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\title{The free boundary Schur process and applications}

\author{Dan Betea\addressmark{1,2}, J\'er\'emie Bouttier\addressmark{1,3}, Peter Nejjar\addressmark{4}\thanks{Partially supported by ERC Advanced Grant No. 338804.} \and Mirjana Vuleti\'c\addressmark{5}}

\address{\addressmark{1}D\'epartement de math\'ematiques et applications, \'Ecole normale sup\'erieure, F--75231 Paris Cedex 05 \\
  \addressmark{2}IRIF, CNRS et Universit\'e Paris Diderot, Case 7014, F--75205 Paris Cedex 13 \\
  \addressmark{3}Institut de Physique Th\'eorique, Universit\'e Paris--Saclay, CEA, CNRS, F--91191 Gif--sur--Yvette \\
  \addressmark{4}IST Austria, 3400 Klosterneuburg, Austria \\
  \addressmark{5}Department of Mathematics, University of Massachusetts Boston, Boston, MA 02125, USA\\}

\received{\today}

\abstract{We study the Schur process with two free boundaries, a generalization of the original Schur process of Okounkov and Reshetikhin. We compute its correlation functions for arbitrary specializations and apply the result to the asymptotics of symmetric last passage percolation models, symmetric plane partitions and plane overpartitions.}

\resume{Nous \'etudions le processus de Schur avec deux bords libres, une g\'en\'eralisation du processus de Schur original de Okounkov et Reshetikhin. Nous calculons ses fonctions de corr\'elation pour des sp\'ecialisations arbitraires et utilisons le r\'esultat pour analyser asymptotiquement les mod\`eles de percolation de dernier passage sym\'etrique, de partitions planes sym\'etriques et de surpartitions planes.}

\keywords{Schur process, free boundaries, pfaffian point process, plane partitions, last passage percolation}


\begin{document}

\maketitle

\section{Introduction and basic definitions}


In this paper, we introduce and study the Schur process with free
boundaries, which is a generalization of the Schur process introduced
by Okounkov and Reshetikhin \cite{or}, and which is a random sequence
of integer partitions whose probability is given as a product of Schur
functions.

Recall that an (integer) \emph{partition} $\lambda$ is a nonincreasing
sequence of nonnegative integers $\lambda_1 \geq \lambda_2 \geq \cdots$ which
vanishes eventually. We denote by
$|\lambda|=\sum_{i \geq 1} \lambda_i$ the \emph{size} of $\lambda$. We write $\mu \subset \lambda$ if $\lambda$ and $\mu$ are two
partitions such that $\lambda_i \geq \mu_i$ for all $i$.
The \emph{free boundary Schur process} of length $N$ is a measure over
the set of sequences of partitions of the form
\begin{equation}
    \label{eq:seq}
    \mu^{(0)} \subset \lambda^{(1)} \supset \mu^{(1)} \subset \cdots
    \supset \mu^{(N-1)} \subset \lambda^{(N)} \supset \mu^{(N)}
\end{equation}
which assigns to any such sequence $(\vec{\lambda},\vec{\mu})$ an
(unnormalized) weight
\begin{equation}
  \label{eq:fbschur}
  \mathcal{W}(\vec{\lambda},\vec{\mu}) := u^{|\mu^{(0)}|} v^{|\mu^{(N)}|}
  \prod_{k=1}^N \left(s_{\lambda^{(k)}/\mu^{(k-1)}}\left(\rho_k^+\right)
  s_{\lambda^{(k)}/\mu^{(k)}}\left(\rho_k^-\right) \right).
\end{equation}
Here $u,v$ are complex parameters, $s_{\lambda/\mu}$ is the skew Schur
function associated with the skew shape $\lambda/\mu$ and the
$\rho_k^\pm$, $k=1,\ldots,N$ are complex specializations of the ring
$\Lambda$ of symmetric functions. We refer the reader to the standard
textbooks \cite[Chapter 1]{mac} and \cite[Chapter 7]{sta} for
definitions. We follow the convention used e.g.\ in \cite{br} of
writing $s_{\lambda/\mu}(\rho)$ in lieu of $\rho(s_{\lambda/\mu})$.

In the case when $u=v=0$, the weight
$\mathcal{W}(\vec{\lambda},\vec{\mu})$ vanishes unless $\mu^{(0)}$ and
$\mu^{(N)}$ are both equal to the empty partition $\emptyset$, and we
recover the original Schur process of \cite{or}. In the case when
$u=1$ and $v=0$, we recover after an inessential change the pfaffian
Schur process of \cite{br}.

Of particular importance for applications is the case where the
specializations are the so-called $\alpha$-, $\beta$- and $\gamma$-
specializations. For $\rho$ an $\alpha$-specialization (i.e.\ a
specialization in a single variable), the quantity
$s_{\lambda/\mu}(\rho)$ is nonzero if and only if $\lambda/\mu$ is a horizontal
strip (which we write $\lambda \succ \mu$ for short). This kind of
specialization is useful for applications to lozenge tilings and plane
partitions \cite{or}. Similarly, for $\rho$ a $\beta$-specialization,
$s_{\lambda/\mu}(\rho)$ is nonzero if and only if $\lambda/\mu$ is a vertical
strip (which we write $\lambda \succ' \mu$). By alternating $\alpha$-
and $\beta$-specializations we obtain sequences in bijection with
domino tilings such as those of the Aztec diamond or plane
overpartitions \cite{bcc}. Finally, $\gamma$- (alias exponential)
specializations arise in applications to last passage percolation
\cite{Joh05}. In all these cases, the weight
$\mathcal{W}(\vec{\lambda},\vec{\mu})$ is typically nonnegative and,
provided that the \emph{partition function} (total mass of the
measure) is finite, we may normalize $\mathcal{W}$ to obtain an actual
probability distribution.

\paragraph{Outline.} In this extended abstract, we present our
fundamental theorem on the free boundary Schur process, which is a
general expression for its \emph{correlation functions}. We first
discuss the partition function in \cref{sec:partitionfun} as a
warm-up, before considering correlation functions in
\cref{sec:correlationfun}. In \cref{sec:lpp,sec:pp} we apply our results to study asymptotics of respectively
last passage percolation and plane partition models with a symmetry
coming from the Schur process with one free boundary, a special case
of our main result. More details, applications and complete proofs
will appear in the journal version of this paper, in preparation.


\section{Partition function}
\label{sec:partitionfun}

A first basic result concerning the free boundary Schur process is an
expression for the \emph{partition function} $Z$, defined as the sum
of the weights \eqref{eq:fbschur} over all sequences of the form
\eqref{eq:seq}. To write it down in general form, we make use of the
following common notations. Recall that the generating function
$H(\rho;t)=\sum_{n\geq 0} h_n(\rho)t^n$, where $h_n$ is the complete
homogeneous symmetric function of degree $n$, determines the
specialization $\rho$.  We use $\rho \cup \rho'$ and $s\rho$ ($s$
being a scalar) to denote the specializations defined by
\begin{equation}
  H(\rho \cup \rho';t) := H(\rho;t) H(\rho';t), \qquad
  H(s\rho;t) := H(\rho;st).
\end{equation}
We also set
\begin{equation}
  \label{eq:hdef}
  H(\rho;\rho') := \sum_{\lambda \in \Par} s_\lambda(\rho) s_\lambda(\rho'),
  \qquad \tilde{H}(\rho) := \sum_{\lambda \in \Par} s_\lambda(\rho),
\end{equation}
where $\Par$ denotes the set of all partitions.

\begin{prop}
  \label{prop:fbz} Assume $|uv|<1$. The partition function of the free boundary Schur process is
  \begin{equation}
   \label{eq:fbz}
    Z = \prod_{1 \leq k \leq \ell \leq N} H(\rho_k^+;\rho_\ell^-)
    \prod_{n \geq 1} \frac{\tilde{H}(u^{n-1}v^{n}\rho^+)
       \tilde{H}(u^{n}v^{n-1}\rho^-) H(u^{2n}\rho^+;v^{2n}\rho^-)}{1-u^n v^n},
  \end{equation}
  where $\rho^\pm = \bigcup_{i=1}^N \rho^{\pm}_i$.
\end{prop}


This result was essentially proved in \cite[Section 5.3]{bcc}, and we
now recall briefly its derivation in our current notations, as this
will be useful for the next section. We consider the
infinite-dimensional Hilbert space $\B$ (\emph{bosonic Fock space})
spanned by the orthonormal basis $\ket{\lambda}, \lambda \in \Par$
(here we make use of the bra-ket notation). For $\rho$ a
specialization, we define the \emph{vertex operators}
$\Gamma_\pm(\rho)$ by
\begin{equation}
  \label{eq:schurelem}
  \bra{\lambda} \Gamma_+(\rho) \ket{\mu} =
  \bra{\mu} \Gamma_-(\rho) \ket{\lambda} =
  s_{\mu/\lambda}(\rho), \qquad \lambda,\mu \in \Par.
\end{equation}
Following \cite[Section 5.3]{bcc}, we also introduce the \emph{free
  boundary states}
\begin{equation}
  \label{eq:fbs}
  \vfv := \sum_{\lambda \in \Par} v^{|\lambda|} \ket{\lambda}, \qquad
  \ufcv := \sum_{\lambda \in \Par} u^{|\lambda|} \bra{\lambda}, \qquad
  u,v \in \C.
\end{equation}
Then, by the transfer-matrix method, the partition function can be
rewritten as
\begin{equation}
  \label{eq:Zmat}
  Z = \ufcv \Gamma_+(\rho_1^+) \Gamma_-(\rho_1^-) \cdots
  \Gamma_+(\rho_N^+) \Gamma_-(\rho_N^-) \vfv.
\end{equation}

To evaluate this product, we make use of the following relations:
\begin{gather}
  \label{eq:gamcomm}
  \Gamma_+(\rho) \Gamma_-(\rho') = H(\rho;\rho')
  \Gamma_-(\rho') \Gamma_+(\rho), \\
  \label{eq:gamcomb}
  \Gamma_\pm(\rho) \Gamma_\pm(\rho') = \Gamma_\pm(\rho \cup \rho'),\\
  \label{eq:refl}
  \Gamma_+(\rho) \vfv = \tilde{H}(v \rho) \Gamma_-(v^2 \rho) \vfv,
  \qquad \ufcv \Gamma_-(\rho) = \tilde{H}(u \rho) \Gamma_+(u^2 \rho)
  \ufcv.
\end{gather}
(These relations are respectively tantamount to the Cauchy identity,
the ``branching rule'' and the Littlewood identity for Schur
functions.)

More precisely, we first ``commute'' in \eqref{eq:Zmat} the $\Gamma_+$
to the right and the $\Gamma_-$ to the left using \eqref{eq:gamcomm}
and \eqref{eq:gamcomb}, to yield
$Z = \prod_{1 \leq k \leq \ell \leq N} H(\rho_k^+;\rho_\ell^-) \cdot
\ufcv \Gamma_-(\rho^-) \Gamma_+(\rho^+) \vfv$.
Next we make use of the ``reflection relations'' \eqref{eq:refl} to
write
\begin{equation*}
  \ufcv \Gamma_-(\rho^-) \Gamma_+(\rho^+) \vfv = \tilde{H}(u \rho^-)
  \tilde{H}(v \rho^+) \ufcv \Gamma_+(u^2 \rho^-) \Gamma_-(v^2 \rho^+)
  \vfv.
\end{equation*}
By iterating the same manipulations, we ``bounce'' the $\Gamma$'s back
and forth on the boundaries and get extra $H$/$\tilde{H}$ factors at
each commutation/reflection, the specializations $\rho^\pm$ being
multiplied by a factor $u^2$ or $v^2$ at each reflection. We end up
with \eqref{eq:fbz} by noting that the $\Gamma$'s tend to the identity
operator as the number of bounces tends to infinity, since $|uv|<1$.


\section{Correlation functions}
\label{sec:correlationfun}

Following \cite{or}, we define the \emph{correlation functions} of the
free boundary Schur process as the probabilities $\rho(U)$ that the
random point configuration
\begin{equation*}
  \mathfrak{S}(\vec{\lambda},\vec{\mu}) := \left\{
    \left(i,\lambda_j-j+\frac{1}{2} \right), 1 \leq i \leq N , j \geq 1
  \right\}
\end{equation*}
contains a given set $U$, with $U$ running over all finite subsets of
$\Z \times \Z'$, $\Z':=\Z+1/2$. (Without loss of generality one can
disregard the partition sequence $\vec{\mu}$.)

For the original Schur process ($u=v=0$), the point process
$\mathfrak{S}(\vec{\lambda},\vec{\mu})$ is known to be determinantal
\cite{or}, while for one free boundary ($uv=0$) it is pfaffian
\cite{br}. For two free boundaries ($u,v$ generic), it is neither in
general, but we shall see just below that a closely related point
process is pfaffian. The situation is reminiscent of the periodic
Schur process, studied by Borodin \cite{BorCyl}, which becomes
determinantal after performing a similar transformation.

Let $t$ be an arbitrary complex parameter. The
\emph{shift--mixed\footnote{This denomination is borrowed from Borodin
    \cite{BorCyl}.}  (free boundary Schur) process} of length $N$ is a
measure over the set of tuples $(\vec{\lambda},\vec{\mu},d)$ with
$(\vec{\lambda},\vec{\mu})$ is a sequence of partitions of the form
\eqref{eq:seq} and $d$ an integer, where to each such tuple we assign
a weight
\begin{equation}
  \label{eq:smfbschur}
  \widehat{\mathcal{W}}(\vec{\lambda},\vec{\mu},d) := t^{2d} (uv)^{2d^2}
  \mathcal{W}(\vec{\lambda},\vec{\mu}).
\end{equation}
We readily see that $d$ is ``independent'' of
$(\vec{\lambda},\vec{\mu})$ and that the partition function of the
shift-mixed process reads $\widehat{Z} = \theta_3(t^2;(uv)^4) Z$, with
$\theta_3(t;q)=\sum_{n \in \Z}t^{n}q^{n^2/2}$ a Jacobi theta
function. To $(\vec{\lambda},\vec{\mu},d)$ we associate the
shift-mixed point configuration
\begin{equation*}
  \mathfrak{S}(\vec{\lambda},\vec{\mu},d) := \mathfrak{S}(\vec{\lambda},\vec{\mu}) + (0,2d) = \left\{
    \left(i,\lambda_j-j+\frac{1}{2} + 2d\right), 1 \leq i \leq N , j \geq 1
  \right\}
\end{equation*}
and we define the shift-mixed correlation function $\hat{\rho}(U)$ as
the sum of the weights of all tuples $(\vec{\lambda},\vec{\mu},d)$
whose associated point configuration contains the subset
$U \subset \Z \times \Z'$, divided by the partition function
$\widehat{Z}$ (when the weights are all real nonnegative, this is just
the probability that $\mathfrak{S}(\vec{\lambda},\vec{\mu},d)$
contains $U$).

The non shift-mixed process sits inside this process as the
restriction to $d=0$. In particular, $\rho(U)$ can be expressed as the
coefficient of $t^0$ in $\theta_3(t^2;(uv)^4) \hat{\rho}(U)$.

\begin{thm} \label{main_thm}
  Assume $|uv| < 1$ and that there exists an $R \in (0,1)$ such that for all specializations $\rho^{\pm}_s, 1 \leq s \leq N$ we have $p_m(\rho^{\pm}_s)/m \in O(R^m)$ as $m \to \infty$. Then for
  $U=\{(i_1,k_1),\ldots,(i_n, k_n)\}$, the $n$--point correlation function of the shift--mixed process is given by the following $2n \times 2n$ pfaffian
  \begin{equation*}
    \label{eq:fbcorr}
    \hat{\rho}(U) = \pf [ K(i_{\alpha},k_{\alpha};i_{\beta},k_{\beta}) ]_{1 \leq \alpha < \beta \leq 2n}
  \end{equation*}
\noindent where $K(i, k; i', k')$ is represented by the $2 \times 2$ matrix kernel
  \begin{equation*}
    K(i, k; i', k') = (K_{a,b}(i, k; i', k'))_{1 \leq a,b \leq 2}
  \end{equation*}

\noindent given by
\begin{equation*}
  \begin{split}
    & \qquad K_{1,1}(i, k; i', k') = \\
    &\int_w \int_z F(i, z) F(i', w) \cdot \frac{((uv)^2; (uv)^2)^2_{\infty} \theta_{(uv)^2}(\frac{w}{z})\theta_3 \left( ( \frac{t zw}{v^2})^2 ; (uv)^4 \right)}  {(- \frac{v}{z}, - \frac{v}{w}, uz, uw; uv)_{\infty} \theta_{(uv)^2} (u^2 zw) (uv; uv)_{\infty}}  \cdot \frac{v^2}{t z^{k+\frac{1}{2}} w^{k' + \frac{3}{2}}} \dx \mathbb{T}, \\
   \\
    & \qquad K_{1,2}(i, k; i', k') = \\
    & \int_w \int_z \frac{F(i, z)}{F(i', w)} \cdot \frac{((uv)^2; (uv)^2)^2_{\infty} \theta_{(uv)^2}(u^2 zw)\theta_3 \left( ( \frac{t w}{z})^2 ; (uv)^4 \right)}  {( \frac{v}{z}, - \frac{v}{w}, - uz, uw; uv)_{\infty} \theta_{(uv)^2} (\frac{w}{z}) (uv; uv)_{\infty}}  \cdot \frac{w^{k'-\frac{1}{2}}}{z^{k-\frac{1}{2}}} \dx \mathbb{T} , \\
 \\
    & \qquad K_{2,1}(i, k; i', k') = \\
    & \int_w \int_z  \frac{F(i', w)}{F(i, z)} \cdot \frac{((uv)^2; (uv)^2)^2_{\infty} \theta_{(uv)^2}(u^2 zw)\theta_3 \left( ( \frac{t w}{z})^2 ; (uv)^4 \right)}  {( \frac{v}{z}, - \frac{v}{w}, - uz, uw; uv)_{\infty} \theta_{(uv)^2} (\frac{w}{z})  (uv; uv)_{\infty}} \cdot \frac{z^{k+\frac{1}{2}}}{w^{k'+\frac{1}{2}}} \dx \mathbb{T}, \\
   \\
    & \qquad K_{2,2}(i, k; i', k') = \\
    & \int_w \int_z  \frac{1}{F(i, z) F(i', w)} \cdot \frac{((uv)^2; (uv)^2)^2_{\infty} \theta_{(uv)^2}(\frac{w}{z})\theta_3 \left( ( \frac{t v^2}{zw})^2 ; (uv)^4 \right)}  {(\frac{v}{z}, \frac{v}{w}, -uz, -uw; uv)_{\infty} \theta_{(uv)^2} (u^2 zw)  (uv; uv)_{\infty}}\cdot v^2 t z^{k-\frac{1}{2}} w^{k'-{\frac{3}{2}}} \dx \mathbb{T}, \\
  \end{split}
\end{equation*}

\noindent where $\dx \mathbb{T} = \frac{\dx z \dx w}{(2 \pi \sqrt{-1})^2 zw}$, and where $F$, for $i \in \N$ and $z \in \C$, is defined by
\begin{align}
\label{eq:F}
F(i, z) & = \frac{H \left(\rho^+_{[1, i]} \cup \left[\bigcup_{l \geq 0} u^2 (uv)^{2l} \rho^- \right] \cup \left[ \bigcup_{l \geq 1} (uv)^{2l} \rho^+ \right] ; z \right)}{H \left(\rho^-_{[i, N]} \cup \left[ \bigcup_{l \geq 0} v^2 (uv)^{2l} \rho^+ \right] \cup \left[ \bigcup_{l \geq 1} (uv)^{2l} \rho^- \right] ; z^{-1} \right)},
\end{align}
with $\rho^\pm_{[a, b]} = \bigcup_{i=a}^b \rho^{\pm}_i$ and  $\rho^\pm= \rho^\pm_{[1, N]}$.
The contours in each of the four integrals are simple positively oriented contours around 0 satisfying $\left| \frac{v}{z} \right| < 1, \left| \frac{v}{w} \right| < 1, |uz| < 1,|uw| < 1$; and in the $K_{1,2}$ and $K_{2,1}$ case they in addition satisfy $\left| \frac{w}{z} \right| < 1$. Here, $\theta_3(t;q)=\sum_{n \in \Z}t^{n}q^{n^2/2}$, $(x;t)_\infty=\prod_{i\geq0}(1-xt^i)$ is the t-Pochhammer symbol and $\theta_t(x)=(x;t)_\infty(t/x;t)_\infty$.
\end{thm}

There are some additional constraints on the contours coming from the
specializations, all in terms of $R$, which are omitted here.

The remainder of this section is devoted to a sketch of the proof of
\cref{main_thm}, which relies on the free fermion/infinite
wedge formalism, see for instance \cite[Chapter 14]{kac}. The
notations that we use here are essentially those of
\cite[Appendix~A]{oko} and \cite{bbccr}.

We say that a subset $S$ of $\Z'$ is \emph{admissible} if both
$S_+ := S \setminus \Z'_{< 0}$ and $S_- := \Z'_{< 0} \setminus S$ are
finite.
There is a well--known bijection between $\Par \times \Z$ and the set
$\s$ of such admissible subsets, given by
$
  \mathfrak{S}(\lambda,c)=\{ \lambda_i - i + 1/2 + c ,\ i \geq 1\}, \quad \lambda \in \Par, c \in \Z.
$
The \emph{fermionic Fock space}, denoted $\F$, is the infinite
dimensional Hilbert space spanned by the orthonormal basis
$\ket{S}, S \in \s$. For $\lambda$ a partition and $c$ an integer we
introduce the shorthand notations
$\ket{\lambda,c} := \ket{\mathfrak{S}(\lambda,c)}$,
$\ket{\lambda} := \ket{\lambda,0}$ and $\ket{c} := \ket{\emptyset,c}$.
The vector $\vv$ is called the \emph{vacuum}.

It appears that the bosonic Fock space $\B$ considered in
\cref{sec:partitionfun} can be seen as a subspace of
$\F$. Actually, we have the orthogonal decomposition
$\F = \oplus_{c \in \Z} \F_c$ where $\F_c \simeq \B$ is the subspace
spanned by the $\ket{\lambda,c}, \lambda \in \Par$. The vertex
operators $\Gamma_\pm(\rho)$ act naturally on $\F$ diagonally with
respect to this decomposition, and the free boundary states
\eqref{eq:fbs} are seen as elements of $\F_0$.
%
For $k \in \Z'$, we define the \emph{fermionic operators} $\psi_k$ and
$\psi_k^*$ by
\begin{equation*}
  \psi_k \ket{S} =
  \begin{cases}
    0 & \text{if $k \in S$} \\
    (-1)^j \ket{S \cup \{k\}} & \text{if $k \notin S$} \\
  \end{cases}, \qquad
  \psi_k^* \ket{S} =
  \begin{cases}
    (-1)^j \ket{S \setminus \{k\}}  & \text{if $k \in S$} \\
    0 & \text{if $k \notin S$} \\
  \end{cases}
\end{equation*}
where $j = |S \cap \Z'_{> k}|$. They satisfy the
canonical anticommutation relations
\begin{equation*}
  \label{eq:car}
  \{ \psi_k, \psi_\ell^* \} = \delta_{k,\ell}, \qquad
  \{ \psi_k, \psi_\ell \} = \{ \psi_k^*, \psi_\ell^* \} = 0, \qquad
  k,\ell \in \Z'
\end{equation*}
where $\{a,b\}:=ab+ba$, and obey $\psi_k \vv = \psi_{-k}^* \vv = 0$
for $k<0$. The generating series
\begin{equation*}
 \psi(z) = \sum_{k \in \Z'} \psi_k z^k, \qquad
 \psi^*(w) = \sum_{k \in \Z'} \psi^*_k w^{-k},
\end{equation*}
are known to satisfy
\begin{equation*}
 \Gamma_\pm(\rho) \psi(z) = H(\rho;z^{\pm 1}) \psi(z) \Gamma_\pm(\rho), \qquad
  \Gamma_\pm(\rho) \psi^*(w) = H(\rho;w^{\pm 1})^{-1} \psi^*(w) \Gamma_\pm(\rho).
\end{equation*}

Let $U=\{(i_1,k_1),\ldots,(i_n, k_n)\}$ where
$1 \leq i_1 \leq i_2 \leq \dots \leq i_n \leq N$ and
$k_1, \dots, k_n \in \Z'$. Using the above definitions and the
transfer-matrix method as in \cref{sec:partitionfun}, we see
that the correlation function $\rho(U)$ can be written as
\begin{equation}
  \label{eq:rhotm}
 \rho(U) = \frac{1}{Z} \ufcv \cdots \Gapl(\rho^+_{i_1}) \psi_{k_1} \psi^*_{k_1} \Gami(\rho^-_{i_1}) \cdots \Gapl(\rho^+_{i_n}) \psi_{k_n} \psi^*_{k_n} \Gami(\rho^-_{i_n})  \cdots \vfv.
\end{equation}
We can try to evaluate this expression along the same lines as in
\cite{or} but we lack a crucial ingredient, namely Wick's formula, which
does not hold anymore in the presence of general free boundary states.
To fix this problem, we are led to introducing the \emph{extended free
  boundary states} defined by
\begin{equation}
  \vtfv := e^{X(v,t)} \vv, \qquad
  \utfcv := \vcv e^{X(\bar{u},\bar{t})^*},
\end{equation}
where $X$ is defined by
\begin{equation*}
  X(v,t) :=
  \sum_{\substack{(k,\ell) \in \Z'^2\\k>\ell}} \tilde{\psi}_k(v,t) \tilde{\psi}_\ell(v,t), \qquad
 \tilde{\psi}_i(v,t) :=
  \begin{cases}
    t^{1/2} v^i \psi_i & \text{for $i \in \Z'_{>0}$,} \\
    (-1)^{i+1/2} t^{-1/2} v^{-i}  \psi_i^* & \text{for $i \in \Z'_{<0}$.} \\
  \end{cases}
\end{equation*}

\begin{prop}
  \label{prop:vfexp}
  We have
  \begin{equation*}
    \vtfv = \sum_{(\lambda,c) \in \Par \times 2\Z} t^{c/2} v^{|\lambda|+c^2/2} \ket{\lambda,c}, \qquad
    \utfcv = \sum_{(\lambda,c) \in \Par \times 2\Z} t^{c/2} u ^{|\lambda|+c^2/2} \bra{\lambda,c},
  \end{equation*}
  and hence $\vfv = \Pi_0 \vtfv$ and $\ufcv = \utfcv
  \Pi_0$, where $\Pi_0$ is the orthogonal projector onto $\F_0$.
\end{prop}

\begin{proof}
  Expand the exponential $e^{X(v,t)}$ as a series and notice that it
  involves a sum of monomials
  $\tilde{\psi}_{i_1} \cdots \tilde{\psi}_{i_{2r}}$. Use the
  anticommutations to show that, by reordering the indices
  $i_1>\cdots>i_{2r}$, each ordered monomial appears with coefficient
  $1$. Observe that, by multiplying by $\vv$ on the right, we obtain a
  sum over all states $\ket{\lambda,c}$ with even $c$.
\end{proof}

Combining \cref{prop:vfexp} with \eqref{eq:rhotm}, we find
that the shift-mixed correlation function reads
\begin{equation}
  \label{eq:rhohtm}
 \hat{\rho}(U) = \frac{1}{\widehat{Z}} \utfcv \cdots \Gapl(\rho^+_{i_1}) \psi_{k_1} \psi^*_{k_1} \Gami(\rho^-_{i_1}) \cdots \Gapl(\rho^+_{i_n}) \psi_{k_n} \psi^*_{k_n} \Gami(\rho^-_{i_n})  \cdots  \vtfv.
\end{equation}
Now we have the previously missing ingredient:

\begin{prop}[Wick's formula for extended free boundary states]
  \label{prop:wickgen}
  Let $\varPsi$ be the vector space spanned by the $\psi_k$
  and $\psi_k^*$, $k \in \Z'$.  For $\phi_1,\ldots,\phi_{2n} \in
  \varPsi$, we have
  \begin{equation*}
   \frac{\utfcv \phi_1 \cdots \phi_{2n} \vtfv}{\utfvtf} = \pf A
  \end{equation*}
  where $A$ is the antisymmetric matrix defined by
  $A_{ij}=\utfcv \phi_i \phi_j \vtfv/\utfvtf$ for $i<j$.
\end{prop}

Using similar tricks as in the proof of \cref{prop:fbz}
first and then \cref{prop:wickgen} (whose proof is omitted
here), we can show that the shift-mixed process is a Pfaffian process
with the correlation kernel $K(i, k; i', k')$ equal to
 \begin{equation*}
 \begin{split}
 \left( \begin{array}{cc} [ z^{k} w^{k'} ] F(i, z) F(i', w) \utfcv \psi(z) \psi(w) \vtfv & \left[ \frac{z^{k}}{w^{k'}} \right] \frac{F(i, z)}{F(i', w)} \utfcv \psi(z) \psi^*(w) \vtfv \\ \left[ \frac{w^{k'}}{z^{k}} \right] \frac{F(i', w)}{F(i, z)} \utfcv \psi^*(z) \psi(w) \vtfv & \left[  \frac{1}{z^{k} w^{k'}} \right] \frac{1}{F(i, z) F(i', w)} \utfcv \psi^*(z) \psi^*(w) \vtfv \end{array} \right),
  \end{split}
\end{equation*}
where $F$ is defined in \eqref{eq:F}. Computing the four expectations of the form $\utfcv \psi(z) \psi(w) \vtfv$ appearing in the kernel we obtain \cref{main_thm}.


\section{Last passage percolation}
\label{sec:lpp}

Last passage percolation (LPP) is a probabilistic model which can be seen as a  particle system (TASEP) (see e.g. \cite{FMP09})  or a stochastic growth model which belongs to the KPZ universality class \cite{Co12}. To define it,
fix $(1,1)$ as starting point, and  an endpoint $P\in \Z^{2}.$  A collection of points $\pi(l)\in \Z^{2}, l=0,\ldots,s$ is an up--right path from $(1,1)$ to $P$ if $\pi(0) =  (1,1)$, $\pi(s) = P $ and $\pi(l+1)-\pi(l)\in\{(1,0),(0,1)\}$.
Attach now to each point $(i,j)\in \Z^{2}$ a random waiting time $\omega_{i,j},$ the  $(\omega_{i,j})_{i,j\in \Z}$ are independent and nonnegative.
The LPP time from $(1,1)$ to $P$ is then the longest way to get from $(1,1)$ to $P$ along an up--right path, i.e. one takes the maximum over all up--right paths from $(1,1) $ to $P$ and sets
\begin{equation}\label{LPP}
L_{(1,1) \to P}=\max_{\pi:(1,1) \to P}\sum_{(i,j)\in\pi} \omega_{i,j}.
\end{equation}
One is interested in the behavior of $L_{(1,1)\to (N,  N)}$ as $N \to \infty$, and expects laws of random matrix theory (RMT) to appear in the limit, see \cref{LPPthm}.
 The link from LPP to  the free boundary Schur process comes from the fact that under the RSK bijection LPP times become the length of the first part  of random partitions, hence the LPP time \eqref{LPP} becomes the gap probability of a Schur process with a free boundary.
When the $(\omega_{i,j})_{i,j\in \Z}$ are geometric random variables as in \eqref{symweights}, the correlation functions are given as pfaffians by  \cref{main_thm}. Then by general theory \cite{Joh05} the  law of an LPP time becomes a series (Fredholm pfaffian), see \eqref{FPF}. To illustrate, consider the case of symmetric weights
\begin{equation}\label{symweights}
  \omega_{j,i}=\omega_{i,j}\sim
\begin{cases}\vspace{0.3cm}
g(q) \quad & \mathrm{if}\quad  i\neq j \\
 g(\alpha \sqrt{q})\quad& \mathrm{if}\quad  i=j
\end{cases}
\end{equation}

\noindent for $q\in (0,1)$ and $\alpha \in (0,1/\sqrt{q})$ and $\Prob(g(q)=k)=q^{k}(1-q)$ for $k\in \N_0$. In this case  the approach outlined above leads to ($K^{\alpha,N}$ an explicit, antisymmetric  kernel, $s\in \N$)
\begin{equation}\label{FPF}
\Prob(L_{(1,1)\to (N,N)}<s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\sum_{x_1 > s-1/2}\cdots \sum_{x_n > s-1/2}\pf(K^{\alpha,N}(x_i,x_j))_{i,j=1,\ldots,n}.
\end{equation}
More generally, we obtain a Fredholm pfaffian formula for (certain) multipoint distributions of LPP. An alternate derivation follows from \cite{br} Theorem 3.3.  This is done in \cite{BBCT16}, where the authors perform standard steepest descent analysis to study asymptotics of the symmetric LPP model with exponential weights (extra care is needed in the GSE case).

The formula \eqref{FPF} allows to do  $N \to \infty$ asymptotics, leading to the following theorem, which was originally established using Riemann--Hilbert techniques in \cite{br2}.
\begin{thm}\label{LPPthm}
Consider the symmetric LPP time $L_{(1,1)\to (N,N)}$ given by  \eqref{LPP} and weights \eqref{symweights} with
$\alpha <1$ and let  $c_q =\frac{1-\sqrt{q}}{q^{1/6}(1+\sqrt{q})^{1/3}}$. Then
\begin{align*}\label{GSE}
\lim_{N \to \infty} \mathrm{Prob}\left( L_{(1,1)\to (N,N) } \leq \frac{2 \sqrt{q}}{1-\sqrt{q}}N+c_{q}^{-1}sN^{1/3}\right)=F_{\GSE}(s).
\end{align*}
If $\alpha=1$, then
\begin{align*}
\lim_{N \to \infty} \mathrm{Prob}\left( L_{(1,1)\to (N,N) } \leq \frac{2 \sqrt{q}}{1-\sqrt{q}}N+c_{q}^{-1}sN^{1/3}\right)=F_{\GOE}(s).
\end{align*}
\end{thm}
Here $F_\GSE , F_\GOE$ are the Tracy--Widom distributions from RMT, expressible through Fredholm pfaffians \cite{TW05}.
One can also study a transitional regime where $\alpha=1- v N^{-1/3}, v \in \R$ in which case one  obtains a transitional distribution $F_{\GOE \to \GSE,v}$.

\section{Plane partitions and plane overpartitions} \label{sec:pp}

Here we describe two models on plane partitions: symmetric plane partitions and plane overpartitions. They are special cases of the free boundary Schur process with $u=0, v=1$ (which pins the left boundary at $\emptyset$).

\noindent {\bf Symmetric plane partitions.}\quad A \textit{free boundary plane partition of length $N$} is an array $(\pi_{i,j})_{ 1 \leq j \leq i \leq N}$ of non--negative integers satisfying the properties $\pi_{i,j} \geq \pi_{i+1, j}$ and $\pi_{i,j} \geq \pi_{i, j+1}$ for all meaningful $i,j$. Its \textit{volume} is the sum of its entries: $|\pi| = \sum_{1 \leq j \leq i \leq N} \pi_{i,j}$. Such an object can be viewed as half of a \textit{symmetric plane partition} with base in the square $N \times N$ -- an array $(\pi_{i,j})_{ 1 \leq i, j\leq N}$ satisfying the above constraints plus the symmetry constraint $\pi_{i,j} = \pi_{j, i}$. An example of length 5 and volume 79 is depicted below.


\begin{figure}[!ht]
\centering
\includegraphics{replacement_figure}
\label{free_pp}
\end{figure}




For a fixed $q \in (0,1)$, we study large free boundary plane partitions weighted according to their volume
$ \Prob(\pi) \propto q^{|\pi|}$ in the limit $q \to 1$ and $N \to \infty$. Similarly as in \cite{or}, for the right choice of specializations, this can be seen as a free boundary Schur process on the sequence  $\lambda^{(N-i)}_k = (\pi_{N-i+k, k})_{k>0}$, $i=1,\dots,N$ and $\lambda^{(N)}:= \emptyset$.

\cref{main_thm} implies that this process $\emptyset \prec \lambda^{(N-1)} \prec \dots \prec \lambda^{(0)}$ is pfaffian and its correlation kernel (via $\lambda^{(i)} \mapsto \{ k^{(i)}_s = \lambda^{(i)}_s - s + \frac{1}{2} \}$) is given by
\begin{align*}
  K_{1,1}(i, k; i', k') &= \int_{z} \int_{w} F(N-i, z) F(N-i', w) \frac{1}{z^{k} w^{k'}} \frac{\sqrt{zw} (z-w)}{(z+1)(w+1)(zw-1)} \dx \mathbb{T}, \\
K_{1,2}(i, k; i', k') &= -K_{1,2}(i',k';i,k)=\int_{z} \int_{w} \frac{F(N-i, z)}{F(N-i', w)} \frac{w^{k'}}{z^{k}} \frac{\sqrt{zw} (zw-1)}{(z+1)(w-1)(z-w)} \dx \mathbb{T}, \\
K_{2,2}(i, k; i', k') &= \int_{z} \int_{w} \frac{1}{F(N-i, z) F(N-i', w)} z^{k} w^{k'} \frac{\sqrt{zw} (z-w)}{(z-1)(w-1)(zw-1)} \dx \mathbb{T},
\end{align*}
where $F(N-i,z) := (q/z;q)_N/(q^{i+1}z; q)_{N-i}$ and the contours are $1+\epsilon > |z| > |w| > 1$ for $\epsilon$ small in the case $i \geq i'$.


We asymptotically analyze the kernel using the steepest descent method much like in \cite{or}. It boils down to analyzing the critical points of the function $F$ in the limit.

Let us define the following curve in the \textit{macroscopic} $(x,y)$ plane:
\begin{align*}
  \mathcal{C} = \{ (x, y) \in \R^2 | x > 0, (\exp(-y/2))^2 = (1 \pm \exp(-x/2))^2 \}.
\end{align*}
With $X = \exp(-x), Y = \exp(-y)$, in the $(\sqrt{X}, \sqrt{Y})$ coordinate system $\mathcal{C}$ is half the boundary of the amoeba of the polynomial $P(Z,W)=1+Z+W$. $\mathcal{C}$ cuts the half plane $x > 0$ into three connected components, which we call $\mathcal{D}_t$ ($y >> 0$), $\mathcal{D}_b$ ($y << 0$) and $\mathcal{D}_m$ for region in the middle (``between'' the two path--connected components of $\mathcal{C}$).

Let $x > 0, y \in \R$ be fixed. As $r \to +0$, for $q=\exp(-r)$, we consider random free boundary plane partitions, volume weighted, and look at microscopic coordinates $(i,k)$ approaching $(x,y)$ macroscopically: $(r i \to x, r k \to y)$ (in terms of lozenge heights $r h \to y - x/2$). We can prove the following.

\begin{thm} \label{thm:pp}
  As $r \to +0$: the probability that there is a horizontal lozenge at position $(i, k-i/2)$ decreases exponentially to 0 if $(x,y) \in \mathcal{D}_t$; it increases exponentially to 1 if $(x,y) \in \mathcal{D}_b$; it is finite and strictly in between 0 and 1 if $(x,y)$ is inside $\mathcal{D}_m$.
\end{thm}

The \textit{liquid region} $\mathcal{D}_m$ and \textit{frozen regions} $\mathcal{D}_t$ and $\mathcal{D}_b$ are the same for both large symmetric and large non--symmetric plane partitions. This is illustrated with two simulated random plane partitions in the figure below. Shown below are a random plane partition for $q = 0.953$ (left) and a random symmetric one for $q^2$ (right). The squaring of $q$ in the symmetric case is necessary to obtain the same asymptotic drawing scale.

\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.023]{pp1.pdf}
\quad \qquad
\includegraphics[scale=0.023]{pp_sym1.pdf}
\end{center}

\label{fig:sym_pp}
\end{figure}

\noindent{\bf Plane overpartitions.}\quad  A plane overpartition is a plane partition where in each row the last occurrence of an integer can be overlined or not and all the other occurrences of this integer are not overlined, and in each column the first occurrence of an integer can be overlined or not and all the other occurrences of this integer are overlined. A plane overpartition with the largest entry at most $N$ and shape $\lambda$ can be recorded as a sequence of partitions $
\es \prec \lambda^{(1)} \prec' \lambda^{(2)} \prec \cdots \prec \lambda^{(2n-1)} \prec' \lambda^{(2N)}=\lambda
$
where $\lambda^{(i)}$ is the partition whose shape is formed by all fillings greater than ${N-i/2}$, where the convention is that $\overline{k}=k-1/2$ (see e.g., \cite{bcc}). An example of a plane overpartition is shown above.
\begin{figure}[htp!]
\centering
 \label{fig:plane-overpartitions}
  { \begin{tikzpicture}[scale=0.5]
\draw[thick, smooth] (0,0)--(0,4)--(5,4);
\draw[thick, smooth] (0,0)--(1,0);
\draw[thick, smooth] (0,1)--(2,1);
\draw[thick, smooth] (0,2)--(4,2);
\draw[thick, smooth] (0,3)--(5,3);
\draw[thick, smooth] (1,0)--(1,4);
\draw[thick, smooth] (2,1)--(2,4);
\draw[thick, smooth] (3,2)--(3,4);
\draw[thick, smooth] (4,2)--(4,4);
\draw[thick, smooth] (5,3)--(5,4);
\node at (0.5,0.5) { 1};
\node at (0.5,1.5) {$\overline{3}$};
\node at (0.5,2.5) { 3};
\node at (0.5,3.5) { 4};
\node at (1.5,1.5) { $\overline{1}$};
\node at (1.5,2.5) { 3};
\node at (1.5,3.5) { $\overline{4}$};
\node at (2.5,2.5) { $\overline{3}$};
\node at (2.5,3.5) { $\overline{3}$};
\node at (3.5,2.5) {$\overline{2}$};
\node at (3.5,3.5) { 2};
\node at (4.5,3.5) {2};
\end{tikzpicture}
}
\end{figure}

As before we study the asymptotics of the $q^{\textrm {volume}}$ measure on plane overpartitions when $q\to 1$ and $N\to \infty$. In this case the liquid region, as before called $\mathcal{D}_m$, under an appropriate change of coordinates,  turns out to be half of the amoeba of the polynomial $-1+Z+W+ZW$, is
\[
\mathcal{D}_m=\left\{(\tau,\chi)|\tau \in \R, \chi \in \R_{\geq 0}, -1\leq f(\tau,\chi)\leq1\right\},
\]
where $f(\tau,\chi)=(e^{\chi}+1)(e^\tau-1)/(2e^{\chi/2}(e^{\tau}+1))$.

In the limit, inside the liquid region $\mathcal{D}_m$ the process becomes determinantal. Its kernel can be written in terms of the incomplete beta--hypergeometric kernel which for $(\tau,\chi) \in \mathcal{D}_m, t, x, y \in \Z$ is  defined with
\[
B_\pm(\tau,\chi,t,x,y)=
\displaystyle \frac{1}{2\pi i}\int_{C^{\pm}(e^{-\chi/2},\theta_c(\tau,\chi))}\frac{1}{z^{t+1}}\frac{(1-z)^{x}}{(1+z)^y}dz,
\]
where $\theta_c(\tau,\chi)=\arccos(f(\tau,\chi))$ and $C^{+}(R,\theta)$($C^-(R,\theta)$) is the counterclockwise (clockwise) oriented arc on $|z|=R$ from $Re^{-i\theta}$ to $Re^{i\theta}$ for $R>0$ and $0\leq\theta\leq \pi$.

In the limit on the free boundary edge the process remains to be pfaffian and its correlation kernel is given also in terms of the kernel $B_\pm$.

\acknowledgements{We thank Philippe Biane, Alexei Borodin, C\'edric Boutillier, Ivan Corwin, Philippe Di Francesco and Patrik Ferrari for useful conversations regarding this note. This work has mostly been completed while P. N. was a postdoctoral fellow in the D\'epartement de math\'ematiques et applications at \'Ecole normale sup\'erieure, Paris.}

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