%% if you are submitting an initial manuscript then you should have submission as an option here
%% if you are submitting a revised manuscript then you should have revision as an option here
%% otherwise options taken by the article class will be accepted
\documentclass[finalversion]{FPSAC2025}
\articlenumber{28}
%% but DO NOT pass any options (or change anything else anywhere) which alters page size / layout / font size etc

%% note that the class file already loads {amsmath, amsthm, amssymb}

\newtheorem{thm}{Theorem}
\newtheorem{lem}{Lemma}


%\usepackage[pdftex,backref=page,colorlinks=true,linkcolor=blue,citecolor=red]{hyperref}
%\usepackage[alphabetic,nobysame]{amsrefs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%main packages
\usepackage{mathtools}
\usepackage{graphicx,color}
\usepackage{upgreek}
\usepackage{esint}
\usepackage[mathscr]{euscript}

%equations
\allowdisplaybreaks
\numberwithin{equation}{section}

%tikz
\usepackage{tikz}
\usetikzlibrary{shapes,arrows,positioning,decorations.markings}

%conveniences
\usepackage{ytableau}
\usepackage{array}
\usepackage{adjustbox}
\usepackage{cleveref}
%\usepackage{enumerate}
\usepackage[margin=10pt]{caption}

%paper geometry
%\usepackage[DIV=12]{typearea}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%draft-specific
\synctex=1
% \usepackage[notcite,notref]{showkeys}
% \usepackage{refcheck,comment}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%this paper specific
\newcommand{\ssp}{\hspace{1pt}}
\renewcommand{\le}{\leqslant}
\renewcommand{\ge}{\geqslant}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\inv}{\operatorname{inv}}
\newcommand{\Prob}{\operatorname{\mathbb{P}}}
\newcommand{\ls}{\mathsf{s}}
\newcommand{\bs}{\mathsf{S}}
\DeclareMathOperator{\PD}{PD}
\DeclareMathOperator{\RPD}{RPD}
\DeclareMathOperator{\BPD}{BPD}
\DeclareMathOperator{\RBPD}{RBPD}
\DeclareMathOperator{\cross}{cross}
\DeclareMathOperator{\SEbump}{SEbump}
\DeclareMathOperator{\NWbump}{NWbump}
\DeclareMathOperator{\emptytile}{empty}
\DeclareMathOperator{\wt}{wt}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{proposition}{Proposition}[section]
\newtheorem{lemma}[proposition]{Lemma}
\newtheorem{corollary}[proposition]{Corollary}
\newtheorem{theorem}[proposition]{Theorem}
\newtheorem{conjecture}[proposition]{Conjecture}
\newtheorem{claim}[proposition]{Claim}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\theoremstyle{definition}
\newtheorem{definition}[proposition]{Definition}
\newtheorem{remark}[proposition]{Remark}
\newtheorem{example}[proposition]{Example}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%% define your title in the usual way
\title[Grothendieck Shenanigans]{Grothendieck Shenanigans:
Permutons from pipe dreams via integrable probability}
%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
%% for more than 3 names, use the Oxford comma: One Author, Two Author, Red Author, \and Blue Author
\author[Morales, Panova, Petrov, Yeliussizov]{\texorpdfstring{Alejandro H. Morales \thanks{\href{mailto:morales_borrero.alejandro@uqam.ca}{morales\_borrero.alejandro@uqam.ca}. Partially supported by NSF, FRQNT, and NSERC}\addressmark{1}, Greta Panova\thanks{\href{mailto:gpanova@usc.edu}{gpanova@usc.edu}. Partially supported by NSF CCF:AF grant}\addressmark{2}, \\Leonid Petrov\thanks{\href{mailto:lenia.petrov@gmail.com}{lenia.petrov@gmail.com}. Partially supported by NSF DMS}\addressmark{3}, \and Damir Yeliussizov\thanks{\href{mailto:yeldamir@gmail.com}{yeldamir@gmail.com}. Partially supported by MSHE RK (grant AP23489146)}\addressmark{4}}{Alejandro H. Morales , Greta Panova, Leonid Petrov, and Damir Yeliussizov}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Université du Québec à Montréal, Montreal, QC, Canada. \\ \addressmark{2}University of Southern California, Los Angeles, USA.\\
\addressmark{3}University of Virginia, Charlottesville, USA.\\
\addressmark{4}Kazakh-British Technical University, Almaty, Kazakhstan.}

%% put the date of submission here
\received{\today}

%% leave this blank until submitting a revised version
%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
%% please don't use custom commands in your abstract / resume, as these will be displayed online
%% likewise for citations -- please don't use \cite, and instead write out your citation as something like (author year)
\abstract{
We study random permutations corresponding to pipe dreams. Our
	main model is motivated by the Grothendieck
	polynomials with parameter $\beta=1$ arising in the
	$K$-theory of the flag variety. The probability weight of a
	permutation is proportional to the principal specialization
	(setting all variables to~$1$)
	of its
	Grothendieck polynomial. By mapping
	this random permutation to a version of TASEP (Totally
	Asymmetric Simple Exclusion Process), we describe the
	limiting permuton and fluctuations around it as the order
	$n$ of the permutation grows to infinity. The fluctuations
	are of order $n^{\frac13}$ and have the Tracy--Widom GUE distribution,
	which places this algebraic ($K$-theoretic) model
	into the Kardar--Parisi--Zhang universality class.
	%As an application, we find the expected
	%number of inversions in this random permutation, and
	%contrast it with the case of
	%non-reduced pipe dreams.

	Inspired by
	Stanley's question for the maximal value of principal
	specializations of Schubert polynomials, we resolve the
	analogous question for $\beta=1$ Grothendieck polynomials,
	and provide bounds for general~$\beta$. This analysis uses a correspondence with the free fermion six-vertex model, and the frozen boundary of the Aztec diamond.
}

%% put your French abstract here, or comment this out if you don't have one
%\resume{\lipsum[2]}

%% put your keywords here, or comment this out if you don't have them yet
\keywords{Grothendieck polynomials, pipe dreams, TASEP, six vertex model}

%% you can include your bibliography however you want, but using an external .bib file is STRONGLY RECOMMENDED and will make the editor's life much easier
%% regardless of how you do it, please use numerical citations; i.e., [xx, yy] in the text

%% this sample uses biblatex, which (among other things) takes care of URLs in a more flexible way than bibtex
%% but you can use bibtex if you want
\usepackage[backend=bibtex, giveninits]{biblatex}
\addbibresource{bib_pipes_fpsac.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands

\begin{document}

\maketitle
%% note that you DO NOT have to put your abstract here -- it is generated by \maketitle and the \abstract and \resume commands above
% \section{Introduction}
% \label{sec:intro}

\section{A story from Algebra to Probability}
\label{sub:overview}

Algebraic Combinatorics established itself as a field that uses combinatorial methods to understand algebraic behavior in problems ranging from Group Theory to Algebraic Geometry. It started with stark exact formulas, like the celebrated hook-length formula for the dimension of irreducible modules of the symmetric group $S_n$; beautiful interpretations, such as the Littlewood--Richardson rule for the structure constants of representations of the general linear group $GL_n$; powerful and intricate bijections, such as the Robinson--Schensted--Knuth correspondence. However, such answers only go so far, leaving room for questions like ``approximately how many'', ``what are the typical objects'', and ``what is the typical behavior''. These questions lead us into the realm of Asymptotic Algebraic Combinatorics, which aims to answer them with the help of tools originating outside of Combinatorics. In the present work, we employ Integrable Probability, a rapidly evolving field focused on developing and analyzing interacting particle systems and random growth models possessing a certain degree of structure or symmetry. The arising probabilistic models exhibit rich structure leading to new permutons representing ``typical permutations''. The connection between the algebraic model and statistical mechanics is multi-fold via a correspondence between the so-called bumpless pipe dream models for Schubert/Grothendieck and the six-vertex model. The well studied free fermion six-vertex model and the frozen boundary of the Aztec diamond are key to understanding the permutations which maximize the principal specialization of Grothendieck polynomials.

% On the algebro-combinatorial side, many significant questions concerning exact formulas and combinatorial interpretations in the cohomology and $K$-theory of the flag variety remain open. Understanding their asymptotic behavior is thus even more natural.
Stanley \cite{stanley2017some} asked the most basic question on the principal specializations of Schubert polynomials $\mathfrak{S}_w$
(representing cohomology classes of the flag variety): does the following limit exist
\begin{equation*}
	\lim_{n\to \infty} \frac{1}{n^2} \log_2 \max_{w \in S_n} \mathfrak{S}_w\bigl(
	\underbrace{1,\ldots,1 }_n\bigr),
\end{equation*}
and if so, what is it and for which permutations $w$ is this achieved.
This question (including the existence of the limit) is still open.
In~\cite{MoralesPakPanova2019}, a lower bound of about $0.29$ was established for {\em layered permutations}. An upper bound of about $0.37$ comes from a remarkable connection with Alternating Sign Matrices and the six-vertex model.

% This problem has an even more interesting interpretation from the Probability/Statistical Mechanics side as it is asking for the asymptotic growth of the energy in a model with \emph{long-range interactions}, and for such models no general tools are known.

 The Grothendieck polynomials
$\mathfrak{G}_w^{\beta}$, which represent $K$-theoretic classes of the flag variety, are a natural one-parameter generalization of the Schubert polynomials. Extending Stanley's questions, we
would like to understand the asymptotic behavior of maximal
 principal specializations $\max_{w\in S_n}\mathfrak{G}_w^{\beta}(1^n)$
of Grothendieck polynomials. This
question was first touched on in
\cite{MPP4GrothExcited}, \cite{dennin2022}.
% Establishing a family of
% such maximal permutations would shed light on the
% Schubert questions as $\beta \to 0$, as well as on the corresponding long-range interaction model.

Thanks to a combinatorial model for Grothendieck and
Schubert polynomials, these questions have very natural
statistical mechanics interpretations.
Namely, both polynomials are partition functions of tilings
into crosses and elbows of a size $n$ triangle (staircase) shape,
which result in a configuration of $n$ ``pipes''. Such pipe configurations
are often called \emph{pipe dreams}.
In the Schubert case, the only valid
tilings are the ones where no two pipes cross more than
once. This is a global (\emph{long-range interaction}) condition, and no tools in Statistical Mechanics are known to compute the asymptotics of the energy (equivalent to Stanley's question and estimating the number of such pipe dreams).
In the Grothendieck case
for $\beta=1$, all tilings are allowed, but the pipes must be
resolved (reduced) to obtain a permutation. One of the key ideas leading to our analysis is that this model can be mapped to a
colored stochastic six-vertex model
(and further to TASEP, the Totally Asymmetric Simple Exclusion Process).
The resulting interacting particle systems have only local (short-range) interactions, and are
amenable to techniques from Integrable Probability.


In this work,
we investigate the asymptotics of the \emph{typical
$\beta=1$ Grothendieck random permutations}
and characterize their limit shape which is described by a permuton.
% In principle, the typical
% permutation (sampled from the limiting permuton) should give the
% asymptotically maximal principal specialization
% $\mathfrak{G}_w^{\beta=1}(1^n)$, but we do not address this question here.
We also study fluctuations of Grothendieck random permutations around the limiting permuton.
They
are of order $n^{\frac13}$ and asymptotically have the Tracy--Widom GUE distribution.
This distribution was
first observed in the fluctuations of
the largest eigenvalue of Gaussian
random matrices
with unitary symmetry. By now, having Tracy--Widom fluctuations is an indication that a
model is within the
Kardar--Parisi--Zhang (KPZ) universality class \cite{CorwinKPZ}, which includes a wide range of
random growth models and interacting particle systems. We also derive the expected number of inversions, which are of order $n^2$. This is in contrast to the model of Colin Defant~\cite{defant2024randomsubwordspipedreams} where no pipes are resolved and the inversions are of order $n^{3/2}$.


% We also study the natural, yet not as algebraically
% motivated, model, where the staircase is tiled with crosses
% with probability $p$ and elbows with probability $1-p$, and
% the pipes follow the tiles without any reductions or
% reassignments. Using simple random walks, we
% show that the resulting random permutation is close to the identity, and, moreover,
% that the total
% displacement
% $\sum_{i=1}^{n}|i-w_i|$
% and length (number of inversions)
% of this permutation
% are of order $n^{3/2}$. This was motivated by, and partially resolves, an open problem of Colin Defant.


Returning to Stanley's original question, when $\beta=1$, we have
$$\log_2 \max_{w \in S_n} \mathfrak{G}_w^\beta(1^n) = \frac{1}{2}n^2 - O(n \log n).$$
% When $\beta=1$, this maximum is known to be $O(2^{\binom{n}{2}})$.
Using the
correspondence with
2-enumerated Alternating Sign Matrices
(equivalently, the six-vertex model with domain wall boundary conditions
and free-fermion weights; or the model of uniform domino tilings of the Aztec diamond),
we show that a large family of layered
permutations also achieve this asymptotic maximum which strengthens the intuition that layered also maximize Schubert polynomials asymptotically~\cite{merzon2016determinantal}. For
general $\beta$, we establish certain bounds for the maximal
principal specialization.

This is the extended abstract of~\cite{morales2024grothendieckshenaniganspermutonspipe}, which contains all the proofs and references, background on TASEP and six-vertex model. Here we will informally describe our main results and sketch the connections.


\section{Pipe dreams and permutations}
\label{sub:pipe_dreams}
We denote by $S_n$ the set permutations of
$\{1,2,\ldots,n\}$ that we write in the one-line notation
$w=w_1w_2\cdots w_n$ unless indicated otherwise.
We also denote the image of $i$ under $w$ by $w(i)$, and use the notation
$w_i=w(i)$ interchangeably when this does not lead to confusion.
The longest
permutation $n\,n-1\ldots 2\,1$ is denoted by $w_0=w_0(n)$.
For a permutation $w$ of length $\ell$, we denote by $R(w)$
the set of \emph{reduced words} of $w$, that is, tuples
$(r_1,\ldots,r_{\ell})$ such that $s_{r_1}\cdots s_{r_\ell}$
is a reduced decomposition of $w$, where $s_i=(i, i+1)$ are the
simple transpositions.

% Grothendieck polynomials can be defined combinatorially
% via pipe dreams (equivalently, rc-graphs), as partition functions of the following model.

A \emph{pipe dream}
of order $n$
is a tiling of the
staircase shape
(having $n-1$ boxes in the first row,
$n-2$ boxes in the second row, and so on,
with boxes left-justified)
by tiles of two types:
\emph{bumps}
\raisebox{-2pt}{\includegraphics[scale=0.4]{images/elbows}}
and
\emph{crossings}
\raisebox{-2pt}{\includegraphics[scale=0.4]{images/cross}}.
The $n$-th diagonal below the staircase is
equipped with half bumps
\raisebox{-2pt}{\includegraphics[scale=0.4]{images/half_elbows}}.
Without any restrictions there
are $2^{\binom{n}{2}}$ pipe dreams of order $n$.
%See \Cref{fig:pipe_dream}, left, for an example of a pipe dream of order $6$.

\begin{figure}[htpb]
	\centering
	\includegraphics[height=1.45in]{images/pipe_dream_ex.pdf}$\quad$ \includegraphics[height=1.5in]{images/pipe_dream_6}$\quad$
	\includegraphics[height=1.5in]{images/bumpless_pipe_dream_ex2.pdf}
	\caption{\textbf{Left}: A reduced pipe dream without any double intersections corresponding to permutation $w=241653$, and an unreduced pipe dream $D$ of order $6$.
		\textbf{Center}: A reduction of the pipe dream leading to the permutation
		$w(D)=241653$. \textbf{Right:} A reduction $D'$ of a bumpless pipe dream $D$ leading to the permutation $w(D)=w(D')=45128637$.
		 Dashing is added for the
		printed version and accessibility.}
	\label{fig:pipe_dream}
		\label{fig:bumpless_pipe_dream}
\end{figure}

A pipe dream (a tiling of the staircase shape) forms
a collection of strands (or \emph{pipes}) labeled~$1$ to $n$ from
the row where they start.
A pipe dream is called \emph{reduced} if
any two pipes cross through each other at most once.

\begin{definition}[Reduction of a pipe dream]
	\label{def:reduction_pipe_dream}
	Given a pipe dream~$D$
	that is not necessarily reduced, the \emph{reduction} of~$D$
	is a unique reduced pipe dream~$D'$ obtained as follows:
	starting at the bottom left tile traverse the pipe dream upwards along columns and to the right. For each encountered crossing,
	replace it with a bump if the pipes have already crossed in the already traversed squares.
	The labeling of pipes together with a reduction
	is indicated by colored paths in
	\Cref{fig:pipe_dream}, center.
\end{definition}

\begin{definition}[Permutation from pipe dream]
	\label{def:permutation_from_pipe_dream}
	One can associate a permutation $w(D)\in S_n$ to a pipe dream
	of order $n$ as
	follows. If $D$ is reduced then $w(D)^{-1}_j$ is the column where
	the pipe $j$ ends up in.\footnote{Throughout
	the paper, the column coordinate~$j$ increases from left to right,
	and the row coordinate~$i$ increases from top to bottom.}
	Equivalently, the column $j$ contains the exiting pipe of color
	$w(D)_j$. %Note that to capture the column number for the pipe $j$, 	we need the inverse permutation $w(D)^{-1}$.
 If $D$ is not reduced, then $w(D)$
	is the permutation associated to the reduction $D'$ of $D$. Alternatively, associating transpositions $s_i$ to each cross and reading from the bottom left to top right, we obtain a (non-necessarily reduced) word. Multiplying the transposition using the \emph{Demazure} (0-Hecke) product rule (where $s_i^2 =s_i$) gives $w$.
\end{definition}

% \begin{remark}
% 	\label{rmk:Demazure_product}
% 	Alternatively, the permutation $w(D)$
% 	from a non-reduced pipe dream
% 	can be defined using the \emph{Demazure product}
% 	\cite{demazure1974desingularisation}.
% 	Namely, for an elementary transposition
% 	$s_i=(i,i+1)$ (with $1\le i\le n-1$)
% 	and a permutation $w\in S_n$, the Demazure
% 	(or 0-Hecke) product
% 	is defined as
% 	\begin{equation}
% 		\label{eq:Demazure_product}
% 		s_i\star w\coloneqq
% 		\begin{cases}
% 			ws_i, & \textnormal{if}\ \ell(ws_i)>\ell(w),\\
% 			w, & \textnormal{otherwise}.
% 		\end{cases}
% 	\end{equation}
% 	Here $\ell(\cdot)$ is the length of a permutation, that is,
% 	its number of inversions.

% 	To each cross in $D$, associate a transposition $s_i$.
% 	Reading from the bottom left to top right, we obtain a
% 	(not necessarily reduced) word.
% 	For example, the pipe dream in \Cref{fig:pipe_dream}, left,
% 	corresponds to the word $s_5s_5s_3s_4s_1s_2s_4s_5s_4$.
% 	Replacing this product
% 	of elementary transpositions
% 	by the Demazure product, we obtain the permutation
% 	$w(D)^{-1}$. For our example, the Demazure product is
% 	\begin{equation*}
% 		s_5\star
% 		s_5\star
% 		s_3\star
% 		s_4\star
% 		s_1\star
% 		s_2\star
% 		s_4\star
% 		s_5\star
% 		s_4=
% 		s_5s_2s_1s_4s_3s_5=316254,
% 	\end{equation*}
% 	which is the inverse of $w(D)$ given in \Cref{fig:pipe_dream}, right.
% \end{remark}

Let $\PD(n)$ and $\RPD(n)$ be the sets of pipe
dreams and reduced pipe dreams of size $n$.
For each $w\in S_n$,
let $\PD(w)$ and
$\RPD(w)$ be, respectively, the sets of pipe dreams and reduced pipe dreams~$D$ such that $w(D)=w$. Note that
$\#\PD(n)=2^{\binom{n}{2}}$, whereas there is no simple
known formula for $\#\RPD(n)$.
%\cite[\href{http://oeis.org/A331920}{A331920}]{oeis}.
Given
a pipe dream $D$, let $\cross(D)$ be the set of
coordinates~$(i,j)$
of the cross tiles.
The \emph{weight} of $D$ is the monomial
$\wt(D)\coloneqq \prod_{(i,j)\in \cross(D)} x_i$.
For example, for the unreduced pipe dream in \Cref{fig:pipe_dream}, left, we have
$\wt(D)=x_1^3 x_2^2 x_3^2 x_4 x_5$. The \emph{Grothendieck} and \emph{Schubert} polynomials are originally defined recursively, via certain divided difference operators, but using \cite{fomin1996yang} %\cite{fomin1994grothendieck},
and \cite{BilleyBergeron} we can define them as partition functions of the pipe dream models.


% The nonsymmetric Grothendieck polynomials generalize the Schubert and the Schur polynomials,
% and capture the $K$-theory of the flag variety.
% They can be defined or interpreted in a number of ways,
% including via
% divided difference operators \cite{Lascoux1990}, bumpless pipe dreams
% \cite{Weigandt2020_bumpless},
% \cite{LamLeeShimozono},
% and solvable lattice models
% \cite{BrubakerALCO}, \cite{BuciumasScrimshaw}.
% We can also define them as the partition functions of the pipe dreams model, via the
% the following
% result due to
% \cite{fomin1996yang}, \cite{fomin1994grothendieck}, and \cite{BilleyBergeron}.
\begin{theorem}
	\label{thm:Grothendieck_pipe_dreams_intro}
	For any $w\in S_n$, we have that the Grothendieck polynomial is given by
\begin{equation} \label{eq: Groth in terms of pd}
	\mathfrak{G}_w^\beta(x_1,\ldots,x_n ) \,=\, \sum_{D \in \PD(w)} \beta^{\#\cross(D)-\ell(w)} \wt(D).
\end{equation} 
In particular, setting $\beta=0$ forbids non-reduced
pipe dreams, so the Schubert polynomial is
\begin{equation}
	\label{eq: schubs in terms of rpd}
	\mathfrak{S}_w({\bf x}) = \sum_{D\in \RPD(w)} \wt(D).
\end{equation}
\end{theorem}



% Grothendieck polynomials were were introduced
% by Lascoux and Sch\"utzenberger
% \cite{lascoux1982structure},
% \cite{Lascoux1990} to study the $K$-theory of
% flag varieties. Their original definition is recursive, via isobaric divided difference operators, which we will not use here.
% Setting $\beta=0$ in $\mathfrak{G}_w^\beta$, we obtain a Schubert
% polynomial $\mathfrak{S}_w$ which represents cohomology
% classes of Schubert cycles in the flag variety
% \cite{bernstein1973schubert},
% \cite{demazure1974desingularisation},
% \cite{lascoux1982structure}.


\section{Asymptotics of Grothendieck random permutations}
\label{sub:random_permutations}
%\label{sub:asymptotics_permutations}

Fix $p\in(0,1)$.
Equip the set of all pipe dreams
with a probability measure by independently
placing the tiles in each box:
\begin{equation}
	\label{eq:pipe_dream_measure}
	\raisebox{-6pt}{\includegraphics[scale=0.6]{images/cross}}
	\ \; \textnormal{with probability}\ p,
	\qquad
	\raisebox{-6pt}{\includegraphics[scale=0.6]{images/elbows}}
	\ \; \textnormal{with probability}\ 1-p.
\end{equation}
In particular, for $p=\frac12$, we have the uniform measure on the set of pipe dreams.


%where the first equality is simply the definition of the measure \eqref{eq:pipe_dream_measure},
%and the second immediately follows from \Cref{thm:Grothendieck_pipe_dreams_intro} with $\beta=1$. This distribution on permutations arises from the six-vertex model with domain wall boundary conditions and free-fermion weights (corresponding to 2-enumeration of Alternating Sign Matrices).
\begin{figure}[htpb]
	\centering
	\includegraphics[width=.28\textwidth]{images/perm_2000_08.pdf} \
	\includegraphics[width=.28\textwidth]{images/perm_2000_05.pdf} \
	\includegraphics[width=.28\textwidth, angle=90]{images/permuton_2000_2000_05_coarse8.pdf}
	\caption{\textbf{Left and center}: A sample of a Grothendieck random permutation of order $n=2000$ with
		$p=\frac{4}{5}$ and $p=\frac{1}{2}$, respectively.
		\textbf{Right}: An average of Grothendieck
		random permutations with $n=2000$ and $p=\frac{1}{2}$
		over $2000$ samples.
		We take a sum of permutation matrices, and coarse-grain the result
		into $8\times 8$ blocks. The plot is the heat map of the resulting matrix,
		which approximates the Grothendieck permuton.
		%The \texttt{C} code for these simulations is available on the \texttt{arXiv} as an ancillary file.
	}
	\label{fig:intro_simulations}
\end{figure}

By reducing this random pipe dream as in \Cref{def:reduction_pipe_dream},
we obtain a
random permutation $\mathbf{w}\in S_n$
which we call the \emph{Grothendieck random permutation} (of order~$n$ and with parameter~$p$;
we suppress the dependence on $n$ and $p$ in the notation).
The name is justified by a connection with the polynomials $\mathfrak{G}_w^{\beta=1}$.
Indeed,
we have for any $w\in S_n$:
\begin{equation}
	\label{eq:pipe_dream_specialization_p_Grothendieck_intro}
	\Prob(\mathbf{w}=w)
	=
	\sum_{D \in \PD(w)} p^{\cross(D)}(1-p)^{\mathrm{elbow}(D)}
	=
	(1-p)^{\binom{n}{2}}\mathfrak{G}_w^{\beta=1}
	\left( {\frac{p}{1\!-\!p},\ldots, \frac{p}{1\!-\!p}} \right).
\end{equation}

% When $p=0$ or $p=1$, we almost surely have
% $\mathbf{w}=\mathrm{id}$ or $\mathbf{w}=(n,n-1,\ldots,1 )$, so we assume that $p\in(0,1)$.
% Individual samples of
% Grothendieck random permutations with $p=\frac{4}{5}$ and $p=\frac{1}{2}$
% and a plot constructed from averaging over many samples
% are given in
% \Cref{fig:intro_simulations}.

%\section{Asymptotics of Grothendieck random permutations}
%\label{sub:asymptotics_permutations}

For $w\in S_n$,
define its \emph{height function} as in~\Cref{fig:height_function_permutation_matrix} by
\begin{equation*}
	H(x,y)\coloneqq
	\operatorname{\#}\left(
		\left\{ w^{-1}(x),w^{-1}(x+1),\ldots,w^{-1}(n) \right\}
		\cap
	\left\{ y,y+1,\ldots,n \right\} \right),
\end{equation*}
where $1\le x,y\le n$. In terms of the pipe dream as in \Cref{fig:pipe_dream}, center,
$H(x,y)$ is the number of pipes of color $\ge x$ which exit through the positions
$\ge y$ at the top. %In the example in \Cref{fig:pipe_dream}, center,
%we have $H(4,3)=2$. In terms of the permutation matrix of $w$ where 
$H(x,y)$ gives also the number of 1s in the rectangle with lower left corner at $(x,y)$.

\begin{figure}[htpb]
	\centering
	\begin{tikzpicture}[scale=.5]
		\foreach \x in {1,2,3,4,5,6} {
		\foreach \y in {1,2,3,4,5,6} {
			\draw (\x-.5,\y-.5) rectangle (\x+.5,\y+.5);
		}
	}

	\fill (1,2) circle (0.2);
	\fill (2,4) circle (0.2);
	\fill (3,1) circle (0.2);
	\fill (4,6) circle (0.2);
	\fill (5,5) circle (0.2);
	\fill (6,3) circle (0.2);

	\foreach \i in {1,2,3,4,5,6} {
	\node at (\i,0) {\i};
	\node at (0,\i) {\i};
}
	\draw[blue,opacity=.3, fill] (2.5,3.5)--++(4,0)--++(0,3)--++(-4,0)--cycle;
\end{tikzpicture} $\qquad \qquad$ \includegraphics[width=2in]{images/h_circ_graph.pdf}

	\caption{Left: Permutation matrix of $w=(2,4,1,6,5,3)$
		coming from the pipe dream in
		\Cref{fig:pipe_dream}, center (dots indicate 1's).
		The highlighted rectangle has
		$H(4,3)=2$ entries. Right: the limit shape $h$ of the rescaled $H$.}
	\label{fig:height_function_permutation_matrix}
\end{figure}


Let now $\mathbf{w}\in S_n$ be the Grothendieck random permutation with the fixed parameter $p\in(0,1)$.
%This makes the height function $H(x,y)$ random, too.
We show that the random $H(x,y)$ satisfies the law of large numbers,
and characterize its asymptotic fluctuations illustrated in \Cref{fig:intro_simulations}.

\begin{theorem}
	\label{thm:Grothendieck_permutations_asymptotics_intro}
	\begin{enumerate}[itemsep=0pt]%[{\bf{}1.\/}]
		\item
			There exists a limiting height function $\mathsf{h}^\circ$ such that
			\begin{equation*}
				\lim_{n\to\infty} n^{-1}\ssp H(\lfloor n\ssp\mathsf{x} \rfloor,
				\lfloor n\ssp\mathsf{y} \rfloor)=\mathsf{h}^\circ(\mathsf{x},\mathsf{y}),
				\qquad
				(\mathsf{x},\mathsf{y})\in[0,1]^2,
			\end{equation*}
			where the convergence is in probability.
			The function $\mathsf{h}^\circ(\mathsf{x},\mathsf{y})$
			is
			explicit, see~\cite{morales2024grothendieckshenaniganspermutonspipe}.
			It is continuous and depends only on $p$.
			The graph of $\mathsf{h}^\circ$
			is given in
			\Cref{fig:height_function_permutation_matrix}, right.
		\item The fluctuations of $H(x,y)$ around $\mathsf{h}^\circ$
			belong to the KPZ universality class:
			\begin{equation*}
				\lim_{n\to\infty}\Prob
				\left( \frac{H(\lfloor n\ssp\mathsf{x} \rfloor,
				\lfloor n\ssp\mathsf{y} \rfloor)-n\ssp\mathsf{h}^\circ(\mathsf{x},\mathsf{y})}
				{\mathsf{v}(\mathsf{x},\mathsf{y})\ssp n^{1/3}}
				\leq r \right)=F_2(r),\qquad r\in \mathbb{R},
			\end{equation*}
			where $F_2$
			is the cumulative distribution function of the Tracy--Widom GUE distribution,
			and $\mathsf{v}(\mathsf{x},\mathsf{y})$ is given in \cite{morales2024grothendieckshenaniganspermutonspipe}.
	\end{enumerate}
\end{theorem}
% \Cref{thm:Grothendieck_permutations_asymptotics_intro}
% justifies
% the simulations in
% \Cref{fig:intro_simulations}.
% We prove it
% by realizing $H(x,y)$ as an observable of a
% discrete time
% TASEP with parallel geometrically distributed jumps started from the densely packed
% (step) initial configuration.
% The corresponds starts by identifying the pipe dreams as a colored vertex model where the colors of the pipes correspond to the labels they carry (so the numbers $1,2,\ldots,n$). The local rules then give the reduced pipe dream model. Passing to a color-blind model where colors $\leq k$ are treated as $0$s and larger colors as $1$s we arrive at the TASEP.
% By employing techniques
% from Integrable Probability to analyze TASEP and passing back to the
% Grothendieck random permutation, we obtain the limit shape and fluctuations.

% \medskip

The law of large numbers in the first part of
\Cref{thm:Grothendieck_permutations_asymptotics_intro}
means that the Grothen\-dieck random permutations $\mathbf{w}\in S_n$
converge to a deterministic \emph{permuton}.
Recall that permutons are Borel probability measures $\mu$
on $[0,1]^2$ with uniform marginals, that
is,
$\mu\left( [0,1]\times [a,b] \right)=
\mu\left( [a,b]\times [0,1] \right)=b-a$.
%For details on permutons
%and their connection to pattern frequencies in permutations, we refer to
%\cite{hoppen2013limits},
%\cite{bassino2019universal}, and the survey
%\cite{grubel2023ranks}.
Our limiting \emph{Grothendieck permuton} is completely determined
by the limiting height function
via
$ \displaystyle \mu\left( [\mathsf{x},1]\times [\mathsf{y},1] \right)
	=\mathsf{h}^\circ(\mathsf{x},\mathsf{y}).$
% Note that the ``complementary''
% function
% $\mu\left(
% 	[0,\mathsf{x}]\times [0,\mathsf{y}]
% \right)$ is often referred to as \emph{copula} in the literature, cf.
% \cite{grubel2023ranks}.
Its singular part (a positive portion of its mass)
is concentrated along the curve
\begin{equation*}
	\mathscr{E}_p\coloneqq
	\left\{
		(\mathsf{x},\mathsf{y})
		\colon
		(\mathsf{y}-\mathsf{x})^2/p+(\mathsf{y}+\mathsf{x}-1)^2/(1-p)=1,\
		1-p\le \mathsf{x}\le 1
	\right\}\subset [0,1]^2.
\end{equation*}
The total mass supported on this curve is equal to $\gamma_p$ defined below in~\eqref{eq:Grothendieck_permuton_inversions_intro}.


As a corollary of the permuton convergence of \Cref{thm:Grothendieck_permutations_asymptotics_intro},
one can also obtain laws of large numbers for
arbitrary pattern counts in Grothendieck random permutations.
The simplest example is the number of inversions:

\begin{proposition}
	\label{prop:Grothendieck_permuton_inversions_intro}
	Let
	$\mathbf{w}=\mathbf{w}(n)\in S_n$ be the Grothendieck random permutations with a
	fixed parameter $p\in(0,1)$.
	We have{  \setlength{\abovedisplayskip}{1pt}
	\begin{equation}
		\label{eq:Grothendieck_permuton_inversions_intro}
		\lim_{n\to\infty}\frac{\inv\left( \mathbf{w}(n) \right)}{\binom n2}=\gamma_p:= 1-\sqrt{\frac{1-p}{p}}\ssp \arccos\sqrt{1-p}.
	\end{equation} }
\end{proposition}
In the original model we have $\gamma_{\frac{1}{2}}=1-\frac{\pi}{4}$.
The fact that the scaled number of inversions converges to $\gamma_p$,
the singular part of the Grothendieck permuton, is surprising.
Besides exact computations
we do not have a conceptual explanation for this phenomenon.

% \begin{remark}
% 	\label{remark:Grothendieck_and_ASM}
% 	In \Cref{sub:square_ice,sub:square_ice_aztec_asymptotics}, we connect
% 	Grothendieck random permutations
% 	to
% 	2-enumerated Alternating Sign Matrices
% 	(bijectively corresponding to the six-vertex model with
% 	domain wall boundary conditions and free-fermion weights, and also to
% 	domino tilings of the Aztec diamond).
% 	The connection between Grothendieck polynomials and the
% 	six-vertex model goes back to
% 	\cite{Lascoux02ice}, see also \cite{Weigandt2020_bumpless}.
% 	Equivalences between 2-enumerated Alternating Sign Matrices,
% 	six-vertex models, and domino tilings of the Aztec diamond
% 	are well-known in statistical mechanics and integrable probability,
% 	see, for example,
% 	\cite{cohn-elki-prop-96}, \cite{CohnKenyonPropp2000}, and
% 	\cite{elkies1992alternating}, \cite{zinn2000six}, \cite{ferrari2006domino}.

% 	Via this connection,
% 	our asymptotic results (\Cref{thm:Grothendieck_permutations_asymptotics_intro})
% 	can be recast as results about a random permutation coming from the free-fermion
% 	six-vertex model
% 	with domain wall boundary conditions.
% 	Moreover, there is a vast amount of asymptotic results
% 	about domino tilings and free-fermion six-vertex models,
% 	and in \Cref{sub:square_ice_aztec_asymptotics}
% 	we show how some of them lead to further asymptotic properties
% 	of Grothendieck random permutations.
% \end{remark}

\subsection{Proofs via colored vertex models and TASEP}

The \emph{colored stochastic six-vertex model} on $\mathbb{Z}^2$ has the following vertex weights
$$
w_p(a,b;c,d)
=
w_p
\bigl(
	\begin{tikzpicture}[baseline=-3,scale=.7,very thick]
			\draw[fill] (0,0) circle [radius=0.025];
			\draw (0.5,0) -- (0.05,0);
			\draw (-0.5,0) -- (-0.05,0);
			\draw (0,0.05) -- (0, 0.5);
			\draw (0,-0.05) -- (0,-0.5);
			\node at (.25,-.4) {\scriptsize $a$};
			\node at (-.7,0) {\scriptsize $b$};
			\node at (-.25,.4) {\scriptsize $c$};
			\node at (.7,0) {\scriptsize $d$};
\end{tikzpicture}
\bigr),$$ where
$a,b,c,d\in \left\{0, 1,\ldots,n \right\}$.
Here~$0$ represents
the absence of a pipe, and
positive numbers indicate the
pipes' colors.
We view $(a,b)$ and $(c,d)$ as incoming and outgoing pipes, respectively.
The weights are defined as follows: \( w_p(a,a;a,a) = 1 \),
\( w_p(b,a;b,a) = p \), \( w_p(b,a;a,b) = 1-p \), \(
w_p(a,b;a,b) = 0 \), and \( w_p(a,b;b,a) = 1 \), where \( 0
\le a < b \le n \) (weights of all other configurations are $0$). The
weights \emph{conserve} the pipes, meaning \( w_p(a,b;c,d) =
0 \) unless \( \{a,b\} = \{c,d\} \) as sets, and are
\emph{stochastic}: \( w_p(a,b;c,d) \ge 0 \) and \(
\sum_{c,d=0}^n w_p(a,b;c,d) = 1 \) for all \( a, b \).
Assign coordinates \( (i,j) \in \mathbb{Z}_{\ge 1}^2 \) to
the boxes of the staircase shape, with \( i \) increasing
downward and \( j \) increasing to the right. The staircase
shape is then \( \updelta_n \coloneqq \{ (i,j) : i + j \le n
\} \).

Place a stochastic vertex with the weight $w_p$ at each box $(i,j)\in \updelta_n$.
Let the initial condition along the left boundary of $\updelta_n$ be
the \emph{rainbow} one, with colors $1$ to $n$ from top to bottom.
Then we can sample a random configuration
of pipes as in \Cref{fig:pipe_dream}, center,
by running a discrete time Markov chain with time $\tau=j-i$, where
$-(n-1)\le \tau\le n-1$. At each step $\tau\to \tau+1$,
the configuration with $j-i\le \tau$
is already sampled, which determines the incoming colors at all boxes
$(i,j)\in \updelta_n$ with $j-i=\tau+1$. The next
step $\tau\to \tau+1$ consists in an independent update of the outgoing colors
at all boxes $(i,j)\in \updelta_n$ with $j-i=\tau+1$,
using the stochastic vertex weights
$w_p(a,b;\cdot,\cdot)$, $1\le a,b\le n$.
Here we view each $w_p(a,b;\cdot,\cdot)$ as a probability distribution on possible outputs,
$(a,b)$ or $(b,a)$.
Reading off the outgoing colors at the top boundary of $\updelta_n$,
we arrive at a random permutation $\mathbf{w}\in S_n$.

\begin{proposition}
	\label{prop:pipe_dreams_to_vertex_model}
	The random permutation $\mathbf{w}\in S_n$ obtained from the colored stochastic six-vertex model
	as described above has the same distribution as the Grothendieck random permutation
	defined in \Cref{sub:random_permutations}.
\end{proposition}

The \emph{permutation height function} $H(x,y)$ is also the height function for that model:
\begin{equation}
	\label{eq:permutation_height_function}
		H(x,y)
		\coloneqq
		\operatorname{\#}\bigl\{\textnormal{pipes of colors $\ge x$
		which exit through positions $j\ge y $ at the top}\bigr\}
\end{equation}
Consider an \emph{uncolored} (\emph{color-blind}) stochastic
vertex model
with all pipes of the same color ($1$) or no pipes ($0$) and the following weights
$w_p^{\bullet}(0,0;0,0)=1,$ $w_p^{\bullet}(1,1;1,1)=1$ 	$w_p^{\bullet}(1,0;1,0)=p$, $w_p^{\bullet}(1,0;0,1)=1-p$,
$w_p^{\bullet}(0,1;0,1)=0$, $w_p^{\bullet}(0,1;1,0)=1$.

\begin{proposition}
	\label{prop:color_forgetting}
	Fix $1\le x\le n$. In the colored stochastic vertex model,
	erase all pipes of colors $<x$,
	and identify all the remaining pipes for colors $\ge x$.
	The resulting random configuration of
	uncolored pipes evolves
	according to
	the color-blind stochastic
	vertex model $w_p^{\bullet}$.
\end{proposition}

 \begin{definition}
	\label{def:TASEP}
	Let $k\ge 1$.
	Let
	$\xi(t)\coloneqq \bigr(\xi_1(t)>\ldots>\xi_k(t) \bigr)\subset \mathbb{Z}$
	be the
	$k$-particle
	discrete time TASEP (Totally Asymmetric Simple Exclusion Process)
	having
	parallel updates
	and geometrically distributed jumps.
	In detail, $\xi(t)$, $t\in \mathbb{Z}_{\ge0}$, is a discrete time Markov chain
	on particle configurations in $\mathbb{Z}$ which at each time step $t\to t+1$ evolves as follows:
	{
  \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
	\begin{equation}
		\label{eq:TASEP_parallel_def}
		\xi_i(t+1)=\xi_i(t)+\min \left( G_{i}(t+1), \xi_{i-1}(t)-\xi_i(t)-1 \right),\qquad
		1\le i\le k,
	\end{equation}}
	where $G_i(t+1)$ are independent geometric random variables with parameter $p\in(0,1)$,
i.e.
	$ \displaystyle	\Prob
		\left( G_i(t+1)=m \right)=(1-p)\ssp p^m,\qquad m\in \mathbb{Z}_{\ge0}.
$
	The update \eqref{eq:TASEP_parallel_def} occurs in parallel for all
	particles $1\le i\le k$, that is,
	the new positions $\xi_i(t+1)$ depend only on the configuration
	$\xi(t)$ at the previous time step, and new independent random variables.
	By agreement, we have $\xi_0(t)\equiv +\infty$,
	so that the first particle $\xi_1(t)$ performs an independent random walk
	with geometrically distributed jumps.
\end{definition}

Start the TASEP from
the \emph{densely packed} (\emph{step})
initial configuration
$\left\{ 1,2,\ldots,k \right\}$, that is,
$\xi_i(0)=k+1-i$, $1\le i\le k$.
Fix $n\ge k$, and introduce a \emph{moving exit boundary} of staircase shape
%which starts at $n+\frac12$, and deterministically moves to the left with speed $1$.
The \emph{exit time} of a particle $\xi_i(t)$ is then
$	\label{eq:exit_time}
	T_{\mathrm{exit}}(i)\coloneqq \min\left\{ t\colon \xi_i(t)\ge n+1-t \right\}.$


\begin{figure}[htpb]
		\centering
		\includegraphics[height=.27\textwidth]{./images/pipe_dream_etas.pdf}
	\qquad \qquad
	\scalebox{.7}{
	\begin{tikzpicture}
		[scale=.8, very thick]
		\draw[->] (.3,0)--++(6.7,0) node [above right] {$\mathbb{Z}$};
		\draw[->] (.5,-1)--++(0,7) node [left] {$t$};
		\foreach \t in {0,1,2,3,4,5}
		{
		\draw (.65,\t)--++(-.3,0) node [left] {\scriptsize$\t$};
		}
		\foreach \pt in {(1,1),(1,2),(1,3),(1,4),(1,5),
			(2,1),(2,2),(3,1),(3,2),(3,3),
			(4,0),(4,1),(4,2),
			(5,0),(5,1),
			(6,0)
			}
		{
		\draw \pt circle[radius=0.2];
		}
		\draw[red,line width=3pt] (6.5,-.25)--++(0,.75)
		--++(-1,0)--++(0,1)
		--++(-1,0)--++(0,1)
		--++(-1,0)--++(0,1)
		--++(-1,0)--++(0,1)
		--++(-1,0)--++(0,1)
		--++(-1.5,0)
		;
		\node at (1,-1.2) {$\xi_3$};
		\node at (2,-1.2) {$\xi_2$};
		\node at (3,-1.2) {$\xi_1$};
		\node at (1,-.6) {\scriptsize$1$};
		\node at (2,-.6) {\scriptsize$2$};
		\node at (3,-.6) {\scriptsize$3$};
		\node at (4,-.6) {\scriptsize$4$};
		\node at (5,-.6) {\scriptsize$5$};
		\node at (6,-.6) {\scriptsize$6$};
		\foreach \pt in {
			(3,0),(2,0),(1,0),
			(2,1),(1,1),
			(1,2),
			(2,3),(3,2),
			(2,4),(5,1)
		}
		{
			\draw[fill] \pt circle[radius=0.2];
		}
		\draw [line width=8pt,opacity=0.4,red] (3,0)--++(2,1);
		\draw [line width=8pt,opacity=0.4,blue] (2,3)--++(0,1);
	\end{tikzpicture}}
	\caption{\textbf{Left}: The evolution of the uncolored pipes $\eta(t)$.
		Here $n=6$ and $x=4$, so $k=3$. At time $t=2$, we have
		$\eta_2(2)=2$, $\eta_3(2)=4$, and the pipe $\eta_1$ has exited before $t=2$.
		We have $H(4,3)=2$. \textbf{Right}:
		The evolution of the process $\xi(t)$, in bijection
		with the pipe configuration on the left (that is, $\xi_m(t)= n+1-t-\eta_m(t)$).
		In detail, a move of the particle $\xi_\ell$ by $r\ge0$ steps
		corresponds to the pipe moving $r+1$ steps up.} 
		 %(at the same time increment).
		%Two pairs of corresponding moves are highlighted.
		\label{fig:vertex_TASEP_matching}
	\end{figure}


\begin{theorem}
	\label{thm:vertex_TASEP_matching}
	For any $1\le x,y\le n$ and $0\le h\le n-x+1$, we have
	{
  \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
	\begin{equation*}
		\label{eq:vertex_TASEP_matching_theorem}
			\Prob_{\mathbf{w}}\bigl(H(x,y)\le h\bigr)
			=
			%\Prob_{\mathrm{TASEP}}\bigl(T_{\mathrm{exit}}(n\!-\!x\!+\!1\!-\!h)\le y\!-\!1\bigr)
			%=
			\Prob_{\mathrm{TASEP}}\bigl(
				\xi_{n-x+1-h}(y-1)\ge n-y+2
			\bigr).
	\end{equation*}}
	Here
	$\Prob_{\mathbf{w}}$
	corresponds to the Grothendieck random permutation of order $n$,
	and
	$\Prob_{\mathrm{TASEP}}$
	is the probability distribution of the TASEP
	with $k=n-x+1$ particles and moving exit boundary.
	%In \eqref{eq:vertex_TASEP_matching_theorem}
	%we have, by agreement,
	%$T_{\mathrm{exit}}(m)= 0$ and
	%$\xi_m(t)= +\infty$ for all $m\le 0$.
\end{theorem}

To prove \Cref{thm:Grothendieck_permutations_asymptotics_intro} we use the integrability of TASEP~\cite{johansson2000shape} with the given moving boundary (via asymptotics of Schur functions) to analyze the asymptotics of $H(x,y)$.

\section{Asymptotics
%and maxima
of Grothendieck principal specializations}
\label{sub:intro_Groth_spec}

We are interested in the \emph{principal specializations} of the Grothendieck polynomials:
{
  \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
\begin{equation}
	\label{eq:Upsilon_w_definition_intro}
	\Upsilon_w(\beta) \coloneqq \mathfrak{G}_w^{\beta}\bigl(\underbrace{1,1,\ldots,1}_n\bigr),
	\qquad w\in S_n.
\end{equation}}
In particular, $\Upsilon_w(0)$ is the principal specialization of the Schubert polynomial studied in~\cite{MoralesPakPanova2019} and~\cite{stanley2017some}. For $\beta=1$, these quantities are the (unnormalized) probability weights of the Grothendieck random permutation~$\mathbf{w}$ with $p=\frac{1}{2}$ (see \eqref{eq:pipe_dream_specialization_p_Grothendieck_intro}). Set
{
  \setlength{\abovedisplayskip}{.5em}
  \setlength{\belowdisplayskip}{.5em}
$$
	v_n(\beta) \coloneqq \sum_{w \in S_n} \Upsilon_w(\beta) \quad \text{and} \quad u_n(\beta) \coloneqq \max_{w\in S_n} \Upsilon_w(\beta).
$$}
From \Cref{thm:Grothendieck_pipe_dreams_intro}, we have
$\displaystyle 
	v_n(1)=2^{\binom{n}{2}} \quad \text{and} \quad u_n(1)=2^{\binom{n}{2}-o(n^2)} \quad \text{as} \quad n\to\infty.$
The first equality is exact, while the second one is asymptotic.
This asymptotic behavior
follows from
the cardinality
$n!\sim e^{n\log n+O(n)}\ll 2^{\binom{n}{2}}$ of the set of all $n!$ permutations.

For $u \in S_k$ and $w \in S_m$, we denote
$\displaystyle u \times w \coloneqq (u(1), \ldots, u(k), w(1)+k, \ldots, w(m) + k)\in S_{k+m}.$
For a composition $b = (b_{\ell}, \ldots, b_1)$ of $n = b_1 + \cdots + b_{\ell}$, the {\it layered permutation} $w(b) \in S_n$ is defined as
$
w(b) \coloneqq w_0(b_{\ell}) \times \cdots \times w_0(b_{1}), $
where $w_0(k)=(k,k-1,\ldots,1)$ is the full reversal permutation in the one-line notation.
Denote by $L_n\subset S_n$ the subset of layered permutations of size $n$,
and let
	$u_n'(1)\coloneqq \max_{w\in L_n} \Upsilon_w(1).$

\begin{figure}[htbp] \scalebox{.9}{
	\raisebox{0.15in}{\includegraphics[width=3in]{images/domino10.pdf}}}
\raisebox{0.3in}{
%{\begin{minipage}{0.3\textwidth}
\includegraphics[width=1.4in]{images/pslayeredsqrt2_18.jpg}}
\raisebox{1in}{
$
\begin{array}{rrr}
\hline
n & b_1 & \qquad \quad f(n)\\ \hline
% 4&2&0.175460\\
% 6&2&0.215599\\
% 9& 3&0.279931\\
% 13& 4&0.328982\\
% 19&6&0.375242\\
% 27&8&0.406540\\
% 39&12&0.433385\\
% 55&16&0.450935\\
%78&23&0.464651\\
%110&32&0.474448\\
%156&46&0.481823\\
\vdots & \vdots & \vdots \\
%220&64&0.486974\\
311&91&0.490735 \\
439&128&0.493404\\
621&182 & 0.495329 \\
877&256 & 0.496684\\ \hline
\end{array}
$}
%\end{minipage} }
\caption{\textbf{Left}: Domino tilings with Schroder paths, frozen boundary indicated as the inscribed circle. \textbf{Right}: Permutation matrix of a layered permutation $w(b)\in S_{877}$, where
		the composition is
		$b=(256, 182, 128, 91, 64, 46, 32, 23, 16, 12, 8, 6, 4, 3, 2, 2, 1, 1)$.
		Note that
		$b_i /b_{i+1} \approx 1/\sqrt{2}$. Table of exact values for $3\le k\le 19$ of layered permutations $w(b)$ with $b_i/b_{i+1} \approx 1/\sqrt{2}$. The third column is
	$f(n)\coloneqq \frac{1}{n^2} \log_2 \Upsilon_{w(b)}(1)$ for $n=\sum_i b_i$.}
	\label{fig:value_ps_Groth_layered}
\end{figure}

Regarding Stanley's problem on permutations achieving the
maximal Schubert specialization $u_n(0)$ (the case
$\beta=0$), the Merzon--Smirnov conjecture
\cite{merzon2016determinantal} states that \emph{the maximum
is attained on layered permutations}. In~\cite{MoralesPakPanova2019}, $\frac{1}{n^2} \max_{w \in L_n} \log_2 \mathfrak{S}_w(1^n) \approx 0.29$ was computed. As some evidence that indeed layered permutations are good candidates, we prove that $\Upsilon_w(\beta)$ at $\beta=1$
attain their asymptotic maximum in $L_n$:

\begin{theorem}
	\label{thm:gmax_intro}
There are sequences of layered permutations $w(b^{(n)})\in S_n$ so that
{
  \setlength{\abovedisplayskip}{1.2pt}
$$
\lim_{n \to \infty} \frac{1}{n^2} \log_2
\Upsilon_{w(b^{(n)})} (1)
= \frac{1}{2}.
$$}
\end{theorem}
% \Cref{thm:gmax_intro} implies that in leading order,
% $u_n'(1)\sim 2^{n^2/2 -O(n \log n)}$, i.e. it asymptotically behaves in the same way as
% $v_n(1)$ and $u_n(1)$ \eqref{eq:Grothendieck_specialization_v_n_u_n_intro}.

Explicit constructions of such sequences of layered
permutations can be obtained with
geometric $b_i \sim
(1- \alpha)\alpha^{i-1} n$ for any $\alpha \in [1/\sqrt{2}, 1)$.
See
\Cref{fig:value_ps_Groth_layered} for an illustration.
Note, however that we do not know in the limit what
compositions $b$ of size $n$ yield the global maximum of
$\Upsilon_{w(b)}$ over all layered permutations. For
Schubert polynomials, the analogous question was settled in
\cite{MoralesPakPanova2019} with an explicit limiting composition. Our
\Cref{thm:gmax_intro}
establishes that an asymptotic analog of the Merzon--Smirnov conjecture holds for
the $\beta=1$
Grothendieck polynomials. Bounds on the quantities
$v_n(\beta)$ and $u_n(\beta)$ for general values of $\beta$ can also be obtained (see \cite[Section 6.5]{morales2024grothendieckshenaniganspermutonspipe}).
%We present numerics
%for principal specializations of the $\beta=1$ Grothendieck polynomials in
%\Cref{app:data_Groth}, and explain the difference between the (numerically)
%optimal layered permutations and the ones constructed %in the proof of \Cref{thm:gmax}.

\medskip
\noindent
\textbf{Sketch of proof of \Cref{thm:gmax_intro}.}
 Let $u \in S_k, w \in S_n$. We have
	$
		\Upsilon_{u \times w}(\beta) = \Upsilon_{u}(\beta) \cdot \Upsilon_{\mathrm{id}_k \times w}(\beta),
	$
		where
		$u\times w$ is the block permutation. Let $\ls_k$ be the little Schr\"oder numbers, which count paths from $(0,0)$ to $(2k,0)$ of steps $(1,1), (1,-1),(2,0)$ and which do not have a horizontal step on the $x$-axis. We have the following identity:
\begin{theorem}[{\cite[Theorem 5.9]{MPP4GrothExcited}}]
	\label{thm:Groth_beta_1}
	For nonnegative integers $k$ and $n$, we have
	{ \setlength{\abovedisplayskip}{.5em}
	\begin{equation}\label{eq:determinant_NILP}
			\Upsilon_{w_0(k;n)}(1) = \mathfrak{G}^{\beta=1}_{w_0(k;n)}(1,\ldots,1) = 2^{-\binom{k}{2}}\det[\ls_{n-2+i+j}]_{i,j=1}^k.
	\end{equation}}
\end{theorem}

\begin{lemma}\label{lemma:fbound}
Let
$k \in (n/\sqrt{2}, n]$. We have
{ \setlength{\abovedisplayskip}{1pt}
\begin{equation}
	\label{eq:fbound}
\log_2 \Upsilon_{w_0(k;n-k)}(1) = \frac{n^2}{2} - \frac{k^2}{2} - O(n), \qquad n \to \infty.
\end{equation}}
\end{lemma}
\begin{proof}[Proof of \Cref{lemma:fbound}]By Lindstrom--Gessel--Viennot,
	the determinant
	\eqref{eq:determinant_NILP}
	counts $k$-tuples of nonintersecting Sch\"oder paths staying above height $0$ and
	starting from $(-2i,0)$
	and ending at
	$(2(n-k+i-1),0)$, respectively, where $i=1,\ldots,k $.
	Denote this space of configurations by $\mathcal{M}_{k,n-k}$.

	A well-known
	correspondence (e.g., see \cite{eu2005simple})
	between domino tilings of the Aztec diamond
	and nonintersecting paths allows to interpret
	{ \setlength{\abovedisplayskip}{.5em}  \setlength{\belowdisplayskip}{.5em}
	\begin{equation}
		\label{eq:determinant_NILP_count}
		\frac{\det [\ls_{n-k-2+i+j}]_{i,j=1}^k}{2^{\binom n2}}
	\end{equation}}
	as a probability of a certain event in the uniformly random
	domino tiling of the Aztec diamond $D_{n-1}$ of order $n-1$.
	The ratio
	$|\mathcal{M}_{k,n-k}|/|\mathcal{S}_n|$
	given by
	\eqref{eq:determinant_NILP_count}
	is equal to the probability that the $n-k$ paths
	in $\mathcal{S}_n\setminus \mathcal{M}_{k,n-k}$
	are in their \emph{lowest possible configuration}.


	The model of uniformly random domino tilings of the Aztec diamond
	develops an arctic circle \cite{jockusch1998random}. In
	particular,
	the configuration outside of the circle
	inscribed in the Aztec diamond (illustrated in \Cref{fig:value_ps_Groth_layered})
	is frozen (nonrandom).
	Qualitatively,
	by \cite[Proposition~13]{cohn-elki-prop-96},
	this means that with probability $1-e^{-O(n)}$,
	the $n-k$ paths in $\mathcal{S}_n\setminus \mathcal{M}_{k,n-k}$
	are indeed in their lowest possible configuration.
	Note that here we rely on the assumption $k>n/\sqrt 2$,
	which guarantees that the top of the $n-k$ paths
	in $\mathcal{S}_n\setminus \mathcal{M}_{k,n-k}$
	does not reach the arctic circle.
	Therefore, we get desired asymptotics 
	\eqref{eq:fbound}.
\end{proof}


Let $w_0(k;n)\coloneqq \mathrm{id}_k \times w_0(n) = (1,\ldots,k, k+n,k+n-1,\ldots,k+1)
	 \in S_{k+n}$. Using~\Cref{lemma:fbound} we can estimate that if $k>n/\sqrt{2}$ then
	 $\displaystyle \log_2 \mathfrak{G}_{w_0(k;n)}(1^n)= \frac{n^2}{2}-\frac{k^2}{2} - O(n).$
	 Via the multiplicativity property $\mathfrak{G}_{u\times v}(1^n) = \mathfrak{G}_u(1^{|u|} ) \cdot \mathfrak{G}_{\mathrm{id}_{|u|}\times v}(1^n)$ we can then estimate the values for a layered permutation recursively as 
	 { \setlength{\abovedisplayskip}{1pt}
	  \setlength{\belowdisplayskip}{1.2pt}
	 $$\log_2 \mathfrak{G}_{w(\ldots, b_2, b_1)}(1^n)= \log_2 \mathfrak{G}_{w(\ldots,b_3,b_2)}(1^{n-b_1}) + \frac{n^2}{2}-\frac{k_1^2}{2} - O(n), $$ }
where $k_i:=n-b_1-\cdots -b_i$ and $k_i/k_{i-1} \in (1/\sqrt{2},1)$ and establish~\Cref{thm:gmax_intro}.


\acknowledgements{We are very grateful to Jehanne Dousse for her valuable insights in this project and help with experiments. We thank Igor Pak and AIM for bringing us together as part of a SQuaRE.
 We thank Dave Anderson, Philippe Biane, Jacopo Borga, Filippo Colomo, Colin Defant, Philippe DiFrancesco, Rick Kenyon, Matthew Nicoletti, Khaydar Nurligareev, Mikhail Tikhonov, and Anna Weigandt for helpful discussions. We thank IPAM for the hospitality as three of the four authors were part of a long program.}
%% if you use biblatex then this generates the bibliography
%% if you use some other method then remove this and do it your own way

\printbibliography

\end{document}
