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\title[DPP and random Young tableaux]{A determinantal point process approach to scaling \\ and local limits
of random Young tableaux}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author[J.~Borga, C.~Boutillier, V.~F\'eray and P.-L.~M\'eliot]{
Jacopo Borga\addressmark{1}, Cédric Boutillier\addressmark{2},
Valentin Féray\thanks{\href{mailto:valentin.feray@univ-lorraine.fr}{valentin.feray@univ-lorraine.fr}. This is an extended abstract of \cite{borga2023determinantal},
submitted elsewhere.}\addressmark{3}, \and Pierre-Loïc Méliot\addressmark{4}
}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1} Department of Mathematics, Stanford University, United States \\
\addressmark{2} LPSM, Sorbonne Université, France \\
\addressmark{3} Université de Lorraine, CNRS, IECL, F-54000 Nancy, France \\
\addressmark{4} Institut de mathématiques d'Orsay, Université Paris-Saclay,  France\\
}

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\abstract{
  We obtain scaling and local limit results for large random multirectangular Young tableaux
  via the asymptotic analysis of a determinantal point process due to Gorin and Rahman (2019). 
  In particular, we find an explicit description of the limiting surface,
  based on solving a complex-valued polynomial equation.
  As a consequence, we find a simple criterion to determine if the limiting surface is continuous in the whole domain,
  implying that, for multirectangular tableaux, the limiting surface is generically discontinuous.
  } 

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\begin{document}

\maketitle
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\section{Introduction}

Random Young diagrams form a classical theme in probability theory, starting with the work of
Logan--Shepp and Vershik--Kerov on the Plancherel measure \cite{LoganShepp1977,VershikKerov1977}.
%motivated by Ulam's problem on the typical length of the longest increasing subsequence in a uniform random permutation.
The topic is deeply connected with random permutations, random matrix theory and particle systems,
and has known an increase of interest
after the discovery of an underlying determinantal point process
for a Poissonized version of the Plancherel measure \cite{BorodinOkounkovOlshanski2000}.
It would be vain to do a complete review of the related literature, and
we refer only to \cite{HoraBookYoung,RomikLIS} for books on the topic. 
%We also refer the reader to \cref{sect:yt_pyt_bc} for precise definitions of the objects mentioned in this section.

In comparison, random Young tableaux %, which are in essence dynamic versions of random diagrams,
have a shorter history.
Motivations to study random Young tableaux range from asymptotic representation theory
to connections with other models of combinatorial probability, such as random permutations
with short monotone subsequences \cite{romik2006RandomES} 
or most notably random sorting networks; see e.g. \cite{angel2007sorting}.

As in most of the literature, we are interested in the simple model where we fix a partition $\la$ 
(or rather a sequence of growing partitions) and consider a uniform random tableau $T$ of shape $\la$. In \cite{pittel2007young}, Pittel and Romik derived a limiting surface result for uniform
random Young tableaux of {rectangular} shapes, based on the hook length formula and analytic arguments.
An earlier result of Biane in asymptotic representation theory \cite{Biane2001}
implies, in fact, the existence of such limiting surfaces for any underlying shape. 
However, getting explicit formulas for these limiting surfaces is difficult
since their description involves the Markov--Krein correspondence and the free compression of probability measures.
 More recently, entropy optimization methods have been applied to prove the existence of limiting surfaces,
extending the result to skew shapes~\cite{sun2018dimer}. These techniques lead to some natural gradient variational problems in $\R^2$ whose solutions are explicitly parameterized by $\kappa$-harmonic functions, as shown in \cite{KenyonPrause2022variational}.

Recently, in \cite{GR19}, a determinantal point process structure was discovered
for a Poissonized version of random Young tableaux.
This determinantal structure was used for a specific problem motivated by the aforementioned sorting networks, namely describing
the local limit of uniform tableaux of staircase shape around their outer diagonal \cite{GR19,gorin2022random}.

The goal of the current paper is to exploit this determinantal point process structure in order to get limiting results
for a large family of shapes.
Namely, we consider shapes obtained as dilatations of any given
Young diagram $\lambda^0$, i.e.~multirectangular diagrams.
Here is an informal description of our results.
\begin{itemize}
  \item We obtain a new description of the limiting surface corresponding to the shape $\lambda^0$, based on solving a complex-valued polynomial equation (\cref{thm:limiting_surface_formula}).
  This new description is more explicit compared to the one obtained through the existence approaches.
  
  \item This first result leads us to a surprising discontinuity phenomenon for the limiting surface corresponding to $\lambda^0$.
    More precisely, we establish a simple criterion -- some equations involving the so-called \emph{interlacing coordinates} of $\lambda^0$ -- to determine whether the limiting surface is continuous (\cref{thm:continuity}). This shows that such limiting surfaces are typically discontinuous for multirectangular tableaux.
  
  \item We also obtain a local limit result in the bulk of random Young tableaux. %(\cref{thm:cv_bead_process})
    Due to space constraints, we do not present this result in this extended abstract and refer the interested reader to the long version of the article \cite{borga2023determinantal}.
%    The limit is a new model of infinite random tableaux (\cref{defn:isyt}), 
%    constructed from the so-called \emph{random infinite bead process} (\cref{defn:bead_process}) introduced by the second author in \cite{boutillier2009bead}.
%    Interestingly (but somehow expectedly by analogy with results
%    on lozenge tilings \cite{petrov2014tilings,aggarwal2019universality}),
%    this local limit depends on the underlying shape $\la$
%    and on the chosen position in $\la$ only through a single parameter $\beta\in(-1,1)$.
\end{itemize}

\begin{remark}
In parallel to this work, explicit formulas for the limiting surfaces
of random Young tableaux have also been obtained by Prause \cite{Prause2023tableaux}
through a different method (solving a variational
problem obtained by the tangent plane method of Kenyon and Prause
\cite{KenyonPrause2022variational}).
\end{remark}

\section{Results}


\subsection{Young tableaux and height function}\label{sect:yt_pyt_bc}
Let us start by fixing terminology and notation.
A \emph{partition} of $n$ is a non-increasing list 
$\la=(\la_1,\la_2, \dots ,\la_l)$ of positive integers with 
$N = \sum_{i=1}^l\la_i$. We write $|\lambda|=N$ for the \emph{size} of the partition and $\ell(\la)=l$ for the {\em length} of the partition
and use the convention $\la_i=0$ when $i>\ell(\la)$.
We represent partitions graphically with the \emph{Russian convention},
i.e.\ for each $i \le \ell(\la)$ and $j \le \la_i$ we have a square box
whose sides are parallel to the lines $x=y$ and $x=-y$
and whose center has coordinates $(j-i,i+j-1)$;
see the left-hand side of \cref{fig:interlacing1}. 
This graphical representation is called \emph{Young diagram} of shape $\lambda$.
\begin{figure}
	\YRussian
	\Yboxdim{23pt}
	\[ \begin{array}{c}
      \newcommand\enta{\scalebox{.9}{(0,1)}}
	\newcommand\entb{\scalebox{.9}{(1,2)}}
	\newcommand\entc{\scalebox{.9}{ (-1,2)}}
	\newcommand\entd{\scalebox{.9}{(2,3)}}
	\newcommand\ente{\scalebox{.9}{(0,3)}}
	\newcommand\entf{\scalebox{.9}{(3,4)}}
	\newcommand\entg{\scalebox{.9}{ (-2,3)}}
	\newcommand\enth{\scalebox{.9}{(1,4)}}
	\newcommand\enti{\scalebox{.9}{ (-3,4)}}
	\newcommand\entj{\scalebox{.9}{(2,5)}}
	\newcommand\entk{\scalebox{.9}{ (-1,4)}}
	\young(\enta\entb\entd\entf,\entc\ente\enth\entj,\entg\entk,\enti)
 \end{array} \qquad \begin{array}{c}
\includegraphics[height=3.7cm]{Interlacing1}
\end{array}\]
	\captionsetup{width=\linewidth}
	\caption{\textbf{Left:} The Young diagram of the partition $(4,4,2,1)$ drawn in Russian convention,
	with the coordinates of each box inside it.
	\textbf{Right:} A Young tableau $T: \la \to [N]$ of shape $\la$ corresponding to the partition $(6,6,6,4,4,4,3,3)$ drawn according to the Russian convention; all the boxes are squares with area $2$. We indicate the interlacing coordinates $a_0 < b_1 < a_1 < b_2 < \dots < b_m < a_m$ below the $x$-axis.}
	\label{fig:interlacing1}
\end{figure}

When looking at a Young diagram $\lambda$, its upper boundary is the graph of a $1$-Lipschitz function, denoted by $\omega_{\lambda}:\R\to\R$,
and the diagram $\lambda$ can be encoded using the local minima and maxima of the function $\omega_{\lambda}$.
Following Kerov \cite{kerov2000anisotropic}, we denote them by
\begin{equation}\label{eq:int_coord}
	a_0 < b_1 < a_1<  b_2 < \dots < b_m < a_m, \qquad a_i,b_i\in\Z,
\end{equation}
and we call them \emph{interlacing coordinates}. See the right-hand side of \cref{fig:interlacing1} for an example.
Note that $a_0=-\ell(\la)$ and $a_m=\la_1$.
Furthermore, interlacing coordinates satisfy
$\sum_{i=0}^m   a_i = \sum_{i=1}^m  b_i$,
see, e.g., \cite[Proposition 2.4]{IvanovOlshanski2002}.


\medskip

A \emph{Young tableau} of shape $\la$ is a filling of the boxes
of $\la$ with the numbers $1, 2, \dots , N$ such that the numbers 
along every row or column are increasing. 
We see Young tableaux as  functions $T: \la \to [N]=\{1, 2, \dots , N\}$, where  the Young diagram $\la$ 
is identified with the set $\{(j-i,i+j-1), i \le \ell(\la), j \le \la_i\}$; see again the right-hand side of \cref{fig:interlacing1} for an example.
The function $T: \la \to [N]$ can be thought of as the graph of a
(non-continuous) surface above the set $\la$.

We also represent a tableau $T$ of size $N$ by its \emph{height function} $H_T$
(normalized in the second argument).
It is a map from $([a_0,a_m]\cap\Z) \times [0,1]$ to $\Z_{\geq 0}$ defined by
\begin{equation}\label{eq:defn_H}
  H_{T}(x,t)= \#\left\{ y : T(x,y) \le Nt \right\},
\end{equation}
i.e.\ $H_{T}(x,t)$ is the number of entries smaller than $Nt$ in the vertical line $\{x\} \times \Z_{\geq 0}$.
Clearly, the tableau $T$ is entirely determined by the height function $H_{T}$. Moreover, we have that
\begin{equation}\label{eq:syt_to_height}
	T(x,y) < N t\quad\text{if and only if}\quad H(x,t) > \tfrac{1}{2}(y-|x|).
\end{equation}


\subsection{Previous results: existence of a limiting height function}
\label{sect:scaling_limit}

We now look at growing Young diagrams, and the associated random tableaux.
We fix a Young diagram $\la^0$, and take our growing sequence of diagrams as dilatations of $\la^0$.
Namely, for $n > 0$, 
we define $N=N(n,\la^0)=n^2 |\la^0|$ and consider the $(n \times n)$-dilated diagram $\la_N$
obtained by replacing each box of $\la^0$ by a square of $n \times n$ boxes.
Note that $\la_N$ has size $N$.
We set $\eta=1/\sqrt{ |\la^0|}$, so that scaling $\la^0$ in both directions
by a factor $\eta$ gives a diagram of area $2$.
Finally, we let $T_N$ be a uniform random Young tableau of shape $\la_N$.
\medskip

The following convergence result for the height function of $T_N$ is proved
in \cite[Theorem 7.15]{sun2018dimer}. 
It also follows indirectly from earlier concentration results on random Young diagrams
by Biane \cite{Biane2001}; see \cite[Proposition 10.1]{sniady2022evacuation}.
\begin{theorem}[{\cite[Theorem 7.15]{sun2018dimer} and \cite[Proposition 10.1]{sniady2022evacuation}}]\label{thm:limiting_surface_existence}
Let $\la^0$ be a fixed Young diagram and $a_0,\dots,a_m$ be its interlacing coordinates
as defined in \eqref{eq:int_coord}.
We let $T_N$
be a uniform random Young tableau
of shape $\lambda_N$.
Then there exists a deterministic function 
$H^\infty: [\eta\, a_0, \eta\, a_m] \times [0,1]\to \R$ such that the following convergence in probability holds:
\begin{equation}\label{eq:lim_height}
  \frac{1}{\sqrt N} \,H_{T_N}\left(\lfloor x\sqrt N \rfloor,t\right) \xrightarrow[N \to +\infty]{} H^\infty(x,t),
\end{equation}
uniformly for all $(x,t)$ in $[\eta\, a_0, \eta\, a_m] \times [0,1]$.
\end{theorem}

In \cite{sun2018dimer}, the limiting function $H^\infty$ is implicitly found to be the unique maximizer
of a certain entropy functional subject to some boundary conditions depending on the diagram $\la^0$.
Using the approach of \cite{Biane2001,sniady2022evacuation}, for each $t\in[0,1]$,
the section $H^\infty(\cdot,t)$ is described using the free cumulants of an associated
probability measure.  Both descriptions are difficult to manipulate.
Our first result gives an alternative and more explicit description of $H^\infty$
through the solution of a polynomial equation, called the \emph{critical equation}.


\subsection{First result: a compact description of the limiting height function}
\label{subsec:def_liquid} 
Let $a_0 < b_1 < a_1<  b_2 < \dots < a_m$ be the interlacing coordinates of $\la^0$, introduced in \eqref{eq:int_coord}.
For $(x,t)$ in $[\eta\, a_0, \eta\, a_m]  \times [0,1]$, we consider the following polynomial equation, referred to throughout the paper as the \emph{critical equation}:
\begin{equation}
\label{eq:critical_intro}
	U \, \prod_{i=1}^m (x-\eta\, b_i+U) = (1-t) \, \prod_{i=0}^m (x-\eta\, a_i+U).
\end{equation}
This is a polynomial equation in the complex variable $U$ of degree $m+1$.
Using the fact that the $a_i$'s and $b_i$'s are alternating,
one can easily prove that \eqref{eq:critical_intro} has at least $m-1$ real solutions; 
see \cite[Lemma 24]{borga2023determinantal} for details. Hence it has either $0$ or $2$ non-real solutions.

\begin{definition}\label{defn:liquid_region}
	We let $L$ be the set of pairs $(x,t)$ in $[\eta\, a_0, \eta\, a_m] \times [0,1]$ such that \eqref{eq:critical_intro} has two non-real solutions and we call it {\em liquid region}.
	The complement of the liquid region in  $[\eta\, a_0, \eta\, a_m] \times [0,1]$
	will be referred to as the \emph{frozen region}.
\end{definition}

%{Several equivalent descriptions of the liquid region are given in \cref{prop:description_liquid_region}. For instance, we will show that $L$ is an open subset of $[\eta\, a_0, \eta\, a_m] \times [0,1]$ and give an explicit parametrization of its boundary, i.e.\ the so-called \emph{frozen boundary curve}.}
%
For $(x,t) \in L$, we denote by $U_c=U_c(x,t)$ the unique solution with a positive imaginary part of the critical equation~\eqref{eq:critical_intro}.
We use the notation $\Re z$ and $\Im z$ for the real and imaginary parts
of the complex number $z$.
It turns out that the limiting height function $H^\infty$ 
is expressed in terms of $U_c$ using the following simple formula.

\begin{theorem}\label{thm:limiting_surface_formula}
With the above notation,
for $(x,t)\in [\eta\, a_0, \eta\, a_m] \times[0,1]$, we have
\begin{equation*}
	H^\infty(x,t) = \frac{1}{\pi}\int_0^t \frac{\Im U_c(x,s)}{1-s} \, \bm{1}[(x,s) \in L] \, \dd{s}.
\end{equation*}
\end{theorem}
Informally, the liquid region is the limit of the region where the height function $H_{T_N}$
 is strictly increasing in the $t$-direction, and the $t$-derivative of $H_{T_N}$
 in this region is roughly given by $\sqrt N\, \Im U_c(x,s)/(\pi (1-s))$.

\subsection{Limiting surfaces and discontinuities} 
\label{ssec:limiting-surface}
It is natural to try to translate the limiting result for the height function
to a limit result for the tableau itself, seen as a discrete surface.
Namely, we set 
\begin{equation}\label{eq:domain_tableaux}
	D_{\lambda^0}:=\left\{(x,y) \in \R^2 : |x| < y < \eta \, \omega_{\la^0}(x/\eta)\right\},
\end{equation}
which is the open domain of $\mathbb R^2$ corresponding to the diagram $\lambda^0$ (in Russian convention),
normalized to have area $2$.
For $(x,y)$ in $D_{\lambda^0}$, letting $T_N$ be a uniform tableau
of shape $\la_N$, we consider 
\begin{equation}\label{eq:rescaling_T}
  \widetilde T_N(x,y):=  \frac1{N}\, T_N\left(\lfloor x \sqrt N \rfloor,\lfloor y \sqrt N \rfloor +\delta\right),
\end{equation}
where $\delta \in \{0,1\}$ is chosen so that the arguments of $T_N$ have distinct parities.
We want to find a scaling limit for the function $\widetilde T_N(x,y)$. 
%(Again, it is simple to see that considering uniform Poissonized or (classical) uniform Young tableaux of shape $\la$ is irrelevant here.)
To this end, we set for all $(x,y)\in D_{\lambda^0}$,
\begin{equation}\label{eq:Tplusminus}
	\begin{array}{l}
	T^\infty_+=T^\infty_+(x,y):= \sup \left\{t\in[0,1] \,:\, H^\infty(x,t) \le {\tfrac 1 2 (y-|x|)} \right\},\vspace{2mm}\\
		T^\infty_-=T^\infty_-(x,y):= \inf \left\{t\in[0,1] \,:\, H^\infty(x,t) \ge {\tfrac 1 2 (y-|x|)} \right\}.\end{array}
\end{equation}

Comparing \cref{eq:syt_to_height,eq:Tplusminus}, the following statement is an easy consequence of \cref{thm:limiting_surface_existence}, see \cite{borga2023determinantal} for details.
\begin{proposition}\label{prop:reg_points}
For all $\eps>0$, the following limit holds uniformly for all $(x,y) \in D_{\lambda^0}$:
\[\lim_{N \to +\infty} \mathbb P\big(\widetilde T_N(x,y) < T^\infty_- -\eps \big) 
=\lim_{N \to +\infty} \mathbb P\big(\widetilde T_N(x,y) > T^\infty_+ + \eps\big) =0.\]
\end{proposition}

%\begin{proof}
%Recalling the relation~\eqref{eq:syt_to_height} and the rescaling~\eqref{eq:rescaling_T}, we note that
%\[ \widetilde T_N(x,y) <  T^\infty_- -\eps \quad\text{if and only if}\quad H_{\la_N}\big(\lfloor x \sqrt N \rfloor,T^\infty_- -\eps\big) > \frac 1 2 (\lfloor y \sqrt N \rfloor +\delta-|\lfloor x \sqrt N \rfloor |).\]
%We claim that the latter event happens with probability tending to $0$. Indeed, \cref{thm:limiting_surface_existence} guarantees 
%the following convergence in probability uniformly for all $(x,y) \in D_{\lambda^0}$:
%\[ \lim_{N \to +\infty} \frac{1}{\sqrt N}H_{\la_N}\big(\lfloor x \sqrt N \rfloor,T^\infty_- -\eps \big) = H^\infty(x,T^\infty_- -\eps) < \tfrac 1 2 (y-|x|),\]
%where the last inequality follows by definition of $T^\infty_-$. 
%The statement for $T^\infty_+$ is proved similarly.
%\end{proof}

We let $D^{\mathrm{reg}}_{\lambda^0}$ be the set of coordinates $(x,y)\in D_{\lambda^0}$ such that 
$T^\infty_-(x,y)=T^\infty_+(x,y)$. For such points, we simply write $T^\infty(x,y)$
for this common value. 
%By definition, on $D^{\mathrm{reg}}_{\lambda^0}$, one has
%	\begin{equation}\label{eq:equation_surface}
%		H^\infty(x,T^\infty(x,y))=\tfrac 1 2 (y-|x|).
%	\end{equation}
Then \cref{prop:reg_points} implies
the following convergence in probability for  $(x,y)\in D^{\mathrm{reg}}_{\lambda^0}$:
	\begin{equation}\label{eq:lim_conv}
		\lim_{N \to +\infty} \widetilde T_N(x,y) = T^\infty(x,y),
	\end{equation}
%Note that this convergence holds uniformly on compact subsets of $D^{\mathrm{reg}}_{\lambda^0}$.
%\begin{remark}\label{rem:cont_points}
%	For any $x$ fixed, since $t\mapsto H^\infty(x,t)$ is non-decreasing, the points $(x,y)$ are in
%	$D^{\mathrm{reg}}_{\lambda^0}$ for all but countably many $y$. As a consequence, $D_{\lambda^0} \setminus D^{\mathrm{reg}}_{\lambda^0}$ has zero (two-dimensional) Lebesgue measure. Moreover, if $(x,y)\in D_{\lambda^0} \setminus D^{\mathrm{reg}}_{\lambda^0}$ and $x-\eps,x+\eps \in D^{\mathrm{reg}}_{\lambda^0}$, then
%	$T^\infty(x-\eps,y)$ and $T^\infty(x+\eps,y)$ converge respectively
%	to $T^\infty_-(x,y)$ and $T^\infty_+(x,y)$ as $\eps$ tends to $0$ with $\eps>0$.
%	Therefore the limiting surface $T_\infty$ is discontinuous at such points $(x,y)$
%	and we do not know whether $\widetilde T_N(x,y)$ converges or not.
%	This discontinuity phenomenon was overlooked in \cite[Theorem 9.1]{sun2018dimer},
%	where it is claimed that the convergence holds for all $(x,y)$ in $D_{\lambda^0}$.
%\end{remark}
On the other hand, for $(x,y)$ in $D_{\lambda^0} \setminus D^{\mathrm{reg}}_{\lambda^0}$,
we do not know whether $\widetilde T_N(x,y)$ converges at all, and the limiting surface $T^\infty$
is discontinuous at such points.

A natural question is whether such discontinuity points $(x,y)$ exist at all in $D_{\lambda^0}$.
Our second main result shows that such points indeed exist unless $\la^0$
is a rectangle, or unless its interlacing coordinates satisfy
some specific equations.
\begin{theorem}\label{thm:continuity}
	For a Young diagram $\la^0$, the following assertions are equivalent:
	\begin{enumerate}
		\item The limiting surface $T^\infty$ is continuous in the whole domain $D_{\lambda^0}$;
		\item The interlacing coordinates defined in \eqref{eq:int_coord} satisfy the system of equations:
		\begin{equation}\label{eq:cont_cond}
			\sum_{\substack{i=0\\
				i\neq i_0}}^m \frac{1}{a_{i_0}-a_i}=\sum_{i=1}^m \frac{1}{a_{i_0}-b_i}, \qquad \text{for all }i_0=1,\dots,m-1.
		\end{equation}
	\end{enumerate}
\end{theorem}

Note that when $m=1$, i.e.\ when $\la^0$ has a rectangular shape, there are no equations in the second item. 
Indeed, the limiting surface $T^\infty$ is always continuous in this case.
For $m>1$ however, the limiting surface is generically discontinuous.

\section{Examples}\label{sect:appl_intro}
In this section, we illustrate our results in the cases $m=1$ (rectangular shapes)
and $m=2$ ($L$-shapes). 
Before starting, let us note that our model and all results are invariant
when multiplying all interlacing coordinates of $\la^0$ by the same positive integers.
We will therefore allow ourselves to work with diagrams $\la^0$
with rational (non-necessarily integer) interlacing coordinates.
%The statements of this section are proved in \cref{sect:applications}.


\subsection{An explicit formula for the rectangular case}
In this section, we consider a rectangular diagram $\la_0$.
Without loss of generality, we assume $a_0=-1$ and write $r=a_1$.
Necessarily, $b_1=r-1$. Solving explicitly the critical equation \eqref{eq:critical_intro}, 
which is in this case a degree 2 polynomial equation, we get:

\begin{proposition}\label{prop:limit_shape}
	The limiting height function corresponding to a $1 \times r$ rectangular shape $\la^0$
    is given by
	\begin{equation}
      \label{eq:limit_H_rectangular}
		H^\infty_r(x,t)=\frac{1}{\pi}\int_0^t\frac{\sqrt{s (4 r - (1 + r)^2 s) + 2 (r-1) \sqrt{r} s x - r x^2}}{2 \sqrt{r} (1 - s) s} \dd{s}
	\end{equation} 
	with the convention that $\sqrt{x}=0$ if $x\leq 0$.
    Furthermore, the limiting surface $T^\infty_r$ is continuous on $D_{\lambda^0}$
    and is therefore implicitly determined by the equation
	\begin{equation}\label{eq:eq_cont_case}
		H^\infty_r(x,T^\infty_r(x,y))=\tfrac 1 2 (y - | x |).
	\end{equation}
\end{proposition}

\begin{remark}\label{rem:square_case}
	In the case $r=1$ (square Young tableaux), we get
	\begin{equation*}
		H^\infty_1(x,t)=\frac{1}{\pi} \int_{0}^{t} \frac{\sqrt{4 s - 4 s^2 - x^2}}{2 s - 2 s^2}\dd{s}.
	\end{equation*}
        The graph of the function $\alpha(x,s)=\frac{\sqrt{4 s - 4 s^2 - x^2}}{2 s - 2 s^2}$ is plotted on the left-hand side of \cref{fig:density_plot}, while the corresponding limiting surface $T^\infty_1$ is on the right.
	The above integral can be explicitly computed, 
    recovering the formula found by Pittel and Romik from \cite{pittel2007young}.
    Pittel and Romik also give formulas for the general rectangular case, which
    should coincide with \eqref{eq:limit_H_rectangular}, though we could not verify directly the equivalence of both formulae.
  \end{remark}
\begin{figure}
	\[\includegraphics[height=4cm]{Graph_density_1}
	\qquad\qquad
	\includegraphics[height=4cm]{LImit_surface}\]
%	\captionsetup{width=\linewidth}
	\caption{\textbf{Left:} The graphs of the function $\alpha(x,s)=\frac{\sqrt{4 s - 4 s^2 - x^2}}{2 s - 2 s^2}$ from \cref{rem:square_case}.
    \textbf{Right:} The corresponding limiting surface $T^\infty_1(x,y)$ for squared diagrams.
	Note that we are using two different orientations of axes in order to improve the visualization.
		\label{fig:density_plot}}
\end{figure}
%Using precisely the same proof of \cref{prop:limit_shape}, 
One can also obtain explicit formulas for $L$-shaped diagrams; since the latter expressions are involved, we decided not to display them, and to discuss examples instead.




\subsection{Two concrete examples of \texorpdfstring{$L$}{L}-shape diagrams}
\label{ssec:examples}
We now consider two specific diagrams $\la^0$ and $\widetilde \la^0$
which are both $L$-shaped (i.e.~$m=2$).
Due to the shape of the corresponding liquid regions (see the pictures in \cref{fig:Example1}),
the first one is called the \emph{heart example} and the second one the \emph{pipe example}. 
\medskip

In the heart example, the Young diagram $\la^0$ has
interlacing coordinates 
\begin{equation}\label{eq:example_1}
	a_0=-5\;<\;b_1=-4\;<\;a_1=-1\;<\;b_2=3\;<\;a_2=5.
\end{equation} 
In this case we have $|\la^0|=13$, so that $\eta=1/\sqrt{ |\la^0|}=1/\sqrt{13}$ and 
$[\eta\, a_0, \eta\, a_m]=[-5/\sqrt{13},5/\sqrt{13}]\approx[-1.39,1.39]$.

In the pipe example, the Young diagram $\widetilde \la^0$
has interlacing coordinates
\begin{equation}\label{eq:example_2}
	\widetilde a_0=-200\;<\;\widetilde b_1=-197\;<\; \widetilde a_1=-90\;<\;\widetilde b_2=10\;<\;\widetilde a_2=103.
\end{equation} 
In this case, we have $|\widetilde \la^0|=9900$, so that $\widetilde\eta=\frac{1}{30 \sqrt{11}}$
and
$[\widetilde \eta\, \widetilde a_0, \widetilde \eta\, \widetilde a_m]=[-\frac{200}{30 \sqrt{11}},\frac{103}{30 \sqrt{11}}]\approx[-2.01,1.04]$. 

\begin{figure}
	\centering
		\includegraphics[width=0.22\linewidth]{Example1-diagram}
%		\includegraphics[width=0.32\linewidth]{Example1}
		\includegraphics[width=0.22\linewidth]{Example1-liquid}
		 \includegraphics[width=0.22\linewidth]{Example1-surface}
%		\includegraphics[width=0.32\linewidth]{Example1-single}
		\includegraphics[width=0.22\linewidth]{Example1-height}
		\includegraphics[width=0.22\linewidth]{Example2-diagram}
%		\includegraphics[width=0.32\linewidth]{Example2} 
		\includegraphics[width=0.22\linewidth]{Example2-liquid}
		\includegraphics[width=0.22\linewidth]{Example2-surface_with_circle}
%		\includegraphics[width=0.32\linewidth]{Example2-single}
		\includegraphics[width=0.22\linewidth]{Example2-height}
		\caption{Figures for the heart example (top row) and for the pipe example (bottom row).
        \textbf{In each row, from left to right:} The Young diagram $\la^0$ or $\widetilde{\la}^0$ with its interlacing coordinates,
        the boundary curve of the corresponding liquid region, 
			a uniform random tableau $T_N$ of shape $\la_N$ (with respectively $N=130000$ and $N=59400$ boxes)
                        and the corresponding height function $H_{T_N}$
            (in 3D plots, the brown colour is used for small values of the surface and blue for large ones). \label{fig:Example1}}
	\end{figure}
For both examples, we have computed %(using the parametrization from \cref{prop:description_liquid_region} below)
the boundary of the liquid region
defined in \cref{defn:liquid_region}. 
Independently, we have also generated a uniform random tableau of shape $\lambda_N$
for large $N$ (using the Greene--Nijenhuis--Wilf
hook walk algorithm \cite{greene1982probabilistic}),
and we present 3D plots both of the tableau $T$ as a function from $\la$ to $[0,1]$
and of its height function $H_T$ (which is a function from $[na_0,na_m]\times [0,1]$ to $\mathbb Z_{\ge 0}$).
In both cases, the domain where the height function $H_T$ is increasing in $t$
fits very well with the liquid region,
as predicted by \cref{thm:limiting_surface_formula}. 
\medskip

An essential difference between the two examples is that  the interlacing coordinates satisfy Condition~\eqref{eq:cont_cond} in the heart example,
while this is not the case in the pipe example.
From \cref{thm:continuity}, the limiting surface is continuous
in the heart example and not in the pipe example. 
This is indeed visible on the pictures, as we now explain.

In the heart example, the intersection of the liquid region with any vertical line
is connected; in other terms, for every $x\in[\eta\, a_0,\eta\, a_m]$, the function $t \mapsto H^\infty(x,t)$ is first constant equal to $0$, then strictly increasing, and then constant equal to its maximal value.
Therefore, with the notation of \eqref{eq:Tplusminus}, 
we have $T^\infty_-(x,y)=T^\infty_+(x,y)$ for all $(x,y)$ in $D_{\la^0}$ 
and the limiting surface $T^{\infty}$ is defined and continuous on the whole set $D_{\la^0}$.
Looking at the random tableau drawn as a discrete surface, it is indeed plausible
that it converges to a continuous surface.

In the pipe example, however, we can find some $x_0$ just on the right of $\eta\, \widetilde{a}_1=-\frac{3}{\sqrt{11}}\approx-0.9$ so that
the liquid region intersects the line $x_0 \times [0,1]$ in two disjoint intervals.
The function $t \mapsto H^\infty(x_0,t)$ is then constant, equal to some value $y_0$ between these two intervals.
It follows that $T^\infty_-(x_0,y_0)<T^\infty_+(x_0,y_0)$ and the limiting surface $T^\infty$ is discontinuous at $(x_0,y_0)$.
This discontinuity can be observed on the 3D plot of the tableau $T_N$ 
(see the zoom inside the red circle on the left-hand side,
where we observe a jump in the values of $T_N$).


\section{Proof strategy}
We now discuss the proof strategy of \cref{thm:limiting_surface_formula,thm:continuity}.
Details can be found in \cite{borga2023determinantal}.

\subsection{Poissonized tableaux and determinantal point process}
Following \cite{GR19}, we define a {\em Poissonized Young tableau} of shape $\la$
as a function $\la \to [0,1]$ satisfying the same monotonicity constraints as standard tableaux.
We encode such a tableau $T$ by a set $M_T$ of particles in $\mathbb Z \times [0,1]$ defined as
\[M_T=\big\{(x,T(x,y)), (x,y) \in  \lambda\big\}.\]
A remarkable result of \cite{GR19} states that, for any shape $\la$,
if $T$ is a uniform random Poissonized tableau of shape $\la$, 
then $M_T$ is a determinantal point process with kernel
\begin{equation}\label{eq:kernel}
\scalebox{1}{$K_\la((x_1,t_1),(x_2,t_2)) =
-\frac{1}{(2\I \pi)^2}\oint_{\gamma_z}\!\oint_{\gamma_w} \frac{F_\la(z)}{F_\la(w)}\,\frac{\Gamma(w-x_1+1)}{\Gamma(z-x_2+1)}\,\frac{(1-t_2)^{z-x_2}\,(1-t_1)^{-w+x_1-1}}{z-w}\dd{w}\dd{z},$}
\end{equation}
where 
\setitemize{itemsep=1mm,topsep=1mm,parsep=0pt,partopsep=0pt,leftmargin=8mm}
\begin{itemize} 
  \item $F_\la(u) := \Gamma(u+1)\,\prod_{i=1}^\infty \frac{u+i}{u-\la_i+i} =
\frac{\prod_{i=0}^m \Gamma(u-a_i+1)}{\prod_{i=1}^m \Gamma(u-b_i+1)}$;
	\item $\gamma_w$ and $\gamma_z$ are counterclockwise contours
	containing all the integers in $[a_0,x_1]$ and in $[x_2 ,a_m]$ respectively;
  \item $\gamma_w$ and is inside (resp.\ outside) $\gamma_z$ if $t_1\geq t_2$ (resp.\ $t_1 < t_2$);
	\item the ratio $\frac{1}{z-w}$ remains uniformly bounded on the contours $\gamma_w$ and $\gamma_z$.
\end{itemize}

\subsection{Asymptotic behaviour of the kernel}
To prove \cref{thm:limiting_surface_formula}, we look for the asymptotic behaviour of the kernel in the regime
\[x_i=x_0 \sqrt N +\xi_i, \quad t_i=t_0 + \tfrac{\tau_i}{\sqrt N} \qquad (i=1,2),\]
where $(x_0,t_0)$ is fixed in $[\eta a_0,\eta a_m] \times [0,1]$.
In particular, the density of particles in $M_T$ around $(x_0\sqrt N,t_0)$, normalized by $1/\sqrt N$, is 
given by $K_\la( (x_0\sqrt N,t_0), (x_0\sqrt N,t_0) )$, corresponding hence to $\xi_1=\xi_2=\tau_1=\tau_2=0$.
In this regime, a careful asymptotic analysis
shows that the integrand in \cref{eq:kernel} behaves as
\begin{equation}
	\label{eq:equiv_integrand}
	\Int_N(W,Z) \simeq (\sqrt{N})^{\xi_2-\xi_1}
	\E^{\sqrt{N}(S(W) - S(Z))}\,h(W,Z),
\end{equation}
for some function $S$ and $h$. 
The critical equation \eqref{eq:critical_intro} corresponds to the equation $S'(U)=0$, 
i.e.~its solutions are critical points of $S$.
The idea is then to move the integration contours so that $S(W)<S(Z)$ on the new contours,
making the integrand and thus the integral tend to $0$. Moving the integration contour
may yield a residue term, which gives the non-trivial asymptotic of $K_\la( (x_0\sqrt N,t_0), (x_0\sqrt N,t_0) )$.

Let us explain briefly how this works when $(x_0,t_0)$ is in the liquid region.
By definition, in this case, $S$ has two non-real critical points, which are necessarily conjugate,
that we denote by $U_c$ and $\bar{U_c}$. Comparing $\Re S(U)$ (for generic $U$) to $\Re S(U_c)$ divides the complex plane
into regions as shown on \cref{fig:integration_contour} (the shape of those regions
is carefully justified in the long version of the paper \cite{borga2023determinantal}).
We then move the integration contours so that $S(W)<S(U_c)<S(Z)$ almost everywhere on the new contours.
Note that the new contour $\gwn$ is not inside $\gzn$, while $\gamma_w$ is inside $\gamma_z$ since $t_1=t_2$ in the case of interest.
Thus moving the contours yields a residue term associated
with the pole $Z=W$ in \cref{eq:kernel}, which can be computed explicitly.
We find
\[ \lim_{N\to +\infty} K_\la( (x_0\sqrt N,t_0), (x_0\sqrt N,t_0) ) =  \frac{\Im U_c(x_0,t_0)}{\pi (1-t_0)},\]
which implies after some work \cref{thm:limiting_surface_formula}.
\begin{figure}
	\[\includegraphics[height=5cm]{Contours_Before} 
	\qquad 
	\includegraphics[height=5cm]{Contours_After} \]
	\captionsetup{width=\linewidth}
	\caption{
	 \textbf{Left:} The yellow regions correspond to $\{\Re S(U) < \Re S(U_c)\}$, 
    while the white regions correspond to $\{\Re S(U) > \Re S(U_c)\}$. We plotted the original integration contours $\gw$ (in green)
    and $\gz$ (in purple) appearing in \cref{eq:kernel}.
The green and purple dots are respectively the $W$-poles and $Z$-poles of the integrand. 	
\textbf{Right:} The new integration contours so that $S(W)<S(Z)$ almost everywhere on the contours.
\label{fig:integration_contour}}
\end{figure}


We note that the general proof strategy is standard in integrable probability but needs many careful estimations
and arguments to justify the existence of the appropriate contours, and the asymptotic behaviour of the various terms (see \cite{borga2023determinantal}).

\subsection{Characterization of continuous limit shapes}
We now discuss the proof of \cref{thm:continuity}.
Looking at the shape of the liquid regions in \cref{fig:Example1} and at the discussions
on the heart and pipe examples, we see that
the limit shape is continuous if and only if the tangent vectors to the boundary
of the liquid region at its cusp points are all vertical. 
The boundary of the liquid region is precisely the set of points $(x,t)$
where the discriminant of the polynomial equation \eqref{eq:critical_intro} vanishes,
see~\cite[Proposition 27]{borga2023determinantal}. 
Using this description, we can obtain an explicit parametrization of
this boundary curve, and compute the tangent vectors at its cusp points.
Each cusp point gives one of the condition given in \cref{eq:cont_cond},
concluding the proof of the theorem.


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\end{document}
