{new}}$ is not inside
$\gamma_{Z}^{\text{new}}$), we have to take care of the pole at $Z=W$.

Fix $Z\in\gamma_{Z}$. Possibly enlarging the contour $\gamma_{Z}$, we can
assume that the point $Z$ is outside both $\gamma_{W}$ and
$\gamma_{W}^{\text{new}}$. Therefore, recalling that
$\operatorname{Int}_{N}(W,Z)$ denotes the integrand in (41), we have

$\oint_{\gamma_{W}}\operatorname{Int}_{N}(W,Z)\,{\differential{W}}=\oint_{\gamma_{W}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\,{\differential{W}},$

which implies that

$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{2},t_{2}))=\frac{-1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}}\oint_{\gamma_{W}}\operatorname{Int}_{N}(W,Z)\differential{W}\differential{Z}=\frac{-1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}}\oint_{\gamma_{W}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{W}\differential{Z}.$

Fix now $W$ in $\gamma_{W}^{\text{new}}\setminus\\{U_{c},\overline{U_{c}}\\}$.
Note that
$\gamma_{W}^{\text{new}}\setminus\\{U_{c},\overline{U_{c}}\\}=\gamma_{W}^{\text{new,left}}\sqcup\gamma_{W}^{\text{new,right}}$,
where $\gamma_{W}^{\text{new,left}}$ and $\gamma_{W}^{\text{new,right}}$
denote respectively the left and right parts of the contour
$\gamma_{W}^{\text{new}}$; c.f. Fig. 13. We then move the $Z$-contour from
$\gamma_{Z}$ (left-hand side of Fig. 13) to $\gamma_{Z}^{\text{new}}$ (right-
hand side of Fig. 13). If $W$ is in $\gamma_{W}^{\text{new,right}}$, then
deforming the $Z$-contour from $\gamma_{Z}$ to $\gamma_{Z}^{\text{new}}$ can
be done without crossing any pole. On the other hand, if $W$ is in
$\gamma_{W}^{\text{new,left}}$, when deforming the $Z$-contour from
$\gamma_{Z}$ to $\gamma_{Z}^{\text{new}}$, we cross the pole $Z=W$. By the
residue theorem, for $W$ fixed in
$\gamma_{W}^{\text{new}}\setminus\\{U_{c},\overline{U_{c}}\\}$, we have

$\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{Z}}\operatorname{Int}_{N}(W,Z)\differential{Z}=\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{Z}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{Z}+\delta_{W\in\gamma_{W}^{\text{new,left}}}\,\Res\big(\operatorname{Int}_{N}(W,Z),Z=W\big{missing}),$

where $\Res\big(\operatorname{Int}_{N}(W,Z),Z=W\big{missing})$ denotes the
residue of $\operatorname{Int}_{N}(W,Z)$ corresponding to the pole $Z=W$.
Integrating over $W$ in $\gamma_{W}^{\text{new}}$ gives

$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{2},t_{2}))=-\frac{1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{W}^{\text{new}}}\oint_{\gamma_{Z}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{Z}\differential{W}\\\
-\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{W}^{\text{new,left}}}\\!\Res\big(\operatorname{Int}_{N}(W,Z),Z=W\big{missing})\differential{W},$
(60)

where the set $\gamma_{W}^{\text{new,left}}$ is interpreted as a path going
from $U_{c}$ to $\overline{U_{c}}$; note that it passes on the left of $0$.
Note that because of the pole in $Z=W$ of $\operatorname{Int}_{N}(W,Z)$, the
term
$\oint_{\gamma_{Z}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{Z}$
has a logarithmic singularity when $W$ tends to $U_{c}$ or $\overline{U_{c}}$;
this singularity is integrable so that the double integral above is well-
defined.

Recalling that
$\operatorname{Int}_{N}(W,Z)=\frac{F_{\lambda_{N}}(\sqrt{N}(Z+x_{0}))}{F_{\lambda_{N}}(\sqrt{N}(W+x_{0}))}\,\frac{\Gamma(\sqrt{N}W-x_{1}+1)}{\Gamma(\sqrt{N}Z-x_{2}+1)}\cdot\frac{(1-\widetilde{t}_{2})^{\sqrt{N}Z-x_{2}}\,(1-\widetilde{t}_{1})^{-\sqrt{N}W+x_{1}-1}}{(Z-W)}$,
the right term in the latter displayed equation can be computed as follows:

$\displaystyle\Res\big(\operatorname{Int}_{N}(W,Z),Z=W\big{missing})$
$\displaystyle=\frac{\Gamma(\sqrt{N}W-x_{1}+1)}{\Gamma(\sqrt{N}W-x_{2}+1)}\,(1-\widetilde{t}_{2})^{\sqrt{N}W-x_{2}}\,(1-\widetilde{t}_{1})^{-\sqrt{N}W+x_{1}-1}$
$\displaystyle\simeq(\sqrt{N}W)^{x_{2}-x_{1}}(1-t_{0})^{x_{1}-x_{2}-1}\exp\left(\frac{W}{1-t_{0}}\,(t_{1}-t_{2})\right),$

where we used Eq. 43 in the last estimation. The estimate is uniform for $W$
in compact subsets of $\mathbb{C}\setminus\\{0\\}$, implying

$-\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{W}^{\text{new,left}}}\\!\Res\big(\operatorname{Int}_{N}(W,Z),Z=W\big{missing})\differential{W}\\\
\simeq\,-\frac{(\sqrt{N})^{x_{2}-x_{1}}}{2\mathrm{i}\pi}\oint_{\gamma_{W}^{\text{new,left}}}\\!\frac{1}{1-t_{0}}\,\mathrm{e}^{\frac{W}{1-t_{0}}(t_{1}-t_{2})}\left(\frac{1-t_{0}}{W}\right)^{x_{1}-x_{2}}\differential{W}.$

We now consider the double integral term
$\oint_{\gamma_{W}^{\text{new}}}\oint_{\gamma_{Z}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{Z}\differential{W}$
in (60). Call $Z_{-}$ and $Z_{+}$ (resp. $W_{-}$ and $W_{+}$) the
intersections points of the contour $\gamma_{Z}^{\text{new}}$ (resp.
$\gamma_{W}^{\text{new}}$) with the real axis, with the convention
$Z_{-}<0<Z_{+}$ (resp. $W_{-}<0<W_{+}$). By construction, $W_{-}$ and $Z_{+}$
can be chosen outside $[\eta a_{0}-x_{0},\eta a_{m}-x_{0}]$ and hence belong
to the set $D_{S}$ (see its definition before Lemma 23). Moreover, up to
deforming slightly $\gamma_{Z}^{\text{new}}$ and $\gamma_{W}^{\text{new}}$
(still keeping the properties of Lemma 30 true), we may assume that $W_{+}$
and $Z_{-}$ are distinct from all $\eta\,a_{i}-x_{0}$ and $\eta\,b_{i}-x_{0}$.

Fixing $\varepsilon>0$, we have that:

  * •

Using the estimate in (46) (which holds uniformly for $Z$ and $W$ in compact
subsets of $D_{S}$), for $Z\in\gamma_{Z}^{\text{new}}\setminus
D(Z_{-},\varepsilon)$ and $W\in\gamma_{W}^{\text{new}}\setminus
D(W_{+},\varepsilon)$, the integrand $\operatorname{Int}_{N}(W,Z)$ is bounded
by the function $2h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$ in (3.2) for all $N$
large enough, and tends to $0$ pointwise when $N$ tends to $+\infty$ because
$\mathfrak{R}S(Z)>\mathfrak{R}S(W)$ by the third property in Lemma 30 for
$\gamma_{Z}^{\text{new}}$ and $\gamma_{W}^{\text{new}}$. (This is true except
when $\\{Z,W\\}\subseteq\\{U_{c},\overline{U_{c}}\\}$, where the bound by
$h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$ still holds but the integrand
$\operatorname{Int}_{N}(W,Z)$ does not converge to zero; since this is a zero
measure subset, this will not be problematic.)

  * •

Using the second and third parts of Lemma 23, the bound by
$h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$ and the convergence of
$\operatorname{Int}_{N}(W,Z)$ to $0$ also hold when $Z\in
D(Z_{-},\varepsilon)$ or $W\in D(W_{+},\varepsilon)$ (or both), for
$\varepsilon$ small enough. Indeed, the extra factor
$\exp(\varepsilon\pi\sqrt{N})$ in this case is compensated by the factor
$\exp(\sqrt{N}(S(W)-S(Z)))$ as soon as $S(W)<S(Z)-\pi\varepsilon$, which
happens for $W\in D(W_{+},\varepsilon)$ and $Z\in\gamma_{Z}^{\text{new}}$ or
for $W\in\gamma_{W}^{\text{new}}$ and $Z\in D(Z_{-},\varepsilon)$, if
$\varepsilon$ is small enough.

  * •

The function $h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$ is integrable for
$(W,Z)$ on the double contour
$(\gamma_{W}^{\text{new}},\gamma_{Z}^{\text{new}})$. Indeed, it has
singularities for $W=Z=U_{c}$ and $W=Z=\overline{U_{c}}$ but behaves as
$O((W-Z)^{-1})$ near these singularities, and a standard computation shows
that $(W-Z)^{-1}$ is integrable on
$\gamma_{W}^{\text{new}}\times\gamma_{Z}^{\text{new}}$, since the paths cross
non-tangentially.

Hence, using the dominated convergence theorem, we know that the double
contour integral
$\oint_{\gamma_{W}^{\text{new}}}\oint_{\gamma_{Z}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{Z}\differential{W}$
goes to $0$ as $N$ tends to infinity.

Letting $\gamma$ to be the reverse path of $\gamma_{W}^{\text{new,left}}$, we
get

$\lim_{N\to+\infty}\frac{\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{2},t_{2}))}{(\sqrt{N})^{x_{2}-x_{1}}}=\frac{1}{2\mathrm{i}\pi}\int_{\gamma}\\!\frac{1}{1-t_{0}}\,\mathrm{e}^{\frac{W}{1-t_{0}}(t_{1}-t_{2})}\left(\frac{1-t_{0}}{W}\right)^{x_{1}-x_{2}}\differential{W}.$
(61)

Note that $\gamma$ is indeed a path form $U_{c}$ to $\overline{U_{c}}$ passing
on the left of $0$, as required in the case $t_{1}\geq t_{2}$.

Let us now consider the case $t_{1}<t_{2}$. We recall that in this case the
initial contour $\gamma_{W}$ and $\gamma_{Z}$ are swapped. An argument similar
to the one above shows that (61) still holds, but with the path
$\gamma=\gamma_{W}^{\text{new,right}}$ passing on the right of $0$.

Recall that, up to the factor $(\sqrt{N})^{x_{2}-x_{1}}$, the left-hand side
of (61) is the kernel of the bead process $\widetilde{M}_{\lambda_{N}}$. But
the factor $(\sqrt{N})^{x_{2}-x_{1}}$ is a conjugation factor (see the last
paragraph in Section 2.2): adding or removing it does not change the
associated bead process.

The expression on the right-hand side of (61) depends continuously on
$(x_{1},t_{1})$ and $(x_{2},t_{2})$ and is therefore locally bounded.
Moreover, all estimates above and in particular the convergence (61) are
locally uniform in these variables. Therefore, from Proposition 22, the bead
process $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ converges in
distribution to a determinantal point process with kernel
$K^{(x_{0},t_{0})}_{\infty}$. ∎

###  4.3. Recovering the bead kernel

We now want to compare the limit kernel $K^{(x_{0},t_{0})}_{\infty}$ from
Proposition 31 with that of the random infinite bead process
$J_{\alpha,\beta}$ from Theorem 12. Recall from (9) that, given the critical
point $U_{c}$ associated to $(x_{0},t_{0})$ in the liquid region,
$\alpha=\frac{\mathfrak{I}U_{c}}{1-t_{0}}$ and
$\beta=\frac{\mathfrak{R}U_{c}}{|U_{c}|}$.

######  Lemma 32.

There exists a function $g$ such that, for $t_{1}\neq t_{2}$

$K^{(x_{0},t_{0})}_{\infty}\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)}=\frac{g(x_{1},t_{1})}{g(x_{2},t_{2})}\,J_{\alpha,\beta}\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)}.$
(62)

Before proving the lemma, we discuss its consequences. We recall that a
conjugation factor of the form $\frac{g(x_{1},t_{1})}{g(x_{2},t_{2})}$ in a
kernel does not affect the associated point process. Also changing the kernel
on a set of measure $0$, e.g.
$\left\\{\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)},t_{1}=t_{2}\right\\}$, does
not change the point process either. Thus, the above lemma implies that
$K_{\infty}$ and $J_{\alpha,\beta}$ are the kernels of the same point process,
i.e. the random infinite bead process of intensity $\alpha$ and skewness
$\beta$ introduced in Definition 13. Together with Proposition 31, this proves
the first item of Theorem 15.

###### Proof of Lemma 32.

First assume $x_{2}\geq x_{1}$ (but we do not assume any comparison between
$t_{1}$ and $t_{2}$, the forthcoming argument works in both cases). Then the
integrand of the kernel
$K^{(x_{0},t_{0})}_{\infty}\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)}$ in (59)
has no poles, and we can replace $\gamma$ by any path from $\overline{U_{c}}$
to $U_{c}$. We will take a vertical segment, which we parametrize as

$\gamma(u):=R(\cos\theta+\mathrm{i}u\sin\theta),\qquad u\in(-1,1),$

where $R=|U_{c}|$ and $\theta=\operatorname{Arg}(U_{c})$. We get

$K_{\infty}\big{(}(x_{1},t_{1}),(x_{2},t_{2})\big{)}=\frac{1}{1-t_{0}}\,\frac{g(x_{1},t_{1})}{g(x_{2},t_{2})}\int_{-1}^{1}\mathrm{e}^{\mathrm{i}u(t_{1}-t_{2})\frac{R\sin\theta}{1-t_{0}}}(\cos\theta+\mathrm{i}u\sin\theta)^{x_{2}-x_{1}}\frac{R\sin\theta\differential{u}}{2\pi},$

with $g(y,\varepsilon)=\exp(\frac{\varepsilon
R\cos\theta}{1-t_{0}})\,(\frac{R}{1-t_{0}})^{x_{2}-x_{1}}$. Comparing with
(21) where we take
$\alpha=\frac{R\sin\theta}{1-t_{0}}=\frac{\mathfrak{I}U_{c}}{1-t_{0}}$ and
$\beta=\cos\theta=\frac{\mathfrak{R}U_{c}}{|U_{c}|}$, we obtain (62) in the
case $x_{2}\geq x_{1}$.

Assume now $x_{2}<x_{1}$ and $t_{1}>t_{2}$. In this case, in Proposition 31,
we integrate over the following path $\gamma$ going from $\overline{U_{c}}$ to
$U_{c}$ (for $A\geq 1$ large enough this path passes on the left of $0$, as
needed):

  1. (1)

take a vertical path $\gamma_{1}$ going down from $\overline{U_{c}}$ to
$\mathfrak{R}U_{c}-\mathrm{i}A$;

  2. (2)

take a horizontal path $\gamma_{2}$ going left from
$\mathfrak{R}U_{c}-\mathrm{i}A$ to $\mathfrak{R}U_{c}-(\log
A)^{2}-\mathrm{i}A$;

  3. (3)

take a vertical path $\gamma_{3}$ going up from $\mathfrak{R}U_{c}-(\log
A)^{2}-\mathrm{i}A$ to $\mathfrak{R}U_{c}-(\log A)^{2}+\mathrm{i}A$;

  4. (4)

take a horizontal path $\gamma_{4}$ going right from $\mathfrak{R}U_{c}-(\log
A)^{2}+\mathrm{i}A$ to $\mathfrak{R}U_{c}+\mathrm{i}A$;

  5. (5)

and finally take a vertical path $\gamma_{5}$ going down from
$\mathfrak{R}U_{c}+\mathrm{i}A$ to $U_{c}$.

Since $t_{1}-t_{2}>0$ and $x_{1}-x_{2}>0$, the integrals over the second and
fourth paths are bounded by

$\frac{(\log
A)^{2}}{2\pi(1-t_{0})}\,\mathrm{e}^{\frac{\mathfrak{R}U_{c}}{1-t_{0}}(t_{1}-t_{2})}\left(\frac{1-t_{0}}{A}\right)^{x_{1}-x_{2}}.$

This upper bound tends to $0$ as $A$ tends to infinity. Similarly , when
$t_{1}>t_{2}$ and $x_{1}-x_{2}>0$, the integral over the third path is bounded
by

$\frac{2A}{2\pi(1-t_{0})}\,\mathrm{e}^{\frac{\mathfrak{R}U_{c}-(\log
A)^{2}}{1-t_{0}}(t_{1}-t_{2})}\left(\frac{1-t_{0}}{A}\right)^{x_{1}-x_{2}},$

which also tends to $0$ as $A$ tends to infinity. Consider the integral on the
first path. Reversing its direction (which yields a minus sign), it can be
parametrized by

$\gamma_{1}(u)=R(\cos\theta+\mathrm{i}u\sin\theta),\quad
u\in(-A/\mathfrak{I}U_{c},-1).$

Thus, doing the same computation as in the case $x_{2}\geq x_{1}$ above, the
integral over $\gamma_{1}$ is equal to

$-\frac{1}{1-t_{0}}\,\frac{g(x_{1},t_{1})}{g(x_{2},t_{2})}\int_{-A/\mathfrak{I}U_{c}}^{-1}\mathrm{e}^{\mathrm{i}u(t_{1}-t_{2})\frac{R\sin\theta}{1-t_{0}}}(\cos\theta+\mathrm{i}u\sin\theta)^{x_{2}-x_{1}}\frac{R\sin\theta\differential{u}}{2\pi},$

As $A$ tends to infinity, this quantity tends to

$-\frac{1}{1-t_{0}}\,\frac{g(x_{1},t_{1})}{g(x_{2},t_{2})}\int_{-\infty}^{-1}\mathrm{e}^{\mathrm{i}u(t_{1}-t_{2})\frac{R\sin\theta}{1-t_{0}}}(\cos\theta+\mathrm{i}u\sin\theta)^{x_{2}-x_{1}}\frac{R\sin\theta\differential{u}}{2\pi}.$

Similarly the integral over $\gamma_{5}$ tends to

$-\frac{1}{1-t_{0}}\,\frac{g(x_{1},t_{1})}{g(x_{2},t_{2})}\int_{1}^{\infty}\mathrm{e}^{\mathrm{i}u(t_{1}-t_{2})\frac{R\sin\theta}{1-t_{0}}}(\cos\theta+\mathrm{i}u\sin\theta)^{x_{2}-x_{1}}\frac{R\sin\theta\differential{u}}{2\pi}.$

Comparing with (21), we obtain (62) in the case $x_{2}<x_{1}$ and
$t_{1}>t_{2}$. The case $x_{2}<x_{1}$ and $t_{1}<t_{2}$ is similar, taking a
path $\gamma$ going through $\mathfrak{R}U_{c}-\mathrm{i}A$,
$\mathfrak{R}U_{c}+(\log A)^{2}\pm\mathrm{i}A$ and
$\mathfrak{R}U_{c}+\mathrm{i}A$ (recall that for $t_{1}<t_{2}$, the path
$\gamma$ in Proposition 31 should pass on the right of $0$). ∎

##  5\. Asymptotics for the kernel in the frozen region

###  5.1. The small $t$ region

In this section, we fix $x_{0}\in[\eta\,a_{0},\eta\,a_{m}]$ and we let
$t_{0}\leq t_{-}$, where $t_{-}=t_{-}(x_{0})$ is given by Proposition 25. We
first assume that $x_{0}\in(\eta\,a_{0},0)$; the necessary modification for
the cases $x_{0}=\eta\,a_{0}$ and $x_{0}\geq 0$ will be explained afterwards.

From Proposition 25 and Remark 26, we know that in this regime the critical
equation (8) has only real solutions and that the smallest two, which will be
denoted by $U_{c}\leq U^{\prime}_{c}$, are in $(-\infty,\eta\,a_{0}-x_{0})$.
In Lemmas 33 and 34, we assume $U_{c}<U^{\prime}_{c}$ and discuss the case of
a double critical $U_{c}=U^{\prime}_{c}$ only in the proof of the main result
of the section, i.e. Proposition 35.

######  Lemma 33.

The critical points $U_{c}$ and $U^{\prime}_{c}$ are respectively a local
maximum and a local minimum of the function $u\to\mathfrak{R}S(u)$ on the real
line and it holds that $\mathfrak{R}S(U_{c})>\mathfrak{R}S(U^{\prime}_{c})$.

###### Proof.

Recall from Section 3.4 that the map $u\mapsto\mathfrak{R}S(u)$;

  * •

is well-defined and continuous on the real line:

  * •

is differentiable as a function from $\mathbb{R}$ to $\mathbb{R}$, except at
the points $\eta\,a_{i}-x_{0}$ (where it has a positive infinite slope) and at
the points $0$ and $\eta\,b_{i}-x_{0}$ (where it has a negative infinite
slope);

  * •

has a vanishing derivative exactly when $u$ satisfies the critical equation
(8) or the companion equation

$u\prod_{i=1}^{m}(u+x_{0}-\eta\,b_{i})=-(1-t_{0})\prod_{i=0}^{m}(u+x_{0}-\eta\,a_{i}).$
(63)

This companion equation always has exactly $m+1$ solutions, in the intervals
$[\eta\,a_{i-1}-x_{0},\eta\,b_{i}-x_{0}]$ for $i\leq i_{0}$ and
$[\eta\,b_{i}-x_{0},\eta\,a_{i}-x_{0}]$ for $i\geq i_{0}$, where $i_{0}$ is
such that $0\in(\eta\,a_{i_{0}-1}-x_{0},\eta\,a_{i_{0}}-x_{0}]$. In
particular, it does not have solutions for $u<\eta\,a_{0}-x_{0}$, and so
$U_{c}$ and $U^{\prime}_{c}$ are the only local extrema of $\mathfrak{R}S(u)$
in $(-\infty,\eta\,a_{0}-x_{0})$. Observing that
$\lim_{u\to-\infty}\mathfrak{R}S(u)=-\infty$ and $\mathfrak{R}S(u)$ has a
positive infinite slope at $\eta\,a_{0}-x_{0}$, this ends the proof of the
lemma. ∎

We now recall from Eqs. 41 and 46 that the renormalized kernel
$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}$ writes as

$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{2},t_{2}))=-\frac{1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}}\\!\oint_{\gamma_{W}}\operatorname{Int}_{N}(W,Z)\differential{W}\differential{Z},$
(64)

where $\operatorname{Int}_{N}(W,Z)$ can be approximated by
$(\sqrt{N})^{x_{2}-x_{1}}\mathrm{e}^{\sqrt{N}(S(W)-S(Z))}\,h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$
uniformly on compact subsets of $D_{S}$ (defined above Lemma 23). We recall
also that the integrand $\operatorname{Int}_{N}(W,Z)$ has poles for $W=Z$, for
some values of $W$ in an interval $I_{W}=[\eta\,a_{0}-x_{0},o(1)]$ and for
some values of $Z$ in an interval $I_{Z}=[o(1),\eta\,a_{m}-x_{0}-1]$.

######  Lemma 34.

There exist two integration contours $\gamma_{W}^{\text{new}}$ and
$\gamma_{Z}^{\text{new}}$ (both followed in counterclockwise order) such that,

  * •

$\gamma_{W}^{\text{new}}$ lies in the interior of $\gamma_{Z}^{\text{new}}$;

  * •

$\gamma_{W}^{\text{new}}$ (resp. $\gamma_{Z}^{\text{new}}$) contains $I_{W}$
(resp. $I_{Z}$) in its interior;

  * •

$\gamma_{W}^{\text{new}}$ (resp. $\gamma_{Z}^{\text{new}}$) lies inside the
region $\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U^{\prime}_{c})\\}$ (resp.
$\\{\mathfrak{R}S(U)\geq\mathfrak{R}S(U_{c})\\}$).

These new contours are shown in the left-hand side of Fig. 14.

Figure 14. From left to right: In green and purple, the new integration
contours $\gamma_{W}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$ constructed
in Lemmas 34, 38 and 41. The three situations correspond to the small $t$
region $\\{0\leq t_{0}\leq t_{-}(x_{0})\\}$, the large $t$ region
$\\{t_{+}(x_{0})\leq t_{0}\leq 1\\}$, and the intermediate $t$ region
$\\{t_{-}(x_{0})<t_{0}<t_{+}(x_{0})\\}$, respectively. In orange, we highlight
the level lines considered in the proofs of the three lemmas.

###### Proof.

We start by noting that the action $S$ can be analytically continued in the
upper half-plane on a neighborhood of $U_{c}\in\mathbb{R}$ and behaves like

$S(U)=S(U_{c})+\frac{S^{\prime\prime}(U_{c})}{2}\,(U-U_{c})^{2}+O((U-U_{c})^{3}).$

Since $\mathfrak{I}S(u)$ is piecewise affine and $\mathfrak{R}S(u)$, seen as a
function of a real variable $u$, reaches a local maximum at $U_{c}$ (Lemma
33), the coefficient $S^{\prime\prime}(U_{c})$ must be real and negative.
Since $S^{\prime\prime}(U_{c})$ is real, by comparison with the map
$F(U)=U^{2}$, there is an imaginary level line of $S$ leaving from $U_{c}$
orthogonally to the real axis and going in the upper half plane; call it
$\ell$. Moreover, since $S^{\prime\prime}(U_{c})$ is negative, the real part
of $S$ increases along $\ell$ (recall that along an imaginary level line, the
function $\mathfrak{R}S(U)$ is strictly monotone by the open mapping theorem).
In particular, $\ell$ is included in the region
$\\{\mathfrak{R}S(U)\geq\mathfrak{R}S(U_{c})\\}$. Since $S$ does not have any
critical point in the upper half-plane, $\ell$ cannot end inside the upper
half-plane, hence it either reaches again the real axis or it goes to
infinity. If it goes to infinity, we claim that $\ell$ can only go to infinity
in the positive real direction. Indeed, the estimate
$S(U)\sim|\log(1-t_{0})|\,U$ for large $|U|$ (recall (48)) forces
$\mathfrak{I}U$ to stay bounded along $\ell$ and $\mathfrak{R}U$ to stay
bounded from below along $\ell$ (by definition $\mathfrak{I}S(U)$ is bounded
along $\ell$ and $\mathfrak{R}S(U)\geq\mathfrak{R}S(U_{c})$ along $\ell$).

Similarly one can consider the imaginary line $\ell^{\prime}$ of $S$ leaving
from $U_{c}^{\prime}$ and going in the upper half plane. The real part of $S$
is decreasing along $\ell^{\prime}$ so that $\ell^{\prime}$ stays in the
region $\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U^{\prime}_{c})\\}$. In
particular $\ell$ and $\ell^{\prime}$ cannot cross. Similarly as above, one
can see that $\ell^{\prime}$ either reaches again the real axis at some point
or go to infinity in the negative real direction.

To determine the behavior of $\ell$ and $\ell^{\prime}$, it is useful to look
at the imaginary part of the action of the real line. We recall from (56) and
the discussion below it, that
$\mathfrak{I}S(U_{c})=\mathfrak{I}S(U^{\prime}_{c})=-\pi x_{0}$, and that
$-\pi x_{0}\in(0,\mathfrak{I}S(0))$. Since $\mathfrak{I}(S(u))=0$ for large
real positive values of $u$, there exists $\overline{x}>0$ such that
$\mathfrak{I}(S(\overline{x}))=-\pi x_{0}$. Considering the shape of the real
map $u\mapsto\mathfrak{I}(S(u))$ (Fig. 11), there are two possibilities:

  * •

If $\overline{x}$ is an interval of the type
$(\eta\,b_{k}-x_{0},\eta\,a_{k}-x_{0})$, then $u\mapsto\mathfrak{I}(S(u))$ is
decreasing at $\overline{x}$, and $\overline{x}$ is uniquely determined by the
conditions $\overline{x}>0$ and $\mathfrak{I}(S(\overline{x}))=-\pi x_{0}$.

  * •

If $\overline{x}$ is an interval of the type
$(\eta\,a_{k-1}-x_{0},\eta\,b_{k}-x_{0})$, then $u\mapsto\mathfrak{I}(S(u))$
is constant on that interval, and any point of this interval can be chosen as
$\overline{x}$. We then choose $\overline{x}$ to be the unique solution of the
critical equation in that interval (the existence being ensured by Lemma 24
and the uniqueness by our assumption that the critical equations has two
solutions in $(-\infty,\eta\,a_{0}-x_{0})$).

_Claim: The imaginary level lines $\ell$ and $\ell^{\prime}$ can only come
back to the real axis at $\overline{x}$._

First note that $\ell$ and $\ell^{\prime}$ can only come back to the real axis
at a point $u\in\mathbb{R}\setminus\\{U_{c},U^{\prime}_{c}\\}$ such that
$\mathfrak{I}S(u)=\mathfrak{I}S(U_{c})=\mathfrak{I}S(U^{\prime}_{c}).$
Moreover, let $y$ be a non-critical point around which
$u\mapsto\mathfrak{I}S(u)$ is constant for real $u$. Since $y$ is non-
critical, the set $\\{\mathfrak{I}S(z)=\mathfrak{I}S(y)\\}$ is a single curve
around $y$ and thus coincide with the real axis. In particular $\ell$ (or
$\ell^{\prime}$) cannot reach the real axis at such $y$.

The case when $y=\eta a_{i}-x_{0}$ or $y=b_{i}-x_{0}$ is a bit more delicate.
Locally around $y$, the function $S$ looks like $\pm(z-y)\log(z-y)$, and thus
the set $\\{\mathfrak{I}S(z)=\mathfrak{I}S(y)\\}$ looks locally like a half-
line. Thus it is locally included in the real line (we know from Fig. 11 that
at each point $y=\eta a_{i}-x_{0}$ or $y=b_{i}-x_{0}$ , there is a direction
along the real line in which $\mathfrak{I}S(u)$ is constant). Again, $\ell$
(or $\ell^{\prime}$) cannot reach the real axis at such $y$.

Finally, we observe that the interval $(-\infty,\eta a_{0}-x_{0})$ contains no
other critical point than $U_{c}$ and $U^{\prime}_{c}$, so, from the previous
discussion, $\ell$ and $\ell^{\prime}$ cannot come back to the real axis
inside this interval. Altogether, this proves the claim, i.e. that $\ell$ and
$\ell^{\prime}$ can only come back to the real axis at $\overline{x}$.

A first consequence is that at most one of $\ell$ and $\ell^{\prime}$ can come
back to the real axis. But they cannot go both to infinity, otherwise they
would cross. So one of $\ell$ or $\ell^{\prime}$ should go to the real axis,
and the other to infinity. The non-crossing constraint and the fact that
$\overline{x}>0$ forces that $\ell$ goes to infinity, while $\ell^{\prime}$
goes to $\overline{x}$.

The contours of the lemma are now obtained as follows. For
$\gamma_{W}^{\text{new}}$, we simply follow $\ell^{\prime}$ and its mirror
image in the lower half plane. By construction, it lies in the region
$\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U^{\prime}_{c})\\}$. Also, since the
return point $\overline{x}$ to the real axis satisfies $\overline{x}>0$,
$\gamma_{W}^{\text{new}}$ encloses $I_{W}$. For $\gamma_{Z}^{\text{new}}$, we
follow $\ell$ until $\mathfrak{R}(U)$ is sufficiently large, and then go to
the real axis following any smooth curve. We then go back to $U_{c}$ with the
mirror image of the first part of the contour. If we follow $\ell$ until when
$\mathfrak{R}(U)$ is sufficiently large, we can ensure that this contour
encloses $I_{Z}$ and lies in the region
$\\{\mathfrak{R}S(U)\geq\mathfrak{R}S(U_{c})\\}$. Finally note that
$\gamma_{W}^{\text{new}}$ lies in the interior of $\gamma_{Z}^{\text{new}}$ by
construction. ∎

######  Proposition 35.

Let $x_{0}<0$ and $0\leq t_{0}\leq t_{-}(x_{0})$. Then, locally uniformly for
$(x_{1},t_{1})\in\mathbb{Z}\times\mathbb{R}$, we have that

$\lim_{N\to+\infty}\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=0.$
(65)

As a consequence, $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ tends in
probability to the empty set.

###### Proof.

We are interested in (64) for $x_{1}=x_{2}$ and $t_{1}=t_{2}$. In this case,
the contours are such that $\gamma_{W}$ contains $I_{W}$, $\gamma_{Z}$
contains $I_{Z}$ and $\gamma_{W}$ lies in the interior of $\gamma_{Z}$.
Therefore, we can transform the contours $\gamma_{W}$ and $\gamma_{Z}$ to the
contours $\gamma_{W}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$ from Lemma 34
without crossing any poles. Thus, by the residue theorem, we get

$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=-\frac{1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}^{\text{new}}}\\!\oint_{\gamma_{W}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{W}\differential{Z}.$
(66)

We recall that
$\operatorname{Int}_{N}(W,Z)\simeq(\sqrt{N})^{x_{2}-x_{1}}\mathrm{e}^{\sqrt{N}(S(W)-S(Z))}\,h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$
on $D_{S}$ and that
$\mathfrak{R}S(W)\leq\mathfrak{R}S(U_{c}^{\prime})<\mathfrak{R}S(U_{c})\leq\mathfrak{R}S(Z)$
for $(W,Z)\in\gamma_{W}^{\text{new}}\times\gamma_{Z}^{\text{new}}$ (see Lemmas
33 and 34). A difficulty comes from the fact that the right-most intersection
point $W^{+}$ of $\gamma_{W}^{\text{new}}$ with the real axis might lie
outside $D_{S}$ ($W^{+}$ coincides with the point $\overline{x}$ in the proof
of Lemma 34). Nevertheless, the integrand can be controlled around this point
thanks to third part of Lemma 23. The same arguments as in the proof of
Proposition 31 shows that $\operatorname{Int}_{N}(W,Z)$ converges pointwise to
$0$ and that the integrand is bounded by the integrable function
$h^{(x_{1},t_{1})}_{(x_{2},t_{2})}(W,Z)$. Dominated convergence applies,
proving (65).

The convergence in distribution to the empty set then follows from the general
identity for determinantal point processes

$\mathbb{E}\\!\left[\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}(A)\right]=\int_{A}\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))\differential{\lambda}(x_{1},t_{1}),$

for any bounded subset $A\subset\mathbb{Z}\times\mathbb{R}$, where we recall
that $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}(A)$ denotes the number of
beads of $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ contained in the set
$A$. By dominated convergence, these expected numbers of beads go to $0$, so
$(\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}(B_{i}))_{1\leq i\leq k}$
converges in probability to $(0,\ldots,0)$ for any collection of bounded
subsets $(B_{i})_{1\leq i\leq k}$. The discussion from Section 2.2 shows that
this is equivalent to the convergence in distribution towards the empty set.

We now discuss the case where the critical equation has a double root
$U_{c}=U^{\prime}_{c}$ in the interval $(-\infty,\eta\,a_{0}-x_{0})$. Then
$u\mapsto\mathfrak{R}S(u)$ is increasing on $(-\infty,\eta\,a_{0}-x_{0})$
(this is an immediate analogue of Lemma 33), and the action writes locally as
$S(z)=S(U_{c})+\frac{S^{\prime\prime\prime}(U_{c})}{6}(z-U_{c})^{3}+O\left((z-U_{c})^{4}\right)$,
with $S^{\prime\prime\prime}(U_{c})>0$. Fig. 15 shows the imaginary level
lines of $S$, and the $\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$ and
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$ regions around $U_{c}$. In
particular, we can find new integration contours as in Lemma 34 except that
they now meet at $U_{c}=U^{\prime}_{c}$. Note that, in this case, the
integrand in (66) has a singularity at $Z=W=U_{c}$, where it behaves as
$O((Z-W)^{-1})$. This is similar to the setting of Proposition 31 and we can
therefore apply dominated convergence using the same arguments. We conclude
that the renormalized correlation kernel tends also to $0$, finishing the
proof of the proposition. ∎

$\begin{array}[]{c}\includegraphics[height=142.26378pt]{figures/Double_Critical_Point}\end{array}\qquad\begin{array}[]{c}\includegraphics[height=142.26378pt]{figures/Contours_After_Frozen_Double_Critical}\end{array}$
Figure 15. Left: The landscape of the action $S$ around a double critical
point on the real line in the case of the small $t$ region $\\{0\leq t_{0}\leq
t_{-}(x_{0})\\}$. Plain lines are imaginary level lines, dotted lines are the
real level lines. We also indicated the alternation of different regions: the
yellow regions correspond to $\\{\mathfrak{R}S(U)<\mathfrak{R}S(U_{c})\\}$,
while the white regions correspond to
$\\{\mathfrak{R}S(U)>\mathfrak{R}S(U_{c})\\}$. Right: The new integration
contour in the case of a double critical point (to be compared with the left-
hand side of Fig. 14).

The case $x_{0}\in(0,\eta\,a_{m})$ is essentially treated in the same way,
with the following modifications. By Proposition 25, the two relevant real
critical points $U_{c}$ and $U_{c}^{\prime}$ live in
$(\eta\,a_{m}-x_{0},+\infty)$. To simplify the discussion we assume $U_{c}\neq
U^{\prime}_{c}$. Using the convention $U_{c}<U_{c}^{\prime}$, Lemma 33 still
holds. An analog of Lemma 34 also holds, with the important change that in
this case, $\gamma_{Z}^{\text{new}}$ lies in the interior of
$\gamma_{W}^{\text{new}}$ (and not the opposite). Since the contours in (46)
satisfy that $\gamma_{W}$ is inside $\gamma_{Z}$ (for $t_{1}=t_{2}$), moving
them to $\gamma_{Z}^{\text{new}}$ and $\gamma_{W}^{\text{new}}$ yield a
residue term related to the pole $Z=W$ (as in (60), except that the residue
term appears for any $W$ in $\gamma_{W}^{\text{new}}$). For $x_{1}=x_{2}$,
$t_{1}=t_{2}$, recalling (41), the residue of the integrand in (66) related to
the pole $Z=W$ is simply $\frac{1}{1-t_{0}-\frac{t_{1}}{\sqrt{N}}}$ .
Therefore, Eq. 66 is replaced by:

$\displaystyle\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=-\frac{1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}^{\text{new}}}\\!\oint_{\gamma_{W}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{W}\differential{Z}-\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{W}^{\text{new}}}\frac{1}{1-t_{0}-\frac{t_{1}}{\sqrt{N}}}\differential{W}.$

But the second integral is obviously $0$, so that (65) holds also in this
case. We conclude as above and obtain that also in this regime, the bead
process $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ tends in distribution to
the empty set.

The same conclusion also holds for the following remaining three cases: when
$x_{0}=0$ (in this case, by Proposition 25, the only admissible value for
$t_{0}$ is $t_{0}=0$), when $x_{0}=\eta\,a_{0}$, and when $x_{0}=\eta\,a_{m}$.
We leave to the reader the details of these three remaining cases, pointing
out that one needs some little modifications in the same spirit as Remark 26.

Combining all the results in this section, we obtain the following final
result for the small $t$ region.

######  Proposition 36.

Let $x_{0}\in[\eta\,a_{0},\eta\,a_{m}]$ and $0\leq t_{0}\leq t_{-}(x_{0})$.
Then the bead process $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ tends in
probability to the empty set.

###  5.2. The large $t$ region

In this section, we fix $x_{0}\in[\eta\,a_{0},\eta\,a_{m}]$ and we let
$t_{+}\leq t_{0}\leq 1$, where $t_{+}=t_{+}(x_{0})$ is given by Proposition
25. For the sake of brevity, we restrict ourselves to the case when there
exists $i_{0}\geq 1$ such that $x_{0}\in(\eta\,a_{i_{0}-1},\eta\,b_{i_{0}})$.
The case $x_{0}\in(\eta\,b_{i_{0}},\eta\,a_{i_{0}})$ for some $i_{0}\geq 1$
and the case $x_{0}=\eta\,a_{i_{0}}$ for some $i_{0}\geq 0$ are similar (for
the case $x_{0}=\eta\,a_{i_{0}}$, we refer to the discussion in Remark 26).
Finally, for $x_{0}=\eta\,b_{i_{0}}$, the only admissible value for $t_{0}$ in
the large $t$ region is $t_{0}=1$ (since Proposition 25 states that
$t_{+}(\eta\,b_{i_{0}})=1$); also in this case we omit the simple necessary
modifications.

From Proposition 25, we know that the critical equation (8) has only real
solutions, two of which, say $U_{c}\leq U^{\prime}_{c}$, are in the interval
$(0,\eta\,b_{i_{0}}-x_{0})$. We will discuss here the case
$U_{c}<U^{\prime}_{c}$, the case of a double critical point
$U_{c}=U^{\prime}_{c}$ being obtained with simple modifications similar to
those discussed in the previous subsection.

######  Lemma 37.

The critical point $U_{c}$ and $U^{\prime}_{c}$ are respectively a local
minimum and a local maximum of the function $u\to\mathfrak{R}S(u)$ on the real
line and we have $\mathfrak{R}S(U_{c})<\mathfrak{R}S(U^{\prime}_{c})$.

###### Proof.

The lemma follows from the fact that $u\to\mathfrak{R}S(u)$ has negative
infinite slopes at the points $0$ and $\eta\,b_{i_{0}}-x_{0}$ and that there
are no local extrema other than $U_{c}$ and $U^{\prime}_{c}$ in the interval
$(0,\eta\,b_{i_{0}}-x_{0})$. ∎

We now see how to move integration contours. To this end, let us first remark
that when $x_{0}\in(\eta\,a_{i_{0}-1},\eta\,b_{i_{0}})$ the $Z$ poles of
$\operatorname{Int}_{N}(W,Z)$ all lie in a strictly smaller sub-interval of
the interval $I_{Z}=[o(1),\eta\,a_{m}-x_{0}-1]$ considered so far. Indeed,
from (41), one can see that the only $Z$-poles of
$\operatorname{Int}_{N}(W,Z)$ are a subset of the poles of
$F_{\lambda_{N}}(\sqrt{N}(Z+x_{0}))$. But, from (28) and Fig. 9, we know that
the latter function has no poles in the intervals of the form
$(\eta\,a_{i-1}-x_{0},\eta\,b_{i}-x_{0})$ and so, in particular, the $Z$ poles
of $\operatorname{Int}_{N}(W,Z)$ all lie in $\widetilde{I}_{Z}=[\eta
b_{i_{0}}-x_{0},\eta\,a_{m}-x_{0}-1]\subset I_{Z}$.

######  Lemma 38.

There exist two integration contours $\gamma_{W}^{\text{new}}$ and
$\gamma_{Z}^{\text{new}}$ (both followed in counterclockwise order) such that

  * •

$\gamma_{W}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$ have disjoint
interiors;

  * •

$\gamma_{W}^{\text{new}}$ (resp. $\gamma_{Z}^{\text{new}}$) contains $I_{W}$
(resp. $\widetilde{I}_{Z}$) in its interior;

  * •

$\gamma_{W}^{\text{new}}$ (resp. $\gamma_{Z}^{\text{new}}$) lies inside the
region $\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U_{c})\\}$ (resp.
$\\{\mathfrak{R}S(U)\geq\mathfrak{R}S(U^{\prime}_{c})\\}$).

These contours are shown in the middle picture of Fig. 14.

###### Proof.

The proof is similar to that of Lemma 34. We consider the imaginary level
lines $\ell$ and $\ell^{\prime}$ leaving from $U_{c}$ and $U^{\prime}_{c}$
orthogonally to the real line and in the upper-half plane. The level line
$\ell$ (resp. $\ell^{\prime}$) lies inside the region
$\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U_{c})\\}$ (resp.
$\\{\mathfrak{R}S(U)\geq\mathfrak{R}S(U^{\prime}_{c})\\}$). Moreover, they
cannot go back to the the real line. Indeed, the value $\mathfrak{I}S(z)$ on
the real line is maximal on the interval $(0,\eta\,b_{i_{0}}-x_{0})$ (see
Section 3.4 and Fig. 11) so that if $\ell$ or $\ell^{\prime}$ goes back to the
real line, it has to be within this interval. But this would mean that $S$ has
a third critical point in this interval, which is impossible by Lemma 24 and
Proposition 25. We conclude that both $\ell$ and $\ell^{\prime}$ go to
infinity. Since $S(U)\sim|\log(1-t_{0})|\,U$ for large $|U|$ by (48),
necessarily $\ell$ goes to infinity in the negative real direction, while
$\ell^{\prime}$ goes to infinity in the positive real direction.

It is then possible to follow $\ell$ long enough, and then join the negative
real axis while staying in the region
$\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U_{c})\\}$. Completing the path by
symmetry gives the contour $\gamma_{W}^{\text{new}}$. The contour
$\gamma_{Z}^{\text{new}}$ is constructed similarly from $\ell^{\prime}$.

We finally note that the claim in the second item follows from the discussion
above the lemma statement. ∎

######  Proposition 39.

Let $x_{0}\in[\eta\,a_{0},\eta\,a_{m}]$ and $t_{+}(x_{0})\leq t_{0}\leq 1$.
Then, locally uniformly for $(x_{1},t_{1})\in\mathbb{Z}\times\mathbb{R}$, we
have that

$\lim_{N\to+\infty}\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=0.$

As a consequence, $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ tends in
probability to the empty set.

###### Proof.

As already explained, we give details only in the case when there exists
$i_{0}\geq 1$ such that $x_{0}\in(\eta\,a_{i_{0}-1},\eta\,b_{i_{0}})$. Again,
we are interested in (46) for $x_{1}=x_{2}$ and $t_{1}=t_{2}$. The original
contours are such that $\gamma_{W}$ lies in the interior of $\gamma_{Z}$. As
in the case $x_{0}>0$ and $0\leq t_{0}\leq t_{-}(x_{0})$ in the previous
section, moving the contour to those of Lemma 38 yields the following:

$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=-\frac{1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}^{\text{new}}}\\!\oint_{\gamma_{W}^{\text{new}}}\operatorname{Int}_{N}(W,Z)\differential{W}\differential{Z}-\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{W}^{\text{new}}}\frac{1}{1-t_{0}-\frac{t_{1}}{\sqrt{N}}}\differential{W}.$

The second integral is identically $0$, while the first tends to $0$ as $N$
tends to $+\infty$ using the same argument as before. This ends the proof of
the proposition. ∎

###  5.3. The intermediate $t$ region

Fix $x_{0}\in[\eta\,a_{0},\eta\,a_{m}]$. As seen in Section 1.4.2, it might
happen that there is some $t_{0}\in(t_{-}(x_{0}),t_{+}(x_{0}))$ in the frozen
region, i.e. such that the critical equation (8) has only real roots. In this
case, thanks to the third item of Proposition 25, there are three roots –
denoted by $U_{c}\leq U_{c}^{\prime}\leq U^{\prime\prime}_{c}$ – that are
either all inside a negative interval
$(\eta\,b_{j_{0}}-x_{0},\eta\,a_{j_{0}}-x_{0})$ for some $j_{0}<i_{0}$, or all
inside a positive interval $(\eta\,a_{j_{0}}-x_{0},\eta\,b_{{j_{0}}+1}-x_{0})$
for some $j_{0}\geq i_{0}$. These two cases are treated similarly, we will
therefore assume that the three roots are in
$(\eta\,b_{j_{0}}-x_{0},\eta\,a_{j_{0}}-x_{0})$ for some $j_{0}<i_{0}$. Also,
we assume $U_{c}<U_{c}^{\prime}<U^{\prime\prime}_{c}$, and let the reader
convince himself that multiple critical points do not create additional
difficulties. The following lemma is proven similarly as before.

######  Lemma 40.

The critical point $U_{c}$, $U^{\prime}_{c}$ $U^{\prime\prime}_{c}$ are
respectively local minimum, maximum and minimum of the function
$u\to\mathfrak{R}S(u)$ on the real line and we have both
$\mathfrak{R}S(U_{c})<\mathfrak{R}S(U^{\prime}_{c})>\mathfrak{R}S(U^{\prime\prime}_{c})$.

This helps us to construct integration contours as follows. With an argument
similar to the one used above Lemma 38, we note that there are no $W$-poles in
the interval $(\eta\,b_{j_{0}}-x_{0},\eta\,a_{j_{0}}-x_{0})$. A major
difference is that now the $W$ integration contour is split into two disjoint
parts, the union of their interiors containing all $W$ poles.

######  Lemma 41.

There exist integration contours $\gamma_{W,1}^{\text{new}}$,
$\gamma_{W,2}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$ (all followed in
counterclockwise order) such that

  * •

$\gamma_{W,1}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$ have disjoint
interiors, and $\gamma_{W,2}^{\text{new}}$ lies inside
$\gamma_{Z}^{\text{new}}$;

  * •

$\gamma_{W,1}^{\text{new}}$ (resp. $\gamma_{W,2}^{\text{new}}$ and
$\gamma_{Z}^{\text{new}}$) contains
$(\eta\,a_{0}-x_{0},\eta\,b_{j_{0}}-x_{0})$ (resp. $(\eta\,a_{j_{0}}-x_{0},0)$
and $I_{Z}$) in its interior;

  * •

$\gamma_{W,1}^{\text{new}}$ (resp. $\gamma_{W,2}^{\text{new}}$) lies inside
the region $\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U_{c})\\}$ (resp.
$\\{\mathfrak{R}S(U)\leq\mathfrak{R}S(U^{\prime\prime}_{c})\\}$);

  * •

$\gamma_{Z}^{\text{new}}$ lies inside the region
$\\{\mathfrak{R}S(U)\geq\mathfrak{R}S(U^{\prime}_{c})\\}$.

These contours are shown in the right-hand side of Fig. 14.

###### Proof.

The strategy of proof is again the same. We consider the three imaginary lines
$\ell$, $\ell^{\prime}$ and $\ell^{\prime\prime}$ leaving from $U_{c}$,
$U^{\prime}_{c}$ and $U^{\prime\prime}_{c}$. One can prove that
$\ell^{\prime\prime}$ goes back to the real axis at some point $x>0$, while
$\ell$ and $\ell^{\prime}$ go to infinity respectively in the negative and
positive directions. Symmetrizing $\ell^{\prime\prime}$ gives the contour
$\gamma_{W,2}^{\text{new}}$. On the other hand, following $\ell$ and
$\ell^{\prime}$ for long enough and then joining the real line (plus
symmetrizing) gives $\gamma_{W,1}^{\text{new}}$ and $\gamma_{Z}^{\text{new}}$.
∎

######  Proposition 42.

Let $x_{0}\in[\eta\,a_{0},\eta\,a_{m}]$ and
$t_{0}\in(t_{-}(x_{0}),t_{+}(x_{0}))$ such that the critical equation (8) has
only real roots. Then, locally uniformly for
$(x_{1},t_{1})\in\mathbb{Z}\times\mathbb{R}$, we have that

$\lim_{N\to+\infty}\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=0.$

As a consequence, $\widetilde{M}_{\lambda_{N}}^{(x_{0},t_{0})}$ tends in
probability to the empty set.

###### Proof.

Again, we consider (46) for $x_{1}=x_{2}$ and $t_{1}=t_{2}$. The original
contours are such that $\gamma_{W}$ lies in the interior of $\gamma_{Z}$.
Moving the contour to those of Lemma 41 (note that $\gamma_{W,1}^{\text{new}}$
and $\gamma_{W,2}^{\text{new}}$ together enclose the same $W$-poles as
$\gamma_{W}$) yields a residue term which is an integral over
$\gamma_{W,1}^{\text{new}}$. We get

$\widetilde{K}_{\lambda_{N}}^{(x_{0},t_{0})}((x_{1},t_{1}),(x_{1},t_{1}))=-\frac{1}{(2\mathrm{i}\pi)^{2}}\oint_{\gamma_{Z}^{\text{new}}}\\!\oint_{\gamma_{W}^{\text{new}}}{\operatorname{Int}_{N}(W,Z)}\differential{W}\differential{Z}-\frac{1}{2\mathrm{i}\pi}\oint_{\gamma_{W,1}^{\text{new}}}{\frac{1}{1-t_{0}-\frac{t_{1}}{\sqrt{N}}}}\differential{W}.$

Again, the second integral is identically $0$, while the first tends to $0$ as
$N$ tends to $+\infty$. This ends the proof of the proposition. ∎

Propositions 36, 39, and 42 completes the proof of the second item of Theorem
15.

##  6\. The limiting height function and the continuity of the limiting
surface

In this section, we prove Theorems 4 and 7. Recall (from Theorem 2) that
$H^{\infty}:[\eta\,a_{0},\eta\,a_{m}]\times[0,1]\to\mathbb{R}$ is the
deterministic limiting height function of the sequence of random height
functions $\frac{1}{\sqrt{N}}H_{\lambda_{N}}(\lfloor x\sqrt{N}\rfloor,t)$
coming from the bead processes associated with a uniform random (Poissonized)
Young tableaux of fixed shape $\lambda^{0}$. We want to show that

$H^{\infty}(x,t)=\frac{1}{\pi}\int_{0}^{t}\alpha(x,s)\differential{s},\quad\text{for
all }(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1].$

###### Proof of Theorem 4.

We fix $(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$. Recalling the boundary
conditions in (5), we note that

$\frac{1}{\sqrt{N}}H_{\lambda_{N}}(\lfloor
x\sqrt{N}\rfloor,t)\leq\frac{1}{2{\sqrt{N}}}\left(\omega_{\lambda_{N}}(\lfloor
x\sqrt{N}\rfloor)-|\lfloor
x\sqrt{N}\rfloor|\right)=\frac{1}{2}(\omega_{\eta\lambda^{0}}(x)-|x|),\quad\text{for
all }N>0.$

Therefore, by dominated convergence theorem, the convergence in Theorem 2
implies that for every fixed $(x,t)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$,

$H^{\infty}(x,t)=\lim_{N\to+\infty}\frac{1}{\sqrt{N}}\mathbb{E}\big{[}H_{\lambda_{N}}(\lfloor
x\sqrt{N}\rfloor,t)\big{]}.$

We will prove that the right-hand side converges to
$\frac{1}{\pi}\int_{0}^{t}\alpha(x,s)ds$, implying Theorem 4 by uniqueness of
the limit. By definition, $H_{\lambda_{N}}(\lfloor x\sqrt{N}\rfloor,t)$ is the
number of beads in the bead process $M_{\lambda_{N}}$ which lie on the thread
at position $\lfloor x\sqrt{N}\rfloor$ and with height in $[0,t]$. Since
$M_{\lambda_{N}}$ is a determinantal point process with kernel
$K_{\lambda_{N}}$, we have

$~\mathbb{E}\big{[}H_{\lambda_{N}}(\lfloor
x\sqrt{N}\rfloor,t)\big{]}=\int_{0}^{t}K_{\lambda_{N}}\big{(}(\lfloor
x\sqrt{N}\rfloor,s),(\lfloor x\sqrt{N}\rfloor,s)\big{)}ds.$

With the notation in (40), we get

$\frac{1}{\sqrt{N}}K_{\lambda_{N}}\big{(}(\lfloor x\sqrt{N}\rfloor,s),(\lfloor
x\sqrt{N}\rfloor,s)\big{)}=\widetilde{K}^{(x,s)}_{\lambda_{N}}((0,0),(0,0)).$

Recall now that
$\alpha(x,s):=\frac{\mathfrak{I}U_{c}}{1-s}\mathds{1}_{(x,s)\in L}$, where $L$
denotes the liquid region. If $(x,s)$ is in the liquid region, Eqs. 59 and 61
implies that

$\lim_{N\to+\infty}\widetilde{K}^{(x,s)}_{\lambda_{N}}((0,0),(0,0))=K_{\infty}((0,0),(0,0))=\frac{1}{2i\pi}\int_{\overline{U_{c}}}^{U_{c}}\frac{dW}{1-s}=\frac{1}{\pi}\frac{\mathfrak{I}U_{c}}{1-s}=\frac{\alpha(x,s)}{\pi},$

where, for the third last equality, we used a vertical path from
$\overline{U_{c}}$ to $U_{c}$, as in the beginning of the proof of Lemma 32.
On the other hand, if $(x,s)$ is outside the liquid region, we have, by
Propositions 36, 39 and 42,

$\lim_{N\to+\infty}\widetilde{K}^{(x,s)}_{\lambda_{N}}((0,0),(0,0))=0=\frac{\alpha(x,s)}{\pi}.$

Hence, we get that
$\lim_{N\to+\infty}\widetilde{K}^{(x,s)}_{\lambda_{N}}((0,0),(0,0))=\frac{\alpha(x,s)}{\pi}$,
for all $(x,s)\in[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$. Moreover, since
integration contours, integrands and critical points are continuous function
of $(x,s)$, and since all asymptotic estimates given are uniform on compact
sets, the kernel convergence is also uniform on compact sets. We conclude that

$\lim_{N\to+\infty}\frac{1}{\sqrt{N}}\,\mathbb{E}\big{[}H_{\lambda_{N}}(\lfloor
x\sqrt{N}\rfloor,t)\big{]}=\lim_{N\to+\infty}\int_{0}^{t}\widetilde{K}^{(x,s)}_{\lambda_{N}}((0,0),(0,0))ds=\frac{1}{\pi}\int_{0}^{t}\alpha(x,s)ds,$

proving Theorem 4. ∎

We now turn to the proof of the continuity criterion for the limiting surface
$T^{\infty}$ stated in Theorem 7.

###### Proof of Theorem 7.

Recall the definition of the quantities $T^{\infty}_{-}(x,y)$ and
$T^{\infty}_{+}(x,y)$ from (12) and recall also the definition of $t_{-}(x)$
and $t_{+}(x)$ from Proposition 25, noting that $t_{-}(x)=\inf\left\\{\partial
L\cap(\\{x\\}\times[0,1])\right\\}$ and $t_{+}(x)=\sup\left\\{\partial
L\cap(\\{x\\}\times[0,1])\right\\}$.

From Remark 6, we know that the limiting surface $T^{\infty}$ is continuous in
its domain $D_{\lambda^{0}}$ if and only if
$T^{\infty}_{-}(x,y)=T^{\infty}_{+}(x,y)$ for all $(x,y)\in D_{\lambda^{0}}$.
The latter condition is equivalent to the following property of the liquid
region $L$: for all $x\in[\eta\,a_{0},\eta\,a_{m}]$,

$[t_{-}(x),t_{+}(x)]\setminus\left\\{L\cap(\\{x\\}\times[0,1])\right\\}$ does
not contain any non-empty open interval. (67)

Indeed, if this is not true, i.e. for some $x\in[\eta\,a_{0},\eta\,a_{m}]$
there exists an non-empty open interval $I$ contained in
$[t_{-}(x),t_{+}(x)]\setminus\left\\{L\cap(\\{x\\}\times[0,1])\right\\}$, then
there exists some constant $c>0$ such that $H^{\infty}(x,t)=c$ for all $t\in
I$ and so $T^{\infty}_{-}(x,2c-|x|)<T^{\infty}_{+}(x,2c-|x|)$.

Now recall the parametrization of the frozen boundary curve $\partial L$ given
in Proposition 27. We observe that a necessary and sufficient condition such
that (67) holds for all $x\in[\eta\,a_{0},\eta\,a_{m}]$ is that the tangent
vector $(\dot{x}(s),\dot{t}(s))$ at the cusp points121212We recall that for a
plane curve defined by analytic parametric equations $x(t)=f(t)$ and
$y(t)=g(t)$, a _cusp_ is a point where both derivatives of $f$ and $g$ are
zero, and the directional derivative, in the direction of the tangent, changes
sign. of $\partial L$ is vertical. Requesting that the tangent vector
$(\dot{x}(s),\dot{t}(s))$ is vertical at a certain point $(x(s),t(s))$ of the
curve $\partial L$ is equivalent to impose that its (absolute) slope is zero,
that is $|G(s)|=+\infty$. The solutions to the latter equation are
$s=\eta\,a_{i}$ for $i=0,\dots,m$. Moreover, the cusp points of $\partial L$
are given by the solutions to the equations:

$\begin{cases}\dot{x}(s)=1+\frac{\dot{\Sigma}(s)}{\Sigma(s)^{2}}=0,\\\
\dot{t}(s)=G(s)\left(1+\frac{\dot{\Sigma}(s)}{\Sigma(s)^{2}}\right)=0,\end{cases}$

where the directional derivative, in the direction of the tangent, changes
sign. One can check that $(x(s),t(s))$, for $s=\eta\,a_{0}$ and
$s=\eta\,a_{m}$, are not cusp points. Indeed, these two points are the two
points where the frozen boundary curve $\partial L$ is tangent to the two
vertical boundaries of $[\eta\,a_{0},\eta\,a_{m}]\times[0,1]$. Hence, it is
enough to impose that for all $i=1,\dots,m-1$,

$\begin{cases}\dot{x}(\eta\,a_{i})=0,\\\ \dot{t}(\eta\,a_{i})=0.\end{cases}$

With some standard computation, one can note that
$\dot{x}(\eta\,a_{i})=\lim_{s\to\eta\,a_{i}}1+\frac{\dot{\Sigma}(s)}{\Sigma(s)^{2}}=0$
for all $i=1,\dots,m-1$. Therefore, the only non-trivial condition is to
impose that $\dot{t}(\eta\,a_{i})=0$ for all $i=1,\dots,m-1$. Again, with some
standard computations, one can check that

$\dot{t}(\eta\,a_{i})=\lim_{s\to\eta\,a_{i}}G(s)\left(1+\frac{\dot{\Sigma}(s)}{\Sigma(s)^{2}}\right)=\frac{2\left(\sum_{j=0,j\neq
i}^{m}\frac{1}{\eta\,a_{i}-\eta\,a_{j}}-\sum_{j=1}^{m}\frac{1}{\eta\,a_{i}-\eta\,b_{j}}\right)\prod_{j=1}^{m}(\eta\,a_{i}-\eta\,b_{j})}{\prod_{j=0,j\neq
i}^{m}(\eta\,a_{i}-\eta\,a_{j})},$

concluding that the necessary and sufficient conditions for the continuity of
$T^{\infty}$ are the ones in (15). ∎

##  7\. Applications

In this section we prove the applications discussed in Section 1.4.

###### Proof of Proposition 8.

We first prove the formula for $H^{\infty}_{r}(x,t)$. From Theorem 4, we have

$H^{\infty}_{r}(x,t)=\frac{1}{\pi}\int_{0}^{t}\alpha_{r}(x,s)\differential{s},$

where $\alpha_{r}(x,s)=\frac{\mathfrak{I}U_{c}}{1-s}\mathds{1}_{(x,s)\in L}$.
In this case the critical equation becomes

$U\,\big{(}x-\sqrt{r}+\tfrac{1}{\sqrt{r}}+U\big{)}=(1-t)\big{(}x+\tfrac{1}{\sqrt{r}}+U\big{)}\big{(}x-\sqrt{r}+U\big{)}.$

Solving this quadratic equation and integrating the imaginary part of the
solution gives the formula (16) for $H^{\infty}_{r}$.

The fact that the limiting surface $T^{\infty}_{r}(x,y)$ is continuous for all
$r>0$ is a consequence of Theorem 7. ∎

###### Proof of Proposition 10.

From Theorem 7 we know that the surface $T^{\infty}_{p,q,r}$ is continuous in
its domain if and only if

$\frac{1}{a_{1}-a_{0}}+\frac{1}{a_{1}-a_{2}}=\frac{1}{a_{1}-b_{1}}+\frac{1}{a_{1}-b_{2}},$

with the coefficients $a_{0},a_{1},a_{2},b_{1},b_{2}$ as in Eq. 20. That is,

$\frac{1}{1+p}-\frac{2}{2+p-q}-\frac{2}{p+q-2r}+\frac{1}{p-r}=0.$

Solving the above equation, one gets the solutions claimed in the proposition
statement. ∎

##  8\. Local limits for random Young tableaux

The goal of this section is to complete the proof of Corollary 18.

###  8.1. Local topology for standard Young tableaux

We start with some definitions. We recall that a _marked standard Young
tableau_ is a triple $(\lambda,T,(x,y))$ where $(\lambda,T)$ is a standard
Young tableau of shape $\lambda$ and $(x,y)$ are the coordinates of a
distinguished box in $\lambda$ (see the left-hand side of Fig. 16 for an
example). A _marked Poissonized Young tableau_ is defined analogously.

We introduce a family of _restriction functions_ for marked standard Young
tableaux/Poissonized Young tableaux. Fix $h\in\mathbb{N}$. Given a marked
standard Young tableau/Poissonized Young tableau $(\lambda,T,(x,y))$, we
denote by $r_{h}\left(\lambda,T,(x,y)\right)$ the standard Young tableau
$(\diamond_{h},R)$, where $\diamond_{h}$ is a square Young diagram of size
$h^{2}$ and $R$ is a filling of the boxes of $\diamond_{h}$ obtained as
follows:131313We highlight the fact that by definition the restriction
functions $r_{h}$ give always back a standard Young tableau (even in the case
of Poissonized Young tableaux). Note that the rescaling procedure in step (2)
is well-defined also when $(\lambda,T,(x,y))$ is a marked Poissonized Young
tableau since we restrict to the case where the mapping $T$ has distinguished
values.

  1. (1)

first, we define
$\widetilde{R}(x^{\prime},y^{\prime})=T(x+x^{\prime},y+y^{\prime})$ for all
$(x^{\prime},y^{\prime})\in\diamond_{h}$, where we set
$R(x^{\prime},y^{\prime})=*$ if $T(x+x^{\prime},y+y^{\prime})$ is not well-
defined (see the middle picture of Fig. 16);

  2. (2)

second, we rescale the values of the map $\widetilde{R}$ obtaining a new map
$R$ so that $(\diamond_{h},R)$ is a standard Young tableau and the values of
$R$ have the same relative order as the values of $\widetilde{R}$, and the
boxes filled by $*$ are kept as they are (see the right-hand side of Fig. 16).

Figure 16. Left: A marked standard Young tableau
$\left(\lambda,T,(x,y)\right)$ with the marked box highlighted in red at
position $(x,y)=(4,5)$. Middle: The pair $(\diamond_{4},\widetilde{R})$
obtained from the first step in the definition of the restriction function
$r_{4}\left(\lambda,T,(x,y)\right)$. Right: The pair $(\diamond_{4},R)$
obtained from the second step in the definition of the restriction function
$r_{4}\left(\lambda,T,(x,y)\right)$.

An _infinite standard Young tableau_ is a pair $(\diamond_{\infty},T)$ where
$\diamond_{\infty}$ is the infinite Young diagram formed by all the boxes at
positions $(x,y)\in\mathbb{Z}^{2}$ such that $x+y$ is odd and $y>|x|$ (recall
we are using the Russian notation; see the left-hand side of Fig. 1, p. 1) and
$T:\diamond_{\infty}\to\mathbb{Z}_{\geq 1}$ is a bijection that is increasing
along rows and columns. An infinite standard Young tableau is always
(implicitly) marked at the box $(0,1)$. An example of an infinite standard
Young tableau is given in Fig. 17.

Figure 17. An example of an infinite standard Young tableau.

######  Definition 43.

Given a sequence of marked standard Young tableaux/Poissonized Young tableaux
$(\lambda_{n},T_{n},(x_{n},y_{n}))_{n}$ and an infinite standard Young tableau
$(\diamond_{\infty},T)$, we say that $(\lambda_{n},T_{n},(x_{n},y_{n}))_{n}$
_locally converges_ to $(\diamond_{\infty},T)$ if for every $h\in\mathbb{N}$,

$r_{h}(\lambda_{n},T_{n},(x_{n},y_{n}))\xrightarrow[n\to\infty]{}r_{h}\left(\diamond_{\infty},T\right).$

The space of marked standard Young tableaux/Poissonized Young tableaux and
infinite standard Young tableaux with the topology induced by the local
convergence defined in Definition 43 is a metrizable Polish space, so one can
consider convergence in distribution with respect to this local topology.

###  8.2. Local convergence for Poissonized Young tableaux

We first prove that the random infinite standard Young tableau from Definition
16 is well-defined. Recall the definitions of the functions $H_{\beta}$ and
$R_{\beta}$ of the infinite bead process $M_{\beta}$ from the discussion above
Definition 16.

######  Proposition 44.

Fix a parameter $\beta\in(-1,1)$. Then the heights

$\left\\{H_{\beta}(x,y),\,(x,y)\in\diamond_{\infty}\right\\}$

are a.s. all distinct and they have a.s. no accumulation points.

###### Proof.

The infinite bead process $M_{\beta}$ is a determinantal point process. Hence
the distribution of the heights has a joint density with respect to the
Lebesgue measure and so, it puts zero measure on vectors with two equal
coordinates. Therefore the heights
$\left\\{H_{\beta}(x,y),\,(x,y)\in\diamond_{\infty}\right\\}$ are a.s. all
distinct.

We now prove that there are a.s. no accumulation points. It is enough to prove
that $H_{\beta}(k,k+1)$ tends to $+\infty$ a.s., as $k$ tends to infinity.
Indeed, assume this is true. By symmetry $H_{\beta}(-k,k+1)$ also tends to
$+\infty$. But the interlacing condition implies that there are at most
$k^{2}$ pairs $(x,y)$ in $\diamond_{\infty}$ with
$H_{\beta}(x,y)\leq\min(H_{\beta}(k,k+1),H_{\beta}(-k,k+1))$. Therefore, if
$\min(H_{\beta}(k,k+1),H_{\beta}(-k,k+1))$ tends to $+\infty$, then the set
$\\{H_{\beta}(x,y),\,(x,y)\in\diamond_{\infty}\\}$ cannot have accumulation
points.

To prove that $H_{\beta}(k,k+1)$ tends to $+\infty$ a.s., we write

$H_{\beta}(k,k+1)-H_{\beta}(0,1)=\sum_{j=1}^{k}H_{\beta}(j,j+1)-H_{\beta}(j-1,j).$

Since bead models are translation invariant, we can use Birkhoff ergodic
theorem and conclude that, a.s.,

$\lim_{k\to\infty}\frac{H_{\beta}(k,k+1)-H_{\beta}(0,1)}{k}=\mathbb{E}\big{[}H_{\beta}(1,2)-H_{\beta}(0,1)\big{]}>0.\qed$

We now complete the proof of of Corollary 18.

###### Proof of Corollary 18.

Fix $h\in\mathbb{N}$. In order to prove the result it is enough to show that
$r_{h}\left(\lambda_{N},T_{N},\Box_{N}\right)$ converges in distribution to
$r_{h}\left(\diamond_{\infty},R_{\beta}\right)$. Recall from Section 1.5.3
that $(\diamond_{\infty},R_{\beta})$ is constructed starting from an infinite
bead process $M_{\beta}$ of skewness $\beta$ (and intensity $\alpha=1$). Let
$X_{h}$ be the height of the $h$-th bead above $0$ in the thread of index $0$
in $M_{\beta}$. We note that the restriction
$r_{h}(\diamond_{\infty},R_{\beta})$ is completely determined by the relative
positions of beads in the window
$\\{-h+1,\dots,-1,0,1,\dots,h-1\\}\times[0,X_{h}]$.

Using Skorohod representation theorem, let us assume that the convergence in
Theorem 15 holds almost surely. Then, for any fixed $A>0$, the positions of
the beads of $\widetilde{M}^{(x_{0},t_{0})}_{\lambda_{N}}$ in
$\\{-h+1,\dots,-1,0,1,\dots,h-1\\}\times(0,A)$ converge to the positions of
the same beads of $M_{\beta}$. If $X_{h}<A$, the relative positions of these
latter beads a.s. determine (thanks to Proposition 44), for $N$ large enough
(but random), both tableaux

$r_{h}(\lambda_{N},T_{N},\Box_{N})\quad\text{and}\quad
r_{h}(\diamond_{\infty},R_{\beta}),$

where we recall that $\oblong_{N}$ is the box corresponding to the first bead
of $M_{\lambda_{N}}$ above height $t_{0}$ in the $x_{0}\sqrt{N}$-th thread,
i.e. the first bead of $\widetilde{M}^{(x_{0},t_{0})}_{\lambda_{N}}$ above
height zero in the zero thread.

Therefore, conditionally on $X_{h}<A$, for $N$ large enough (but random), we
a.s. have

$r_{h}(\lambda_{N},T_{N},\Box_{N})=r_{h}(\diamond_{\infty},R_{\beta}).$

Since this holds for all $A>0$ and since $X_{h}$ is a.s. finite, we have the
almost sure convergence of $r_{h}(\lambda_{N},T_{N},\Box_{N})$ to
$r_{h}(\diamond_{\infty},R_{\beta})$ in the probability space constructed by
Skorohod representation theorem. Almost sure convergence implies convergence
in distribution of $r_{h}(\lambda_{N},T_{N},\Box_{N})$ to
$r_{h}(\diamond_{\infty},R_{\beta})$, which concludes the proof. ∎

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