%% 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]{FPSAC2021}
\articlenumber{34}
%% 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}



\usepackage{amsmath,amssymb,amsthm,amsfonts}
\usepackage{mathtools}%Mathematical tools to use with amsmath

%\usepackage[english]{babel}%Multilingual support for Plain TeX or LaTeX
\usepackage{comment}%Selectively include/exclude portions of text
% \usepackage[hypertex]{hyperref}
\usepackage{hyperref}%Extensive support for hypertext in LaTeX
\usepackage{bbm}%"Blackboard-style" cm fonts
\usepackage{bm}%"Blackboard-style" cm fonts
\usepackage{mathrsfs}%Support for using RSFS fonts in maths (\mathcal fonts via \mathscr)

% ========================================graphics packages:
\usepackage{tikz,graphicx,color}
\usepackage{tikz-cd}%Create commutative diagrams with TikZ
\usepackage{tikz-3dplot}%Coordinate transformation styles for 3d plotting in TikZ
% % for more on tikz libs, see 
% % http://tex.stackexchange.com/questions/42611/list-of-available-tikz-libraries-with-a-short-introduction
\usetikzlibrary{calc}%to make complex coordinate caculations.
\usetikzlibrary{arrows}%Provides various new and costumizable arrow tips
\usetikzlibrary{shapes}%Shape library, used to define shapes other than rectangle, circle and co-ordinate
% \usetikzlibrary{snakes}%curved lines, and are used either between nodes or as a border to a shape , or as independent shapes. DOESN'T COMPILE
\usetikzlibrary{patterns}%This package "defines patterns for filling areas".
\usetikzlibrary{positioning}%a way of positioning one node in respect with several others
\usetikzlibrary{arrows.meta}
\usetikzlibrary{decorations.markings}
\usepackage{xspace}

% \usepackage{showframe}

% \usetikzlibrary{external}
% \tikzexternalize[prefix=figures/]

\pdfpageattr{/Group <</S /Transparency /I true /CS /DeviceRGB>>}

\usepackage{epstopdf}%Convert EPS to PDF using Ghostscript

% ========================================extra packages:
\usepackage{enumerate}%Enumerate with redefinable labels. Precedes enumitem
% \usepackage{enumitem}%Control layout of itemize, enumerate, description. Supersedes enumerate
\usepackage[arrow]{xy}%Typesetting Graphs and Diagrams in TeX 
\usepackage{diagbox}%Table heads with diagonal lines
\usepackage{subfig}%Figures broken into subfigures
% \usepackage{chngcntr} %Change the resetting of counters
\usepackage{arcs}%Draw arcs over and under text. \overarc and \underarc
\usepackage{xcolor}%Driver-independent color extensions for LaTeX and pdfLaTeX
\usepackage{tabu}
\usepackage{booktabs}% for better rules in the table

% ========================================not recommended:
% \usepackage[margin=1in]{geometry}%Flexible and complete interface to document dimensions DOESN'T COMPILE
% \usepackage{epsf}%Simple macros for EPS inclusion DOESN'T COMPILE
% \usepackage{epsfig}%Include Encapsulated PostScript in LaTeX documents DOESN'T COMPILE
% \usepackage{graphics}%Standard LaTeX graphics
% \usepackage{pgffor}%smth similar to pstricks
% \usepackage{tabularx}
% % \usepackage[table]{xcolor}    % loads also »colortbl« DOESN'T COMPILE
% % \usepackage[enableskew]{youngtab}%Typeset Young-Tableaux DOESN'T COMPILE
% % \usepackage{cite}%Improved citation handling in LaTeX
\usepackage{lipsum}


% Typography
\usepackage{enumitem}
% \usepackage{enumerate}
\usepackage{letltxmacro}
\usepackage{thmtools,etoolbox}

%TODO ADD TO TEMPLATE
\def\myarabic#1{\normalfont(\roman{#1})}
\newlist{theoremlist}{enumerate}{1}
\setlist[theoremlist]{label=\myarabic{theoremlisti},ref={\myarabic{theoremlisti}},itemindent=0pt,labelindent=0pt,
leftmargin=*,noitemsep}

%TODO ADD TO TEMPLATE
\makeatletter
\renewcommand{\p@theoremlisti}{\perh@ps{\thetheorem}}
\protected\def\perh@ps#1#2{\textup{#1#2}}
\newcommand{\itemrefperh@ps}[2]{\textup{#2}}
\newcommand{\itemref}[1]{\begingroup\let\perh@ps\itemrefperh@ps\ref{#1}\endgroup}
\makeatother


\usepackage{nameref,hyperref}%TODO ADD TO TEMPLATE
% \usepackage[capitalize]{cleveref}%TODO ADD TO TEMPLATE




%\usepackage{autonum}%TODO ADD TO TEMPLATE
\newcommand{\retainlabel}[1]{\label{#1}\sbox0{\ref{#1}}}
% \newcommand{\retainlabel}[1]{}




\newtheorem{theorem}{Theorem}[section]
\newtheorem*{statement}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{remark}[theorem]{Remark}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{question}[theorem]{Question}
\theoremstyle{definition}
\newtheorem{problem}[theorem]{Problem}
\theoremstyle{definition}
\newtheorem{example}[theorem]{Example}

%TODO ADD TO TEMPLATE
\addtotheorempostheadhook[theorem]{\crefalias{theoremlisti}{theorem}}
\addtotheorempostheadhook[lemma]{\crefalias{theoremlisti}{lemma}}
\addtotheorempostheadhook[proposition]{\crefalias{theoremlisti}{proposition}}




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

\usepackage{lipsum}
% \usepackage[margin=1in,showframe]{geometry}





% ================mathcal symbols
\def\Acal{\mathcal{A}}\def\Bcal{\mathcal{B}}\def\Ccal{\mathcal{C}}\def\Dcal{\mathcal{D}}\def\Ecal{\mathcal{E}}\def\Fcal{\mathcal{F}}\def\Gcal{\mathcal{G}}\def\Hcal{\mathcal{H}}\def\Ical{\mathcal{I}}\def\Jcal{\mathcal{J}}\def\Kcal{\mathcal{K}}\def\Lcal{\mathcal{L}}\def\Mcal{\mathcal{M}}\def\Ncal{\mathcal{N}}\def\Ocal{\mathcal{O}}\def\Pcal{\mathcal{P}}\def\Qcal{\mathcal{Q}}\def\Rcal{\mathcal{R}}\def\Scal{\mathcal{S}}\def\Tcal{\mathcal{T}}\def\Ucal{\mathcal{U}}\def\Vcal{\mathcal{V}}\def\Wcal{\mathcal{W}}\def\Xcal{\mathcal{X}}\def\Ycal{\mathcal{Y}}\def\Zcal{\mathcal{Z}}


% ================mathbf symbols
\def\abf{\mathbf{a}}\def\bbf{\mathbf{b}}\def\cbf{\mathbf{c}}\def\dbf{\mathbf{d}}\def\ebf{\mathbf{e}}\def\fbf{\mathbf{f}}\def\gbf{\mathbf{g}}\def\hbf{\mathbf{h}}\def\ibf{\mathbf{i}}\def\jbf{\mathbf{j}}\def\kbf{\mathbf{k}}\def\lbf{\mathbf{l}}\def\mbf{\mathbf{m}}\def\nbf{\mathbf{n}}\def\obf{\mathbf{o}}\def\pbf{\mathbf{p}}\def\qbf{\mathbf{q}}\def\rbf{\mathbf{r}}\def\sbf{\mathbf{s}}\def\tbf{\mathbf{t}}\def\ubf{\mathbf{u}}\def\vbf{\mathbf{v}}\def\wbf{\mathbf{w}}\def\xbf{\mathbf{x}}\def\ybf{\mathbf{y}}\def\zbf{\mathbf{z}}

% ================mathfrak symbols
\def\afr{\mathfrak{a}}
\def\Sfr{{ \mathfrak{S}}}
% ...

% ================mathbb symbols
\def\one{{\mathbbm{1}}}
\def\C{\mathbb{C}}
\def\R{\mathbb{R}}
\def\N{\mathbb{N}}
\def\Z{\mathbb{Z}}
\def\Q{\mathbb{Q}}
\def\P{\mathbb{P}}
% ...

% ==============Hat, bar, tilde, par
\def\Ahat{\hat{A}} 
\def\ipar{{(i)}}
\def\kpar{{(k)}}
\newcommand\parr[1]{{({#1})}}
% ...


% ==============Math notation
\def\<{{\langle}}
\def\>{{\rangle}}
\def\e{{\epsilon}}
\def\l{{\lambda}}
\def\m{{\mu}}
\def\lm{{\l/\m}}
\def\RP{{\R P}}
\def\CP{{\C P}}
\def\multiset#1#2{\left(\!\left({#1\atopwithdelims..#2}\right)\!\right)}
\def\toi{{\xhookrightarrow{i}}}

% ==============Operators
\def\Id{\operatorname{Id}}
\def\Vol{\operatorname{Vol}}
\def\det{{ \operatorname{det}}}
\def\tr{{ \operatorname{tr}}}
\def\Ker{{ \operatorname{Ker}}}
\def\Im{{ \operatorname{Im}}}
\def\diag{{ \operatorname{diag}}}
\def\rank{{ \operatorname{rank}}}
\def\rk{{\mathrm{rk}}}
\def\cork{ \operatorname{cork}}
\def\codim{ \operatorname{codim}}
\def\Vert{{ \operatorname{Vert}}}
\def\op{{ \operatorname{op}}}
\def\type{{\operatorname{type}}}
\def\sh{{ \operatorname{sh}}}
\def\Conv{ \operatorname{Conv}}
\def\Span{ \operatorname{Span}}
\def\proj{ \operatorname{proj}}
\def\sgn{ \operatorname{sgn}}
\def\wt{\operatorname{wt}}
\def\inv{\operatorname{inv}}

\def\RR{{\mathbb R}}
\def\CC{{\mathbb C}}
\def\RP{{\RR\mathbb P}}
\def\CP{{\CC\mathbb P}}

% ==============Total positivity
\def\GL{\operatorname{GL}}
\def\SL{\operatorname{SL}}
\def\Fl{\operatorname{Fl}}
\def\Mat{\operatorname{Mat}}
\def\Gr{\operatorname{Gr}}
\def\Grtnn{\Gr_{\ge 0}}
\def\Grtp{\Gr_{>0}}
\def\alt{\operatorname{alt}}
\def\Mattp{\operatorname{Mat}_{>0}}
\def\Mattnn{\operatorname{Mat}_{\ge0}}

\def\medpa{\tau}
\def\xing{{\operatorname{xing}}}

\newcommand{\TODO}[1]{\textcolor{brown}{\bf TODO: #1}}
\def\todo{\textcolor{brown}{\bf [TODO]}\xspace}
\def\todoe{\textcolor{brown}{\bf [TODO:ex]}\xspace}
\def\todof{\textcolor{brown}{\bf [TODO:fig]}\xspace}
\def\todoc{\textcolor{brown}{\bf [TODO:cit]}\xspace}
% \def\todof{\textcolor{purple}{\bf [TODO-fig]}\xspace}%TODO ADD TO TEMPLATE
 \newcommand{\PG}[1]{\textcolor{red}{PG: #1}}

\newcounter{todofigure}
\def\todof{\stepcounter{todofigure}\textcolor{purple}{\bf [TODO-fig-\thetodofigure]}\xspace}%TODO ADD TO TEMPLATE

\def\n{N}
\def\pc{\Pi^{>0}}

% \newcommand{\tarc}{\mbox{\large$\frown$}}
% \newcommand{\arc}[1]{\stackrel{\tarc}{#1}}

\def\Gmed{{G^\times}}
\def\Gbip{{G^{\square}}}
% \def\Gdub{\stackrel{\frown}{G^\square}}
% \def\Edub{\stackrel{\frown}{E^\square}}
% \def\Vdub{\stackrel{\frown}{V^\square}}
\def\Gdub{\widehat{G}^\square}
\def\Edub{\widehat{E}^\square}
\def\Vdub{\widehat{V}^\square}
\def\bbip{d}

\def\vscl{0.3}
\def\vsclG{0.3}
\def\scsc{0.6}
\def\tikzscl{1}
\def\bscl{0.6}
\def\sclbx{1.4}
\newcommand\edgepl[2]{\draw[line width=1pt,blue] (#1) -- (#2);}
\newcommand\edgemed[2]{\draw[line width=1.5pt,brown] (#1) -- (#2);}

\newcommand\edge[2]{\draw[line width=1.5pt] (#1) -- (#2);}
\newcommand\edgedir[2]{\draw[-{Stealth[scale=1.0]},line width=1.5pt] (#1) -- (#2);}
\newcommand\edgeop[2]{\draw[dashed, line width=1.5pt,opacity=0.4] (#1) -- (#2);}
\newcommand\disk[1]{\draw[dashed, opacity=0.3] (#1) circle (1);}
\newcommand\vertx[2]{\node[draw,circle,fill=black,scale=\vsclG] (#1) at (#2) {};}
\newcommand\vwh[2]{\node[draw,circle,fill=white,scale=\vscl] (#1) at (#2) {};}
\newcommand\vbl[2]{\node[draw,circle,fill=black!50,scale=\vscl] (#1) at (#2) {};}



\newcommand\edgeplop[2]{\draw[line width=0.3pt,black,opacity=0.5] (#1) -- (#2);}
\newcommand\edgeplmatch[2]{\draw[line width=2pt,blue] (#1) -- (#2);}
\newcommand\edgelabel[3]{\draw[line width=1.5pt] (#1) -- (#2) node[midway,above,scale=\scsc] {$#3$};}

\def\side{0.3}


  
% \def\Prob{\mathbf{P}}
\def\Prob{\operatorname{Prob}}
\def\Zpart{Z}
\def\M{M}
\def\Space{\Xcal}
\def\Closure{\overline{\Space}}
\def\OG{\operatorname{OG}}
\def\OGtnn{\OG_{\ge0}}
\def\OGtp{\OG_{>0}}
\def\doublemap{\phi}
\newcommand\double[1]{\widetilde{#1}}
\newcommand\Matsymx[1]{\Mat^{\operatorname{sym}}_{#1}(\R,1)}
\def\Matsym{\Matsymx{n}}

\def\insideprod{q}

% \def\Matsym{\Mat^{{\operatorname{sym}}}_n(\R)}
% \newcommand\Matsymx[1]{\Mat^{\operatorname{sym}}_{#1}(\R)}
% \def\doublemap{\operatorname{X}}

\def\P{P}






\def\Reg{R}
% \def\Reg{\Omega}
\def\free{{\operatorname{free}}}
\def\Hbb{{\mathbb{H}}}
\def\Dbb{{\mathbb{D}}}
\def\arg{\operatorname{arg}}


\def\curve{\bm{\gamma}}
\def\bth{{\bm{\theta}}}
\def\btht{{\tilde{\bm{\theta}}}}

\def\bv{{\bm{v}}}

\def\g{\gamma}
\def\J{J}

\def\bx{{\mathbf{x}}}
% \def\Tiling{{\Tcal}}
\def\Tiling{{\mathbb{T}}}

\crefname{figure}{Figure}{Figures}



\def\Arr{{\Acal}}
\def\ska#1{s_k\cdot #1}
\def\ska#1{#1\cdot s_k}
\def\coef{f}

\def\taut{{\tilde\tau}}
\def\Jt{{\tilde\J}}

\def\v{v}
\def\th{\theta}
\def\tht{{\tilde\theta}}
\def\vt{{\tilde\v}}
\def\eps{\varepsilon}
\def\bas{e}

\def\DUMMY{\Phi(t)}

% \def\gg{\bar\g}
\def\gg{\Gamma}
\def\GG{{\bm{\Gamma}}}
\def\supp{\operatorname{supp}_\tau}%todo add to template

\def\u#1{{u_\Reg^\parr{#1}}}


\def\RegN{\Reg_{\parr N}}

\def\bthN{\bth_{\parr N}}
\def\dd#1{\bar\partial_{#1}}
\def\Jp{J'}
\def\Jpp{J''}
\def\w#1{w^\parr{#1,N}}

% \def\GGB{\GG^{[b]}_{\RegN}}
\def\GGB{\GG^{\flat}_{\RegN}}
\def\GGBT{\GGB(T)}

\def\SpanC{\Span_\C}


% \def\figref#1(#2){Figure~\hyperref[#1]{\ref*{#1}(#2)}}
\def\figref#1(#2){\hyperref[#1]{Figure~\ref*{#1}(#2)}}


%% define your title in the usual way
\title{Critical varieties in the Grassmannian}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{Pavel Galashin\thanks{\href{mailto:galashin@math.ucla.edu}{galashin@math.ucla.edu}. P.G.\ was supported by an Alfred P. Sloan Research Fellowship and by the National Science Foundation under Grants No.~DMS-1954121 and No.~DMS-2046915.}\addressmark{1}}

%% then use \addressmark to match authors to institutions here
\address{\addressmark{1}Department of Mathematics, University of California, Los Angeles}%\addressmark{1}

%% 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 introduce a family of spaces called \emph{critical varieties}. The positive part of each critical variety is a subset of one of Postnikov's positroid cells inside the totally nonnegative Grassmannian. The combinatorics of positroid cells is captured by the dimer model on a planar bipartite graph $G$, and the critical variety is obtained by restricting to Kenyon's critical dimer model associated to a family of isoradial embeddings of $G$. This model is invariant under square/spider moves on $G$, and we give an explicit boundary measurement formula for critical varieties which does not depend on the choice of $G$. Special cases include critical electrical networks and Baxter's critical $Z$-invariant Ising model associated to rhombus tilings of polygons in the plane. In the case of regular polygons, our formula yields new simple expressions for response matrices of electrical networks and for correlation matrices of the Ising model.}

%% 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{\todo}

%% 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, ie. [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]{biblatex}
 \addbibresource{34Galashin.bib}
%% note the \printbibliography command at the end of the file which goes with these biblatex commands




\def\Ising{{\operatorname{Ising}}}
\def\elec{{\operatorname{elec}}}
\def\La{\Lambda_\elec}
\def\la{\Lambda}
\def\M{M_{\Ising}}
\def\match{\Acal}
\def\pmatch{I_\Acal}

\def\RowSpan{\operatorname{RowSpan}}
\def\Skn{S_{k,n}}

\def\Ptp_#1{\Pi^{>0}_{#1}}

% \def\perm{\pi}
\def\perm{f}
\def\t{u}
\def\tt{{\mathbf{u}}}


\def\Meas{\operatorname{Meas}}

\def\ft{f}
\def\Bkn{\Bcal_{k,n}}

\def\fb{\bar f}
\def\m#1{{#1}_{-}}
\def\p#1{{#1}_{+}}
\def\Crit{\operatorname{Crit}}
\def\Ctp{\Crit^{>0}}
\def\Cio{\Crit^{\C}}% this is closed
\def\Cioo{\Crit^{\circ}}% this is open
\def\D{D}
\def\J{J}
\def\L{L}
\def\eps{\epsilon}


\def\Ctnn{\Crit^{\geq0}}
\def\GT{G_\Tiling}
% \def\EGT{{E(G_\Tiling)}}
% \def\VGT{{V(G_\Tiling)}}
\def\EGT{E}
\def\VGT{V}
\def\N{N}
\def\fel{f^\elec_\Reg}
\def\fis{f^\Ising_\Reg}
\def\pel{\phi^\elec}
\def\pis{\phi^\Ising}


\def\br[#1]{|#1|}

% ==============================DOCUMENT BEGINS
\begin{document}

\maketitle

% \section{Introduction}
% \todo

% This paper is an extended abstract to~\cite

This is an extended abstract summarizing the results of~\cite{ising_crit,crit}. See those papers for the proofs and further details.

\section{Dimer model}
% \section{Background on the totally nonnegative Grassmannian}
Let $G$ be a planar bipartite graph embedded in a disk. We assume that $G$ has $n$ black boundary vertices, each of degree $1$, labeled $b_1,b_2,\dots,b_n$ in clockwise order. A \emph{strand} (or a \emph{zig-zag path}) in $G$ is a path that makes a sharp right turn at each black vertex and a sharp left turn at each white vertex. Thus $G$ gives rise to a \emph{strand permutation} $\perm_G\in S_n$: for each $1\leq p\leq n$, the strand that starts at $b_p$ terminates at $b_{\perm_G(p)}$. See \figref{fig:plabic}(b). We say that $G$ is \emph{reduced}~\cite{Pos} if it has the minimal number of faces among all graphs with the same strand permutation. It is known that a reduced graph contains no closed strands, thus each strand starts and ends at the boundary of $G$. For simplicity, we always assume that $\perm_G$ has no fixed points ($\perm_G(p)=p$), although the theory extends to that case in a straightforward fashion. 

Given a reduced planar bipartite graph $G$, one can consider the dimer model on it. Let us assign a positive real weight $\wt(e)$ to each edge $e$ of $G$. An \emph{almost perfect matching} $\match$ of $G$ is a collection of edges of $G$ that uses each interior vertex exactly once, and uses some subset of boundary vertices. We denote by $\{b_p\}_{p\in\pmatch}$ for $\pmatch\subset[n]:=\{1,2,\dots,n\}$ the set of boundary vertices used by $\match$. It is easy to check that there exists an integer $0\leq k\leq n$ such that any almost perfect matching of $G$ satisfies $|\pmatch|=k$. The number $k$ depends only on $\perm_G$: we have 
\begin{equation}\label{eq:perm_to_k}
k=\#\{1\leq p\leq n\mid \perm_G(p)<p\}.  
\end{equation}


% and we let $\Skn\subset S_n$ denote the set of permutations $f$ satisfying this condition. 
% We denote
% \[\Skn:=\{\perm\in S_n\mid k=\#\{1\leq i\leq n\mid \perm_G(i)<i\} \}. \]
 Denote the set of $k$-element subsets of $[n]$ by ${[n]\choose k}$, and for $I\in{[n]\choose k}$, define
\[  \Delta_I(G,\wt):=\sum_{\match:\;\pmatch=I} \wt(\match),\quad\text{where}\quad \wt(\match):=\prod_{e\in\match} \wt(e),\]
where the summation runs over almost perfect matchings of $G$. As we explain in \cref{sec:totally-nonn-grassm}, the \emph{boundary measurements} $\Meas(G,\wt):=(\Delta_I(G,\wt))_{I\in{[n]\choose k}}$ give rise to a point in the \emph{totally nonnegative Grassmannian}~\cite{Lus98,Pos}. We consider the tuple $(\Delta_I(G,\wt))_{I\in{[n]\choose k}}$ to be defined up to multiplication by a common scalar.

One can apply certain \emph{moves} to $(G,\wt)$ which preserve the boundary measurements. The most interesting transformation is known as the \emph{square move}: given a bipartite square face $F$ of $G$, one can uncontract some edges so that all vertices of $F$ become trivalent, then swap the colors of the vertices of $F$, and then contract the unicolored edges to obtain a new planar bipartite graph $G'$; see \figref{fig:move}(a) for an example. The weights of the edges are changed appropriately, and the resulting weighted graph $(G',\wt')$ satisfies $\Meas(G,\wt)=\Meas(G',\wt')$ and $\perm_G=\perm_{G'}$. Conversely, any two reduced bipartite graphs with the same strand permutation can be related by a sequence of square moves.
% for a bipartite square consisting of trivalent vertices, one can 


% \TODO{\cite{Pos}}
% \TODO{figures, examples}

\begin{figure}

\def\wh#1(#2){\node[draw,circle,scale=0.4,fill=white] (#1) at (#2) {};}
\def\bl#1(#2){\node[draw,circle,scale=0.4,fill=black!40] (#1) at (#2) {};}
\def\blbnd#1(#2){\node[draw,circle,scale=0.25,fill=black!60] (#1) at (#2) {};}
\def\rad{3}
\def\bnd#1#2{
\blbnd{B#2}(#1:\rad)
\node[anchor=180+#1,scale=0.6] (T#2) at (#1:\rad) {$b_{#2}$};
}
\def\lw{1pt}
\def\edgecl{black}
\def\edg#1#2{\draw[line width=\lw,\edgecl] (#1)--(#2);}
\def\RAD{2.4}
\def\wei#1#2(#3)#4{
\node[scale=0.5,blue,anchor=#4,inner sep=2pt] (ZZZ) at (#3) {$\br[#1#2]$};
}
\def\dang{10}
\def\strandlw{1pt}
\def\tikzscl{0.5}
\scalebox{1}{
\makebox[1.0\textwidth]{
\begin{tabular}{cccc}

\begin{tikzpicture}[scale=\tikzscl,baseline=(ZUZU.base)]
\coordinate(ZUZU) at (0,0);
\draw[line width=0.5,black!20,dashed] (0,0) circle (\rad);
\wh A(-1,1)
\bl B(1,1)
\wh C(1,-1)
\bl D(-1,-1)
\bnd{135}{1}
\bnd{45}{2}
\bnd{-45}{3}
\bnd{-135}{4}
\draw[line width=\lw,\edgecl] (B1)--(A)--(B)--(C)--(D)--(A);
\draw[line width=\lw,\edgecl] (B)--(B2);
\draw[line width=\lw,\edgecl] (C)--(B3);
\draw[line width=\lw,\edgecl] (D)--(B4);


\wh X(45:\RAD)
\wh Y(-135:\RAD)
\end{tikzpicture}

 & 
\def\lw{0.5pt}
\def\edgecl{black!50}
\begin{tikzpicture}[scale=\tikzscl,baseline=(ZUZU.base)]
\coordinate(ZUZU) at (0,0);
\draw[line width=0.5,black!20,dashed] (0,0) circle (\rad);
\wh A(-1,1)
\bl B(1,1)
\wh C(1,-1)
\bl D(-1,-1)
\bnd{135}{1}
\bnd{45}{2}
\bnd{-45}{3}
\bnd{-135}{4}
\draw[line width=\lw,\edgecl] (B1)--(A)--(B)--(C)--(D)--(A);
\draw[line width=\lw,\edgecl] (B)--(B2);
\draw[line width=\lw,\edgecl] (C)--(B3);
\draw[line width=\lw,\edgecl] (D)--(B4);

\draw [blue,->,line width=\strandlw] plot [smooth] coordinates {(135-\dang:\rad) (-0.6,1.2) (0.7,0.7) (1.2,-0.6) (-45+\dang:\rad)};
\draw [red,->,line width=\strandlw] plot [smooth] coordinates {(-135-\dang:\rad) (-135-\dang:\RAD) (-0.6,-1.2) (0.7,-0.7) (1.2,0.6) (45+\dang:\RAD) (45+\dang:\rad) };

\begin{scope}[xscale=-1,yscale=-1]
\draw [brown,->,line width=\strandlw] plot [smooth] coordinates {(135-\dang:\rad) (-0.6,1.2) (0.7,0.7) (1.2,-0.6) (-45+\dang:\rad)};
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\wh X(45:\RAD)
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\begin{tabular}{l}
$\Delta_{12}=\br[23]\cdot \br[24]$ \\
$\Delta_{23}=\br[34]\cdot \br[24]$ \\
$\Delta_{34}=\br[14]\cdot \br[24]$ \\
$\Delta_{14}=\br[12]\cdot \br[24]$ \\
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                           by Ptolemy's \\ theorem
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(a) & (b) & (c) & (d)
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  \caption{\label{fig:plabic} (a) A (reduced) planar bipartite graph $G$; (b) strands in $G$; (c) edge weights $\wt_\tt$, where the unmarked edges have weight $1$ and we abbreviate $\br[pq]:=|\t_q-\t_p|$; (d) the boundary measurements~${\Delta_I(G,\wt_\tt)}$.}
\end{figure}




\begin{figure}

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\\
(a)  & (b) 
\end{tabular}


  \caption{\label{fig:move} (a) A square move and its effect on $\wt_\tt$; (b) converting a plabic graph $G$ into a circular diagram of $\perm_G$ from \cref{sec:critical-cells}.}
\end{figure}


\section{Critical dimer model}\label{sec:critical-dimer-model}
 Let $G$ be a reduced bipartite graph with strand permutation $\perm$. 
% There exists a particularly nice (invariant under square moves) way to choose the assignment $\wt$ of positive weights to the edges of $G$ as we now explain.
 Choose $n$ points $\tt:=(\t_1,\t_2,\dots,\t_n)$ on the unit circle. For simplicity, assume that they are listed in counterclockwise order (more generally, see \cref{dfn:theta}). The main idea of \emph{critical varieties} defined below is to consider not just an arbitrary weight function $\wt:E(G)\to\R_{>0}$, but a particular weight function $\wt_\tt$ defined as follows. Observe that every edge $e$ of $G$ belongs to exactly two strands. Suppose that one strand terminates at $b_p$ and the other strand terminates at $b_q$ for some $1\leq p,q\leq n$. If $e$ is not adjacent to a boundary vertex, we let 
\begin{equation}\label{eq:wt_|tt|}
  \wt_\tt(e):=|\t_q-\t_p|
\end{equation}
 be the distance between $\t_p$ and $\t_q$. If $e$ is adjacent to the boundary, we set $\wt_\tt(e):=1$.%\footnote{\TODO{duality}} 

A crucial property of such a choice is that the result is invariant under square moves: for any two reduced graphs $G,G'$ with the same strand permutation $\perm$, we have $\Meas(G,\wt_\tt)=\Meas(G',\wt'_\tt)$. For instance, the two graphs in \figref{fig:move}(a) produce the same boundary measurements (up to a common scalar). 
% Here $\wt_\tt$ and $\wt'_\tt$ are defined by~\eqref{eq:wt_|tt|} on the edges of $G$ and $G'$, respectively.
 Thus $\Meas(G,\wt_\tt)$ depends only on $\perm$ and on $\tt$, therefore it  makes sense to denote $\Meas(\perm,\tt):=\Meas(G,\wt_\tt)$. Our main result is an explicit simple formula for $\Meas(\perm,\tt)$ which depends only on $\perm$ and $\tt$, and does not involve choosing a reduced graph $G$.
\begin{remark}
The origin of~\eqref{eq:wt_|tt|} comes from the notion of a \emph{critical dimer model}~\cite{KenCrit} associated to \emph{isoradial graphs}~\cite{Mercat}. An embedded planar graph $G$ is called \emph{isoradial} if each interior face of $G$ is inscribed in a circle of radius one. To a given reduced bipartite graph $G$ and a collection $\tt$ of points on a circle one can associate a planar embedding of $G$ (as well as of its planar dual) known under the name \emph{plabic tiling}~\cite{OPS}. This embedding is easily seen to be isoradial. The assignment of weights~\cite{KenCrit} to the edges of $G$ simplifies considerably in the case of plabic tilings, giving rise to the expression~\eqref{eq:wt_|tt|} in terms of strands. While the critical dimer model and its connections to electrical networks and the Ising model are well known~\cite{KenCrit,KLRR}, the specialization to plabic tilings and the totally nonnegative Grassmannian appears to not have been studied before.
\end{remark}



% The following is the main definition of this paper.
% \begin{definition}
% Let $G$ be a reduced bipartite graph with strand permutation $\perm$. Choose $n$ points $\t_1,\t_2,\dots,\t_n$ on the circle in counterclockwise order. 
% \end{definition}

\section{Electrical networks and the Ising model}\label{sec:electr-netw-ising}
We shall come back to the dimer model and explain our boundary measurement formula for critical varieties in the next sections. First, as a warm up, we review two particularly neat applications of the formula in the ``cyclically symmetric'' case.

 Both critical electrical networks and the critical Ising model are defined on isoradial graphs. To obtain an isoradial graph, take a rhombus tiling $\Tiling$ of a polygonal region $\Reg$, such as the one in~\figref{fig:intro1}(a), color its vertices black and white in a bipartite way, and let $G_\Tiling$ be the graph consisting of all diagonals of rhombi that connect their black vertices; see \figref{fig:intro1}(c). The graph $G_\Tiling$ is isoradial, with white vertices of $\Tiling$ being the centers of the corresponding unit circles. One then associates a weight to each edge $e$ of $G_\Tiling$ that depends on the geometry of the rhombus containing $e$, and treats this weight as either the resistance (in the case of electrical networks) or as the interaction constant (in the case of the Ising model) associated to $e$, as explained below.






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\draw [line width=1pt,black,opacity=1.00] (X1)-- (X0);
\draw [line width=1pt,black,opacity=1.00] (X0)-- (X4);
\fill [opacity=0.10,blue] (X1)-- (X6)-- (X7)-- (X2) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X1)-- (X6);
\draw [line width=1pt,black,opacity=1.00] (X6)-- (X7);
\draw [line width=1pt,black,opacity=1.00] (X7)-- (X2);
\draw [line width=1pt,black,opacity=1.00] (X2)-- (X1);
\fill [opacity=0.10,blue] (X8)-- (X9)-- (X3)-- (X2) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X8)-- (X9);
\draw [line width=1pt,black,opacity=1.00] (X9)-- (X3);
\draw [line width=1pt,black,opacity=1.00] (X3)-- (X2);
\draw [line width=1pt,black,opacity=1.00] (X2)-- (X8);
\fill [opacity=0.10,blue] (X5)-- (X10)-- (X6)-- (X1) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X5)-- (X10);
\draw [line width=1pt,black,opacity=1.00] (X10)-- (X6);
\draw [line width=1pt,black,opacity=1.00] (X6)-- (X1);
\draw [line width=1pt,black,opacity=1.00] (X1)-- (X5);
\fill [opacity=0.10,blue] (X11)-- (X12)-- (X0)-- (X3) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X11)-- (X12);
\draw [line width=1pt,black,opacity=1.00] (X12)-- (X0);
\draw [line width=1pt,black,opacity=1.00] (X0)-- (X3);
\draw [line width=1pt,black,opacity=1.00] (X3)-- (X11);
\fill [opacity=0.10,blue] (X7)-- (X13)-- (X8)-- (X2) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X7)-- (X13);
\draw [line width=1pt,black,opacity=1.00] (X13)-- (X8);
\draw [line width=1pt,black,opacity=1.00] (X8)-- (X2);
\draw [line width=1pt,black,opacity=1.00] (X2)-- (X7);
\fill [opacity=0.10,blue] (X12)-- (X14)-- (X4)-- (X0) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X12)-- (X14);
\draw [line width=1pt,black,opacity=1.00] (X14)-- (X4);
\draw [line width=1pt,black,opacity=1.00] (X4)-- (X0);
\draw [line width=1pt,black,opacity=1.00] (X0)-- (X12);
\fill [opacity=0.10,blue] (X10)-- (X15)-- (X7)-- (X6) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X10)-- (X15);
\draw [line width=1pt,black,opacity=1.00] (X15)-- (X7);
\draw [line width=1pt,black,opacity=1.00] (X7)-- (X6);
\draw [line width=1pt,black,opacity=1.00] (X6)-- (X10);
\fill [opacity=0.10,blue] (X9)-- (X16)-- (X11)-- (X3) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X9)-- (X16);
\draw [line width=1pt,black,opacity=1.00] (X16)-- (X11);
\draw [line width=1pt,black,opacity=1.00] (X11)-- (X3);
\draw [line width=1pt,black,opacity=1.00] (X3)-- (X9);
\fill [opacity=0.10,blue] (X16)-- (X17)-- (X12)-- (X11) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X16)-- (X17);
\draw [line width=1pt,black,opacity=1.00] (X17)-- (X12);
\draw [line width=1pt,black,opacity=1.00] (X12)-- (X11);
\draw [line width=1pt,black,opacity=1.00] (X11)-- (X16);
\fill [opacity=0.10,blue] (X14)-- (X18)-- (X5)-- (X4) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X14)-- (X18);
\draw [line width=1pt,black,opacity=1.00] (X18)-- (X5);
\draw [line width=1pt,black,opacity=1.00] (X5)-- (X4);
\draw [line width=1pt,black,opacity=1.00] (X4)-- (X14);
\fill [opacity=0.10,blue] (X17)-- (X19)-- (X14)-- (X12) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X17)-- (X19);
\draw [line width=1pt,black,opacity=1.00] (X19)-- (X14);
\draw [line width=1pt,black,opacity=1.00] (X14)-- (X12);
\draw [line width=1pt,black,opacity=1.00] (X12)-- (X17);
\fill [opacity=0.10,blue] (X13)-- (X20)-- (X9)-- (X8) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X13)-- (X20);
\draw [line width=1pt,black,opacity=1.00] (X20)-- (X9);
\draw [line width=1pt,black,opacity=1.00] (X9)-- (X8);
\draw [line width=1pt,black,opacity=1.00] (X8)-- (X13);
\fill [opacity=0.10,blue] (X18)-- (X21)-- (X10)-- (X5) -- cycle;
\draw [line width=1pt,black,opacity=1.00] (X18)-- (X21);
\draw [line width=1pt,black,opacity=1.00] (X21)-- (X10);
\draw [line width=1pt,black,opacity=1.00] (X10)-- (X5);
\draw [line width=1pt,black,opacity=1.00] (X5)-- (X18);
\end{tikzpicture}
}
 & \scalebox{0.65}{
\begin{tikzpicture}[scale=1.30 ,baseline=(ZUZU.base)]\coordinate(ZUZU) at (0,0); 
\coordinate (X0) at (-1.000,2.000);
\coordinate (X1) at (-0.134,1.500);
\coordinate (X2) at (0.366,2.366);
\coordinate (X3) at (-0.500,2.866);
\coordinate (X4) at (-1.000,1.000);
\coordinate (X5) at (-0.134,0.500);
\coordinate (X6) at (0.866,1.500);
\coordinate (X7) at (1.366,2.366);
\coordinate (X8) at (-0.134,3.232);
\coordinate (X9) at (-1.000,3.732);
\coordinate (X10) at (0.866,0.500);
\coordinate (X11) at (-1.366,2.366);
\coordinate (X12) at (-1.866,1.500);
\coordinate (X13) at (0.866,3.232);
\coordinate (X14) at (-1.866,0.500);
\coordinate (X15) at (1.366,1.366);
\coordinate (X16) at (-1.866,3.232);
\coordinate (X17) at (-2.366,2.366);
\coordinate (X18) at (-1.000,0.000);
\coordinate (X19) at (-2.366,1.366);
\coordinate (X20) at (0.000,3.732);
\coordinate (X21) at (-0.000,0.000);
\coordinate (X22) at (0.000,0.000);
\coordinate (X23) at (0.433,0.250);
\coordinate (X24) at (1.116,0.933);
\coordinate (X25) at (1.366,1.866);
\coordinate (X26) at (1.116,2.799);
\coordinate (X27) at (0.433,3.482);
\coordinate (X28) at (-0.500,3.732);
\coordinate (X29) at (-1.433,3.482);
\coordinate (X30) at (-2.116,2.799);
\coordinate (X31) at (-2.366,1.866);
\coordinate (X32) at (-2.116,0.933);
\coordinate (X33) at (-1.433,0.250);
\coordinate (X34) at (-0.500,0.000);
\draw [dashed, line width=0.5pt,black] (X0)-- (X1);
\draw [dashed, line width=0.5pt,black] (X1)-- (X2);
\draw [dashed, line width=0.5pt,black] (X2)-- (X3);
\draw [dashed, line width=0.5pt,black] (X3)-- (X0);
\fill [opacity=0.10,blue] (X0)-- (X1)-- (X2)-- (X3) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A0) at (X1) {};
\node[scale=0.50,fill=black,draw,circle] (C0) at (X3) {};
\node[scale=0.50,fill=white,draw,circle] (B0) at (X2) {};
\node[scale=0.50,fill=white,draw,circle] (D0) at (X0) {};
\draw[line width=1.5pt,black] (A0)--(C0);
\draw [dashed, line width=0.5pt,black] (X4)-- (X5);
\draw [dashed, line width=0.5pt,black] (X5)-- (X1);
\draw [dashed, line width=0.5pt,black] (X0)-- (X4);
\fill [opacity=0.10,blue] (X4)-- (X5)-- (X1)-- (X0) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A1) at (X4) {};
\node[scale=0.50,fill=black,draw,circle] (C1) at (X1) {};
\node[scale=0.50,fill=white,draw,circle] (B1) at (X5) {};
\node[scale=0.50,fill=white,draw,circle] (D1) at (X0) {};
\draw[line width=1.5pt,black] (A1)--(C1);
\draw [dashed, line width=0.5pt,black] (X1)-- (X6);
\draw [dashed, line width=0.5pt,black] (X6)-- (X7);
\draw [dashed, line width=0.5pt,black] (X7)-- (X2);
\fill [opacity=0.10,blue] (X1)-- (X6)-- (X7)-- (X2) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A2) at (X1) {};
\node[scale=0.50,fill=black,draw,circle] (C2) at (X7) {};
\node[scale=0.50,fill=white,draw,circle] (B2) at (X6) {};
\node[scale=0.50,fill=white,draw,circle] (D2) at (X2) {};
\draw[line width=1.5pt,black] (A2)--(C2);
\draw [dashed, line width=0.5pt,black] (X8)-- (X9);
\draw [dashed, line width=0.5pt,black] (X9)-- (X3);
\draw [dashed, line width=0.5pt,black] (X2)-- (X8);
\fill [opacity=0.10,blue] (X8)-- (X9)-- (X3)-- (X2) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A3) at (X8) {};
\node[scale=0.50,fill=black,draw,circle] (C3) at (X3) {};
\node[scale=0.50,fill=white,draw,circle] (B3) at (X9) {};
\node[scale=0.50,fill=white,draw,circle] (D3) at (X2) {};
\draw[line width=1.5pt,black] (A3)--(C3);
\draw [dashed, line width=0.5pt,black] (X5)-- (X10);
\draw [dashed, line width=0.5pt,black] (X10)-- (X6);
\fill [opacity=0.10,blue] (X5)-- (X10)-- (X6)-- (X1) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A4) at (X10) {};
\node[scale=0.50,fill=black,draw,circle] (C4) at (X1) {};
\node[scale=0.50,fill=white,draw,circle] (B4) at (X6) {};
\node[scale=0.50,fill=white,draw,circle] (D4) at (X5) {};
\draw[line width=1.5pt,black] (A4)--(C4);
\draw [dashed, line width=0.5pt,black] (X11)-- (X12);
\draw [dashed, line width=0.5pt,black] (X12)-- (X0);
\draw [dashed, line width=0.5pt,black] (X3)-- (X11);
\fill [opacity=0.10,blue] (X11)-- (X12)-- (X0)-- (X3) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A5) at (X12) {};
\node[scale=0.50,fill=black,draw,circle] (C5) at (X3) {};
\node[scale=0.50,fill=white,draw,circle] (B5) at (X0) {};
\node[scale=0.50,fill=white,draw,circle] (D5) at (X11) {};
\draw[line width=1.5pt,black] (A5)--(C5);
\draw [dashed, line width=0.5pt,black] (X7)-- (X13);
\draw [dashed, line width=0.5pt,black] (X13)-- (X8);
\fill [opacity=0.10,blue] (X7)-- (X13)-- (X8)-- (X2) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A6) at (X7) {};
\node[scale=0.50,fill=black,draw,circle] (C6) at (X8) {};
\node[scale=0.50,fill=white,draw,circle] (B6) at (X13) {};
\node[scale=0.50,fill=white,draw,circle] (D6) at (X2) {};
\draw[line width=1.5pt,black] (A6)--(C6);
\draw [dashed, line width=0.5pt,black] (X12)-- (X14);
\draw [dashed, line width=0.5pt,black] (X14)-- (X4);
\fill [opacity=0.10,blue] (X12)-- (X14)-- (X4)-- (X0) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A7) at (X12) {};
\node[scale=0.50,fill=black,draw,circle] (C7) at (X4) {};
\node[scale=0.50,fill=white,draw,circle] (B7) at (X14) {};
\node[scale=0.50,fill=white,draw,circle] (D7) at (X0) {};
\draw[line width=1.5pt,black] (A7)--(C7);
\draw [dashed, line width=0.5pt,black] (X10)-- (X15);
\draw [dashed, line width=0.5pt,black] (X15)-- (X7);
\fill [opacity=0.10,blue] (X10)-- (X15)-- (X7)-- (X6) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A8) at (X10) {};
\node[scale=0.50,fill=black,draw,circle] (C8) at (X7) {};
\node[scale=0.50,fill=white,draw,circle] (B8) at (X15) {};
\node[scale=0.50,fill=white,draw,circle] (D8) at (X6) {};
\draw[line width=1.5pt,black] (A8)--(C8);
\draw [dashed, line width=0.5pt,black] (X9)-- (X16);
\draw [dashed, line width=0.5pt,black] (X16)-- (X11);
\fill [opacity=0.10,blue] (X9)-- (X16)-- (X11)-- (X3) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A9) at (X16) {};
\node[scale=0.50,fill=black,draw,circle] (C9) at (X3) {};
\node[scale=0.50,fill=white,draw,circle] (B9) at (X11) {};
\node[scale=0.50,fill=white,draw,circle] (D9) at (X9) {};
\draw[line width=1.5pt,black] (A9)--(C9);
\draw [dashed, line width=0.5pt,black] (X16)-- (X17);
\draw [dashed, line width=0.5pt,black] (X17)-- (X12);
\fill [opacity=0.10,blue] (X16)-- (X17)-- (X12)-- (X11) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A10) at (X16) {};
\node[scale=0.50,fill=black,draw,circle] (C10) at (X12) {};
\node[scale=0.50,fill=white,draw,circle] (B10) at (X17) {};
\node[scale=0.50,fill=white,draw,circle] (D10) at (X11) {};
\draw[line width=1.5pt,black] (A10)--(C10);
\draw [dashed, line width=0.5pt,black] (X14)-- (X18);
\draw [dashed, line width=0.5pt,black] (X18)-- (X5);
\fill [opacity=0.10,blue] (X14)-- (X18)-- (X5)-- (X4) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A11) at (X18) {};
\node[scale=0.50,fill=black,draw,circle] (C11) at (X4) {};
\node[scale=0.50,fill=white,draw,circle] (B11) at (X5) {};
\node[scale=0.50,fill=white,draw,circle] (D11) at (X14) {};
\draw[line width=1.5pt,black] (A11)--(C11);
\draw [dashed, line width=0.5pt,black] (X17)-- (X19);
\draw [dashed, line width=0.5pt,black] (X19)-- (X14);
\fill [opacity=0.10,blue] (X17)-- (X19)-- (X14)-- (X12) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A12) at (X19) {};
\node[scale=0.50,fill=black,draw,circle] (C12) at (X12) {};
\node[scale=0.50,fill=white,draw,circle] (B12) at (X14) {};
\node[scale=0.50,fill=white,draw,circle] (D12) at (X17) {};
\draw[line width=1.5pt,black] (A12)--(C12);
\draw [dashed, line width=0.5pt,black] (X13)-- (X20);
\draw [dashed, line width=0.5pt,black] (X20)-- (X9);
\fill [opacity=0.10,blue] (X13)-- (X20)-- (X9)-- (X8) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A13) at (X20) {};
\node[scale=0.50,fill=black,draw,circle] (C13) at (X8) {};
\node[scale=0.50,fill=white,draw,circle] (B13) at (X9) {};
\node[scale=0.50,fill=white,draw,circle] (D13) at (X13) {};
\draw[line width=1.5pt,black] (A13)--(C13);
\draw [dashed, line width=0.5pt,black] (X18)-- (X21);
\draw [dashed, line width=0.5pt,black] (X21)-- (X10);
\fill [opacity=0.10,blue] (X18)-- (X21)-- (X10)-- (X5) -- cycle;
\node[scale=0.50,fill=black,draw,circle] (A14) at (X18) {};
\node[scale=0.50,fill=black,draw,circle] (C14) at (X10) {};
\node[scale=0.50,fill=white,draw,circle] (B14) at (X21) {};
\node[scale=0.50,fill=white,draw,circle] (D14) at (X5) {};
\draw[line width=1.5pt,black] (A14)--(C14);
\node[inner sep=3pt,scale=1.20000000000000,anchor=134] (B1) at (X10) {$b_{6}$};
\node[inner sep=3pt,scale=1.20000000000000,anchor=194] (B2) at (X7) {$b_{5}$};
\node[inner sep=3pt,scale=1.20000000000000,anchor=254] (B3) at (X20) {$b_{4}$};
\node[inner sep=3pt,scale=1.20000000000000,anchor=314] (B4) at (X16) {$b_{3}$};
\node[inner sep=3pt,scale=1.20000000000000,anchor=374] (B5) at (X19) {$b_{2}$};
\node[inner sep=3pt,scale=1.20000000000000,anchor=435] (B6) at (X18) {$b_{1}$};
\end{tikzpicture}
}
 & \setlength{\tabcolsep}{0pt}\begin{tikzpicture}[baseline=(ZUZU.base)]\coordinate(ZUZU) at (0,0);\node(A) at (0,1.5){\begin{tabular}{ccc}

&&\\
\scalebox{0.80}{
\scalebox{0.90}{
\begin{tikzpicture}[scale=1.20 ,baseline=(ZUZU.base)]\coordinate(ZUZU) at (0,0); 
\coordinate (X0) at (0.000,-0.700);
\coordinate (X1) at (0.500,0.166);
\coordinate (X2) at (0.277,1.141);
\coordinate (X3) at (-0.223,0.275);
\coordinate (X4) at (1.000,-0.700);
\coordinate (X5) at (1.500,0.166);
\coordinate (X6) at (1.277,1.141);
\coordinate (X7) at (0.500,-0.700);
\coordinate (X8) at (1.250,-0.267);
\coordinate (X9) at (1.389,0.653);
\coordinate (X10) at (0.777,1.141);
\coordinate (X11) at (0.027,0.708);
\coordinate (X12) at (-0.000,-0.700);
\coordinate (X13) at (-0.111,-0.213);
\draw [dashed, line width=0.5pt,black] (X0)-- (X1);
\draw [dashed, line width=0.5pt,black] (X1)-- (X2);
\draw [dashed, line width=0.5pt,black] (X2)-- (X3);
\draw [dashed, line width=0.5pt,black] (X3)-- (X0);
\fill [opacity=0.10,blue] (X0)-- (X1)-- (X2)-- (X3) -- cycle;
\node[scale=0.40,fill=black,draw,circle] (A0) at (X1) {};
\node[scale=0.40,fill=black,draw,circle] (C0) at (X3) {};
\node[scale=0.40,fill=white,draw,circle] (B0) at (X2) {};
\node[scale=0.40,fill=white,draw,circle] (D0) at (X0) {};
\draw[line width=1.4pt,black] (A0)--(C0);
\draw [dashed, line width=0.5pt,black] (X4)-- (X5);
\draw [dashed, line width=0.5pt,black] (X5)-- (X1);
\draw [dashed, line width=0.5pt,black] (X0)-- (X4);
\fill [opacity=0.10,blue] (X4)-- (X5)-- (X1)-- (X0) -- cycle;
\node[scale=0.40,fill=black,draw,circle] (A1) at (X4) {};
\node[scale=0.40,fill=black,draw,circle] (C1) at (X1) {};
\node[scale=0.40,fill=white,draw,circle] (B1) at (X5) {};
\node[scale=0.40,fill=white,draw,circle] (D1) at (X0) {};
\draw[line width=1.4pt,black] (A1)--(C1);
\draw [dashed, line width=0.5pt,black] (X5)-- (X6);
\draw [dashed, line width=0.5pt,black] (X6)-- (X2);
\fill [opacity=0.10,blue] (X5)-- (X6)-- (X2)-- (X1) -- cycle;
\node[scale=0.40,fill=black,draw,circle] (A2) at (X6) {};
\node[scale=0.40,fill=black,draw,circle] (C2) at (X1) {};
\node[scale=0.40,fill=white,draw,circle] (B2) at (X2) {};
\node[scale=0.40,fill=white,draw,circle] (D2) at (X5) {};
\draw[line width=1.4pt,black] (A2)--(C2);
\end{tikzpicture}
}

}
  & $\longleftrightarrow$ &
\scalebox{0.80}{
\scalebox{0.90}{
\begin{tikzpicture}[scale=1.20 ,baseline=(ZUZU.base)]\coordinate(ZUZU) at (0,0); 
\coordinate (X0) at (1.000,-0.700);
\coordinate (X1) at (0.777,0.275);
\coordinate (X2) at (-0.223,0.275);
\coordinate (X3) at (0.000,-0.700);
\coordinate (X4) at (1.277,1.141);
\coordinate (X5) at (0.277,1.141);
\coordinate (X6) at (1.500,0.166);
\coordinate (X7) at (0.500,-0.700);
\coordinate (X8) at (1.250,-0.267);
\coordinate (X9) at (1.389,0.653);
\coordinate (X10) at (0.777,1.141);
\coordinate (X11) at (0.027,0.708);
\coordinate (X12) at (-0.000,-0.700);
\coordinate (X13) at (-0.111,-0.213);
\draw [dashed, line width=0.5pt,black] (X0)-- (X1);
\draw [dashed, line width=0.5pt,black] (X1)-- (X2);
\draw [dashed, line width=0.5pt,black] (X2)-- (X3);
\draw [dashed, line width=0.5pt,black] (X3)-- (X0);
\fill [opacity=0.10,blue] (X0)-- (X1)-- (X2)-- (X3) -- cycle;
\node[scale=0.40,fill=black,draw,circle] (A0) at (X0) {};
\node[scale=0.40,fill=black,draw,circle] (C0) at (X2) {};
\node[scale=0.40,fill=white,draw,circle] (B0) at (X1) {};
\node[scale=0.40,fill=white,draw,circle] (D0) at (X3) {};
\draw[line width=1.4pt,black] (A0)--(C0);
\draw [dashed, line width=0.5pt,black] (X1)-- (X4);
\draw [dashed, line width=0.5pt,black] (X4)-- (X5);
\draw [dashed, line width=0.5pt,black] (X5)-- (X2);
\fill [opacity=0.10,blue] (X1)-- (X4)-- (X5)-- (X2) -- cycle;
\node[scale=0.40,fill=black,draw,circle] (A1) at (X4) {};
\node[scale=0.40,fill=black,draw,circle] (C1) at (X2) {};
\node[scale=0.40,fill=white,draw,circle] (B1) at (X5) {};
\node[scale=0.40,fill=white,draw,circle] (D1) at (X1) {};
\draw[line width=1.4pt,black] (A1)--(C1);
\draw [dashed, line width=0.5pt,black] (X0)-- (X6);
\draw [dashed, line width=0.5pt,black] (X6)-- (X4);
\fill [opacity=0.10,blue] (X0)-- (X6)-- (X4)-- (X1) -- cycle;
\node[scale=0.40,fill=black,draw,circle] (A2) at (X0) {};
\node[scale=0.40,fill=black,draw,circle] (C2) at (X4) {};
\node[scale=0.40,fill=white,draw,circle] (B2) at (X6) {};
\node[scale=0.40,fill=white,draw,circle] (D2) at (X1) {};
\draw[line width=1.4pt,black] (A2)--(C2);
\end{tikzpicture}
}

}
\\
&&\\

\end{tabular}};\end{tikzpicture}
 \\

\scalebox{1.00}{\begin{tabular}{c}(a) arbitrary \\ polygon $\Reg$\end{tabular}} & \scalebox{1.00}{\begin{tabular}{c}(b) regular \\ polygon $\Reg_N$\end{tabular}} & \scalebox{1.00}{(c) isoradial graph $G_{\Tiling}$} & \scalebox{1.00}{(d) star-triangle move} 

\end{tabular}}


\caption{\label{fig:intro1} 
(a) A rhombus tiling of an arbitrary polygon $\Reg$; (b) a rhombus tiling of a regular polygon $\Reg_\N$ for $\N=6$; (c) the associated isoradial graph $G_\Tiling$ consists of black vertices and black solid edges; (d) a flip of a rhombus tiling resulting in a star-triangle move on $G_\Tiling$.}
\end{figure}




Denote by $b_1,b_2,\dots,b_\N$ the vertices of $G_\Tiling$ that belong to the boundary of $\Reg$, listed in clockwise order. We now give some background on electrical networks and the Ising model, and explain how in each case one can apply natural boundary measurements to pairs $(b_p,b_q)$ of boundary vertices.

\paragraph{Electrical networks.}
We consider $G_\Tiling$ as an electrical resistor network, replacing every edge by a resistor. 
 % We consider every edge of $G_\Tiling$ to be a resistor.
 Given a rhombus $ABCD$ of $\Tiling$ with black vertices $A$ and $C$, the edge $AC$ of $G_\Tiling$ is treated as a resistor whose resistance equals the ratio of the diagonals $\frac{|AC|}{|BD|}$. 

Let us apply the voltage of $1$ to $b_p$ and the voltage of $0$ to all other boundary vertices. Then the voltages at all interior vertices, as well as the currents through all edges, can be computed from the Ohm's and Kirchhoff's laws. Denote by $\la^{\Tiling}_{p,q}$ the current that flows out of $b_q$.\footnote{By linearity of  the Ohm's and Kirchhoff's laws, knowing $\la^{\Tiling}_{p,q}$ for all $1\leq p,q\leq \N$ allows one to solve the more general problem: for any known voltages that are applied to the boundary vertices, one finds the resulting currents flowing through each boundary vertex.} The matrix $\La^{\Tiling}=(\la^{\Tiling}_{p,q})$ is known as the \emph{response matrix} of the electrical network associated with $G_\Tiling$.



\paragraph{Ising model.}
Denote by $V:=V(\GT)$ and $E:=E(\GT)$ the vertex and edge sets of $\GT$. A \emph{spin configuration} is an assignment $\sigma=(\sigma_v)_{v\in V}\in\{\pm1\}^\VGT$ of spins to the vertices of $\GT$, where we have $\sigma_v=\pm1$ for each $v\in \VGT$. For an edge $e\in \EGT$, we denote $x_e:=\cot(\th_e/2)$, where $2\th_e\in(0,\pi)$ is the angle at a white vertex of the rhombus of $\Tiling$ containing $e$. 
The \emph{critical $Z$-invariant Ising model}~\cite{Bax,Bax2} is a probability distribution on the set $\{\pm1\}^\VGT$ of all spin configurations: the probability of a given spin configuration $\sigma$ equals
\begin{equation*}%\label{eq:Z_dfn}
  \Prob_\Tiling(\sigma):=\frac1{Z_\Tiling}\prod_{\{u,v\}\in \EGT:\, \sigma_u=\sigma_v} x_{\{u,v\}},\quad\text{where}\quad Z_\Tiling:=\sum_{\sigma\in\{\pm1\}^\VGT} \prod_{\{u,v\}\in \EGT:\, \sigma_u=\sigma_v} x_{\{u,v\}}
\end{equation*}
is the \emph{partition function}. Given two boundary vertices $b_p,b_q$, we define their \emph{correlation} as
\begin{equation*}%\label{eq:corr_dfn}
\<\sigma_p\sigma_q\>_\Tiling:=\Prob_\Tiling(\sigma_p=\sigma_q)-\Prob_\Tiling(\sigma_p\neq\sigma_q).
\end{equation*}
 % In the case of the Ising model, one gets a probability measure on the space $\{\pm1\}^{V(G_\Tiling)}$ of spin configurations on the vertices of $G_\Tiling$. Each spin is either a plus or a minus one, and for each pair $1\leq p,q\leq n$, one obtains the associated \emph{boundary correlation} $\<\sigma_p\sigma_q\>_\Tiling\in[-1,1]$. 
% By definition, $\<\sigma_p\sigma_q\>_\Tiling$ is the difference between the probability that the spins at $b_p$ and $b_q$ are equal and the probability that these spins are different. 
The matrix $\M^\Tiling=(\<\sigma_p\sigma_q\>_\Tiling)$ is known as the \emph{boundary correlation matrix} of the Ising model on $G_\Tiling$.


\paragraph{Star-triangle moves.} It is known~\cite{KenyonAlg} that any two rhombus tilings of the same region can be related by a sequence of \emph{flips} as in \figref{fig:intro1}(d). Applying a flip to a rhombus tiling results in applying a \emph{star-triangle move} to the weighted graph $G_\Tiling$. A very well known property of the electrical response matrix $\La^\Tiling$ is that it is preserved by such moves. The same property also holds for the critical $Z$-invariant Ising model: the boundary correlation matrix $\M^\Tiling$ is preserved when $\Tiling$ changes by a flip. Therefore both $\La^\Tiling$ and $\M^\Tiling$ depend only on the region $\Reg$ itself, and not on the particular choice of a rhombus tiling $\Tiling$. 
It is thus natural to denote $\La^\Reg:=\La^\Tiling$ and $\M^\Reg:=\M^\Tiling$.
% , and look for an expression for the entries of these matrices purely in terms of $\Reg$. 
A consequence of our main results is a formula for $\La^\Reg$ and $\M^\Reg$ that depends manifestly only on the region~$\Reg$. 
% In fact, our formula holds more generally for critical varieties in the totally nonnegative Grassmannian $\Grtnn(k,n)$ described in \todo, and these applications are obtained by utilizing the embeddings of electrical networks~\cite{Lam} and the Ising model~\cite{GP} into $\Grtnn(k,n)$.


% \section{Regular polygons}\label{sec:regular-polygons}
\paragraph{Regular polygons.} For arbitrary regions $\Reg$, our formula (\cref{cor:main}) involves computing the inverse of an $\N\times \N$ matrix. However, in the most symmetric case when the region is a regular $2\N$-gon, the matrix can be inverted explicitly, which gives rise to the following results. Let us denote the regular $2\N$-gon by $\Reg_\N$.

\begin{theorem}\label{thm:reg_Elec}
For $1\leq p,q\leq \N$ and $d:=|p-q|$, we have
\begin{equation}\label{eq:reg_Elec}
\la^{\Reg_\N}_{p,q}=\frac{\sin(\pi/\N)}{\N\cdot \sin((2d-1)\pi/2\N)\cdot \sin((2d+1)\pi/2\N)}.
% \frac2n \left(\frac1{\sin \left((2k-1)\pi/2n\right)}-\frac1{\sin \left((2k-3)\pi/2n\right)}+\dots\pm \frac1{\sin \left(\pi/2n\right)} \right)\mp 1.
\end{equation}
\end{theorem}
\begin{example}
Consider the star electrical network as in \figref{fig:intro1}(d) inside a regular hexagon $\Reg_3$. Then the resistance of each edge equals $\frac1{\sqrt3}$. Applying the voltage of $1$ to $b_1$ and the voltage of $0$ to $b_2$ and $b_3$, we calculate that the resulting voltage at the unique interior vertex is $\frac13$, and thus the currents through $b_2$ and $b_3$ are both equal to $\frac1{\sqrt3}$. This agrees with~\eqref{eq:reg_Elec} for $\N=3$ and $d=1,2$. For $d=0$, we also obtain the correct value $-\frac2{\sqrt3}$ for the current through $b_1$, the negative sign representing the fact that the current flows \emph{into} the network.
\end{example}


\begin{theorem}\label{thm:reg_Ising}
For $1\leq p,q\leq \N$ and $d:=|p-q|$, we have
\begin{equation*}\label{eq:reg_Ising}
  \<\sigma_p\sigma_q\>_{\Reg_\N}=\frac2\N \left(\frac1{\sin \left((2d-1)\pi/2\N\right)}-\frac1{\sin \left((2d-3)\pi/2\N\right)}+\dots\pm \frac1{\sin \left(\pi/2\N\right)} \right)\mp 1.
\end{equation*}
\end{theorem}

% \cref{rmk:KW}
\noindent As explained in \cite[Remark~1.15]{ising_crit}, the formula in \cref{thm:reg_Ising} describes the unique $\N\times \N$ boundary correlation matrix of the Ising model that is invariant under the Kramers--Wannier duality~\cite{KrWa}. 

\begin{example}
For the square $\Reg_2$, the graph $\GT$ consists of a single edge $e$ with $x_e=\cot(\pi/8)=\sqrt2+1$. By definition, we find $\<\sigma_1\sigma_2\>_{\Reg_2}=\sqrt2-1$, which agrees with \cref{thm:reg_Ising} for $\N=2$ and $d=1$.
\end{example}
% The proof of \cref{thm:reg} and its asymptotic consequences are presented in \cref{sec:regular-polygons}.

\begin{remark}
Despite the simplicity of these two results, both of them are apparently new. They also lead to new asymptotic consequences (including a convergence result for the Ising model in a disk to a conformally invariant limit~\cite{ising_crit}) which fall outside the scope of the present extended abstract.
\end{remark}



% \section{Positroid cells}
% \subsection{Bounded affine permutations}
% Positroid cells give a stratification of the \emph{totally nonnegative Grassmannian}, defined below. They are indexed by the following combinatorial objects.
% \begin{definition}[\cite{KLS}]
% A bijection $\ft:\Z\to\Z$ is called a \emph{$(k,n)$-bounded affine permutation} if it satisfies
% \begin{itemize}
% \item $\ft(j+n)=\ft(j)$ for all $j\in\Z$,
% \item $\sum_{j=1}^{n} (\ft(j)-j)=kn$, and
% \item $j\leq \ft(j)\leq j+n$ for all $j\in\Z$.
% \end{itemize}
% \end{definition}
% \noindent We let $\Bkn$ denote the (finite) set of $(k,n)$-bounded affine permutations. If we consider $\ft\in\Bkn$ modulo $n$, we get the usual permutation $\fb\in S_n$. If $\fb$ has no fixed points, then $\ft$ can be uniquely recovered form $\fb$. \TODO{fixed points footnote}
% % For a fixed point $i\in[n]$ (such that $\fb(i)=i$), we have two choices: either $\ft(i)=i$ or $\ft(i)=i+n$. We say that a fixed point $i$ is \emph{black} if $\ft(i)=i$ and \emph{white} if $\ft(i)=i+n$. Thus $\ft$ can be uniquely recovered from a \emph{decorated} permutation $\fb$, where a \emph{decoration} of $\fb$ is a coloring of fixed points of $\fb$ into 

% Let us now consider a reduced planar bipartite graph $G$. Recall that there is a number $k$ such that $|\pmatch|=k$ for all almost perfect matchings of $G$. \TODO{fixed points}
% % If $\perm_G(i)=i$ for some $i$ then it is known that either the black boundary vertex $b_i$ is isolated, or it is connected to an interior white vertex of degree $1$. In the former case, we say that $i$ is \emph{black}, and in the latter case, we say that $i$ is \emph{white}. 
%  There exists a unique bounded affine permutation $\ft_G\in\Bkn$ such that $\fb_G=\perm_G$. 


\section{The totally nonnegative Grassmannian}\label{sec:totally-nonn-grassm}

% \subsection{The totally nonnegative Grassmannian}
 The \emph{Grassmannian} $\Gr(k,n)$ is the space of $k$-dimensional linear subspaces of $\R^n$. It can also be identified with the set of full rank $k\times n$ real matrices modulo row operations, via the bijection sending a matrix $M$ to $\RowSpan(M)$. Given such a matrix $M$ and a $k$-element set $I\in {[n]\choose k}$ of columns, let $\Delta_I(M)$ denote the maximal minor of $M$ with column set $I$. The collection $(\Delta_I(M))_{I\in{[n]\choose k}}$ of \emph{Pl\"ucker coordinates} of $M$ uniquely determines the point $\RowSpan(M)\in\Gr(k,n)$, and is defined up to a multiplication by a common scalar. The \emph{totally nonnegative Grassmannian} $\Grtnn(k,n)$ is the subset of $\Gr(k,n)$ where all nonzero Pl\"ucker coordinates have the same sign. Loosely speaking, we can write
\[\Grtnn(k,n):=\left\{X\in\Gr(k,n)\middle| \Delta_I(X)\geq 0 \text{ for all $I\in {[n]\choose k}$}\right\}. \]
The space $\Grtnn(k,n)$ is naturally stratified into \emph{positroid cells} according to which Pl\"ucker coordinates are zero and which are strictly positive. 
% The positroid cells turn out to be in bijection with bounded affine permutations as we now explain.

Given a reduced bipartite planar graph $G$ with weight function $\wt$, there exists a (necessarily unique) point $X\in\Grtnn(k,n)$ such that $\Delta_I(X)=\Delta_I(G,\wt)$ for all $I\in{[n]\choose k}$. By an abuse of notation, we also denote this point by $\Meas(G,\wt)$. Fixing $G$ and letting $\wt\in\R^{E(G)}_{>0}$ vary over all possible assignments of positive real edge weights, we obtain the \emph{positroid cell} $\Ptp_G\subset \Grtnn(k,n)$ defined by
\[\Ptp_G:=\{\Meas(G,\wt)\mid \wt\in \R^{E(G)}_{>0}\}.\]
The set $\Ptp_G$ depends only on $\perm_G$ and is denoted $\Ptp_{\perm_G}$. The positroid cells give a cell decomposition\footnote{To obtain the whole $\Grtnn(k,n)$, one needs to consider \emph{decorated permutations} with fixed points~\cite{Pos}.} of $\Grtnn(k,n)$.
% Roughly speaking, every point in $\Grtnn(k,n)$ belongs to $\Ptp_\ft$ for a unique permutation $\perm$.
% \[ \Grtnn(k,n)=\bigsqcup_{\ft\in\Bkn} \Ptp_\ft. \]


Thus in particular we may take $\wt:=\wt_\tt$ to come from the critical dimer model, in which case we get a point $\Meas(\perm_G,\tt):=\Meas(G,\wt_\tt)\in\Ptp_{\perm_G}$.
\begin{remark}
Taking $\perm$ to be the permutation sending $p\mapsto p+k$ modulo $n$ for all $p\in[n]$ (corresponding to the \emph{top-dimensional positroid cell}), and taking the points in $\tt$ to be equally spaced on the circle, $\Meas(\perm,\tt)$ becomes the unique cyclically symmetric point of $\Grtnn(k,n)$ studied in~\cite{GKL,Karp}.
\end{remark}


\begin{comment}
\todo


\footnote{More precisely, positroid cells are in bijection with \emph{decorated} permutations, where a decoration of $\perm$ is an arbitrary coloring of fixed points of $\perm$ into black and white colors. The actual set $\Skn$ consists of decorated permutations $\perm$ satisfying $\#\{1\leq i\leq n\mid \perm(i)<i\}+\#\{\text{black fixed points of $\perm$}\}=k$.}


For each $f\in\Bkn$, we have a \emph{positroid cell} $\Ptp_\ft\subset\Grtnn(k,n)$ (see below for a definition), which together give a cell decomposition



 The \emph{Grassmannian} $\Gr(k,n)$ is the space of $k$-dimensional linear subspaces of $\R^n$. It can also be identified with the set of full rank $k\times n$ real matrices modulo row operations, via the bijection sending a matrix $M$ to $\RowSpan(M)$. Given such a matrix $M$ and a $k$-element set $I\in {[n]\choose k}$ of columns, let $\Delta_I(M)$ denote the maximal minor of $M$ with column set $I$. The collection $(\Delta_I(M))_{I\in{[n]\choose k}}$ of \emph{Pl\"ucker coordinates} of $M$ uniquely determines the point $\RowSpan(M)\in\Gr(k,n)$, and is defined up to a multiplication by a common scalar. The \emph{totally nonnegative Grassmannian} $\Grtnn(k,n)$ is the subset of $\Gr(k,n)$ where all nonzero Pl\"ucker coordinates have the same sign. Loosely speaking, we can write
\[\Grtnn(k,n):=\left\{X\in\Gr(k,n)\middle| \Delta_I(X)\geq 0 \text{ for all $I\in {[n]\choose k}$}\right\}. \]

The space $\Grtnn(k,n)$ is naturally stratified into \emph{positroid cells} by looking at which Pl\"ucker coordinates are zero and which are strictly positive. The positroid cells turn out to be in bijection with the following combinatorial objects.
\begin{definition}[\cite{KLS}]
A bijection $\ft:\Z\to\Z$ is called a \emph{$(k,n)$-bounded affine permutation} if it satisfies
\begin{itemize}
\item $\ft(j+n)=\ft(j)$ for all $j\in\Z$,
\item $\sum_{j=1}^{n} (\ft(j)-j)=kn$, and
\item $j\leq \ft(j)\leq j+n$ for all $j\in\Z$.
\end{itemize}
\end{definition}
\noindent We let $\Bkn$ denote the (finite) set of $(k,n)$-bounded affine permutations. For each $f\in\Bkn$, we have a \emph{positroid cell} $\Ptp_\ft\subset\Grtnn(k,n)$ (see below for a definition), which together give a cell decomposition
% We have a cell decomposition
\[\Grtnn(k,n)=\bigsqcup_{\ft\in \Bkn} \Ptp_{\ft}.\]
% for certain combinatorially defined subsets $\Ptp_\ft\subset\Grtnn(k,n)$.

% For a full rank $k\times n$ matrix $A$, we let $\ft_A:\Z\to\Z$ be given by
% \[
%   \ft_A(i)=\min\{j\geq i\mid A_i\in \Span \left(A_{i+1},A_{i+2},\dots,A_j\right)\} \quad\text{for $i\in\Z$.}
% \]
% For example, if $A_i$ is a zero column then $\ft_A(i)=i$, and if $A_i$ is not in the span of other columns then $\ft_A(i)=i+n$. It is known~\cite{KLS} that $\ft_A$ is a $(k,n)$-bounded affine permutation which depends only on the row span of $A$. 
% % Therefore to an element $X\in\Gr(k,n)$ we may associate $\ft_X\in\Bkn$ by choosing any $k\times n$ matrix $A$ whose row span equals $X$ and then setting $\ft_X:=\ft_A$.
%  The \emph{positroid stratification} of $\Gr(k,n)$ is given by
% % \begin{equation*}%\label{eq:*}
% \[
%   \Grtnn(k,n)=\bigsqcup_{\ft\in \Bkn} \Ptp_{\ft}, \quad\text{where}\quad \Ptp_{\ft}:=\{\RowSpan(A)\in\Grtnn(k,n)\mid \ft_A=\ft\}.
% \]
% % \end{equation*}
% \TODO{end}

\end{comment}

% placing $2n$ points labeled $\m1,\p1,\dots,\m n,\p n$ on the circle in clockwise order, and then

\section{Critical cells}\label{sec:critical-cells}
We have defined positroid cells above. Their Zariski closures in the complex Grassmannian are called \emph{positroid varieties}~\cite{KLS}. Similarly, we will first introduce \emph{critical cells}; critical varieties will be their Zariski closures, and critical cells will be the totally positive parts of critical varieties.

Let $\perm\in S_n$ be a permutation (without fixed points), and let $k\in[n]$ be defined by~\eqref{eq:perm_to_k}. Place the points $b_1,b_2,\dots, b_n$ on a circle, and for each $p\in[n]$, let $\m p$  (resp., $\p p$) be the point slightly before (resp., after) $b_p$ in clockwise order. The \emph{reduced strand diagram} of $\perm$ 
is obtained by drawing a straight arrow $\p a \to \m p$ whenever $\perm(a)=p$; see \figref{fig:move}(b). We say that $p\neq q\in[n]$ \emph{form an $f$-crossing} if the arrows $\p a\to \m {p}$ and $\p {b}\to\m{q}$ cross, where $a:=\perm^{-1}(p)$ and $b:=\perm^{-1}(q)$.  For instance, in \figref{fig:move}(b), $1$ and $2$ form an $f$-crossing but $1$ and $3$ do not.

The critical cell $\Ctp_\perm$ is parametrized by the following set.
\begin{definition}\label{dfn:theta}
A tuple $\bth=(\th_1,\th_2,\dots,\th_n)$ is called \emph{$\perm$-admissible} if for all $1\leq p<q\leq n$ such that $p$ and $q$ form an $f$-crossing, we have $\th_p<\th_q<\th_p+\pi.$
% \[ \th_p<\th_q<\th_p+\pi.\]
\end{definition}
\noindent Thus in the case of \figref{fig:move}(b), $\bth$ is $\perm$-admissible if and only if  $\th_1<\th_2<\th_3<\th_4<\th_1+\pi$.

Let now $G$ be a reduced bipartite planar graph with strand permutation $\perm$ and let $\bth=(\th_1,\th_2,\dots,\th_n)\in\R^n$ be a tuple of angles. We define a weight function $\wt_\bth$ on the edges of $G$ as follows. When $e\in E(G)$ is adjacent to a boundary vertex, we set $\wt_\bth(e):=1$, and otherwise we set
\begin{equation}\label{eq:wt_bth}
  \wt_\bth(e):=\sin(\th_q-\th_p),
\end{equation}
where $1\leq p<q\leq n$ are the indices such that the two strands passing through $e$ terminate at $b_p$ and $b_q$. It turns out~\cite[Proposition~4.2]{crit} that all edge weights are positive if and only if $\bth$ is $\perm$-admissible.
% since $\bth$ is $\perm$-admissible, all edge weights will be positive. 

\begin{remark}
In \cref{sec:critical-dimer-model}, we worked with a tuple of $n$ points $\tt:=(\t_1,\t_2,\dots,\t_n)$ on the unit circle. Let us regard these points as complex numbers of modulus $1$. Then the connection between~\eqref{eq:wt_|tt|} and~\eqref{eq:wt_bth} is obtained by setting $\t_q:=\exp(2i\th_q)$ for all $q\in[n]$. Then we have   $|\t_q-\t_p|=\sin(\th_q-\th_p)$.
% \[\t_q:=\exp(2i\th_q) \quad\text{for all $q\in[n]$.} \]
\end{remark}

 Recall from \cref{sec:critical-dimer-model} that $\Meas(G,\wt_\bth)$ depends only on $\perm_G$ and $\bth$, and is denoted $\Meas(\perm_G,\bth)$. The following is the main definition of this paper.
\begin{definition}
For a permutation $\perm$, the \emph{critical cell} $\Ctp_\perm\subset\Ptp_\perm$ is given by
\begin{equation*}%\label{eq:*}
  \Ctp_\perm:=\{\Meas(\perm,\bth)\mid \text{$\bth=(\th_1,\th_2,\dots,\th_n)$ is an $\perm$-admissible tuple}\}.
\end{equation*}
The \emph{critical variety} $\Cio_\perm$ is the Zariski closure of $\Ctp_\perm$ inside the complex Grassmannian.
\end{definition}
% \footnote{One can also introduce an \emph{open critical variety} $\Cioo_\perm\subset \Cio_\perm$ contained in the corresponding \emph{open positroid variety} of~\cite{KLS}; see~\cite[Section~5]{crit}.}



\section{Boundary measurement formula}
Currently, in order to compute $\Meas(\perm,\bth)$, one needs to choose a reduced planar bipartite graph $G$. And then the result does not depend on this choice. It is therefore natural to look for an expression for $\Meas(\perm,\bth)$ purely in terms of $\perm$ and $\bth$. Our main result gives a solution to this problem.


Take the reduced strand diagram of $\perm$ as in \figref{fig:move}(b). For each $r\in[n]$, introduce the set
\begin{equation*}%\label{eq:*}
  \J_r:=\{q\in[n]\mid \text{$b_r$ is to the left of the arrow $\p p\to \m q$}\}.
\end{equation*}
Here  we assume $p=\perm^{-1}(q)$. 
% For each arrow $\p p\to \m q$ (with $q=\perm(p)$), define $\L(q)\subset[n]$ to be the set of $r\in[n]$ such that the point $b_r$ is to the left of the arrow $\p p\to \m q$. For $r\in[n]$, denote 
% \begin{equation}\label{eq:*}
%   \J_r:=\{q\in[n]\mid r\in \L(q)\}.
% \end{equation}
% \begin{remark}
 Observe that we always have $r\notin \J_r$. The collection $(\J_r\sqcup\{r\})_{r\in[n]}$ is known as the \emph{Grassmann necklace}~\cite{Pos} of $\perm$. 
% \end{remark}


%  when $\bth$ is \emph{non-degenerate}, in the following sense. Given $\th_0\in\R$, let us define 
% \[\D(\th_0,\bth):=\{p\in[n]\mid \th_p\equiv\th_0\pmod\pi\}.\]
% % points $1,2,\dots,n$ on the circle, where each point $p$ is between $\m p$ and $\p p$. 
% \begin{definition}
% We say that a tuple $\bth$ is \emph{degenerate} if there exists $\th_0\in\R$ such that
% \begin{equation}\label{eq:*}
%   \bigcup_{q\in \D(\th_0,\bth)} \L(q)=[n].
% \end{equation}
% % Otherwise, $\bth$ is called \emph{non-degenerate}.
% \end{definition}
% \noindent In other words, $\bth$ is degenerate when ``too many'' $\th_p$-s are congruent modulo $\pi$. For instance, when all angles in $\bth$ are pairwise non-congruent modulo $\pi$, we see that $\bth$ is non-degenerate. It is quite hard (but possible) to come up with examples of $\perm$-admissible but degenerate tuples $\bth$.


For an index $r\in [n]$, let $\eps_r\in\{\pm1\}$ be given by $\eps_r:=(-1)^{\#\{p\in[n]\mid \perm(p)<p<r\}}$.
% \[\eps_r:=(-1)^{\#\{p\in[n]\mid \perm(p)<p<r\}}.\]
\begin{definition}\label{dfn:curve}
Let $\bth=(\th_1,\th_2,\dots,\th_n)$ be a tuple of angles. Consider a curve $\curve_{\perm,\bth}:\R\to\R^{n}$ with coordinates $\curve_{\perm,\bth}(t)=(\g_1(t),\g_2(t),\dots,\g_{n}(t))$ given by
\begin{equation*}\label{eq:curve}
  \g_r(t):= \eps_r\prod_{p\in \J_r} \sin(t-\th_p) \qquad\text{for $r\in[n]$.}
\end{equation*}
\end{definition}

It is possible to give a formula for an arbitrary $\perm$-admissible tuple $\bth$, but for simplicity, we restrict to the case when $\bth$ is \emph{generic}, that is, when all angles in $\bth$ are pairwise non-congruent modulo $\pi$. 
We are ready to state our main result.
\begin{theorem}
Suppose that $\bth$ is a generic $\perm$-admissible tuple. Then the linear span $\Span(\curve_{\perm,\bth})\subset\R^n$ has dimension $k$ and we have 
\begin{equation*}%\label{eq:*}
  \Meas(\perm,\bth)=\Span(\curve_{\perm,\bth}) \quad\text{inside $\Gr(k,n)$.}
\end{equation*}
\end{theorem}

\begin{example}
Let $k=2$, $n=4$, and $\perm$ be the permutation sending $p\mapsto p+2$ modulo~$4$. The boundary measurement map $\Meas(\perm,\bth)$ was computed in \figref{fig:plabic}(d). Since the Pl\"ucker coordinates are defined up to a common scalar, the term $\br[24]$ cancels out. The sets $\J_r$ are given by $\J_1=\{2\}$, $\J_2=\{3\}$, $\J_3=\{4\}$, and $\J_4=\{1\}$, so $\curve_{\perm,\bth}$ has coordinates
\[\curve_{\perm,\bth}(t)=(\sin(t-\th_2),\sin(t-\th_3),\sin(t-\th_4),-\sin(t-\th_1)).\]
We can choose a basis consisting of e.g.\ $\curve_{\perm,\bth}(0)$ and $\curve_{\perm,\bth}(\pi/2)$, which we can write in the rows of the following matrix:
\[A=\begin{pmatrix}
-\sin(\th_2) & -\sin(\th_3) & -\sin(\th_4) & \sin(\th_1)\\
\cos(\th_2) &\cos(\th_3) &\cos(\th_4) & -\cos(\th_1) 
\end{pmatrix}. \]
We see that the maximal minors of $A$ coincide with the values computed in \figref{fig:plabic}(d).
\end{example}



\noindent A few remarks are in order.
\begin{itemize}
% \item  When $\bth$ is degenerate, $\Span(\curve_{\perm,\bth})$ has dimension strictly less than $k$.
\item  For a generic $\bth$, an explicit basis of $\Span(\curve_{\perm,\bth})$ can be chosen by taking any $k$ distinct points on the curve $\curve_{\perm,\bth}$.
\item A more canonical (and computationally robust) way to produce a basis of $\Span(\curve_{\perm,\bth})$ is to observe that each coordinate $\g_r(t)$ is a trigonometric polynomial of degree $k-1$. Therefore it has precisely $k$ non-trivial Fourier coefficients. The rows of the resulting $k\times n$ matrix of Fourier coefficients form a basis of $\Span(\curve_{\perm,\bth})$ which does not depend on anything besides $\perm$ and the angles $\th_1,\th_2,\dots,\th_{n}$.

\item After a change of variables $T_p:=\exp(i\th_p)$, $T:=\exp(it)$, our boundary measurement formula becomes an algebraic expression in $T_1,T_2,\dots,T_n$. We have $\sin(\th_q-\th_p)=\frac1{2i} \left(T_q/T_p-T_p/T_q\right)$, thus the edge weights of $G$ are also algebraic. This can be used to introduce \emph{open critical varieties} $\Cioo_\perm\subset\Cio_\perm$; see~\cite[Section~5]{crit}.
% Allowing $T_p$ to run over complex numbers, we obtain a Zariski open subset of $\Cio_\perm$.

\item Let $\Ctnn_\perm$ be the closure of $\Ctp_\perm$ inside $\Grtnn(k,n)$. An immediate next direction is to determine the boundary cell structure of $\Ctnn_\perm$ in the spirit of the positroid cell decomposition of $\Grtnn(k,n)$. This will be explored in a forthcoming paper.

% \item Another potentially fruitful direction is the formal similarity between critical varieties and the Peterson variety.
\end{itemize}


\section{Back to electrical networks and the Ising model}
We promised in \cref{sec:electr-netw-ising} to give a formula for the electrical response matrix $\La^\Reg$ and the boundary correlation matrix $\M^\Reg$ associated to a rhombus tiling $\Tiling$ of a polygonal region $\Reg$. Recall that $\N$ is the number of boundary vertices of $G_\Tiling$, and let $n:=2\N$ be the number of boundary edges of $\Reg$. Suppose that the boundary vectors of $\Reg$ are $\t_1,\t_2,\dots,\t_{2\N}$, listed and directed in clockwise order (and treated as complex numbers of modulus one). The tiling $\Tiling$ gives rise to a fixed-point-free involution $\tau_\Reg:[2\N]\to[2\N]$, as follows. For each $p\in[2\N]$, the boundary vector $\t_p$ is contained in a unique rhombus. The opposite side of that rhombus is contained in a unique other rhombus. We continue in this fashion until we arrive at another boundary vector $\t_q$ (which necessarily satisfies $\t_q=-\t_p$), and we set $\tau_\Reg(p):=q$ and $\tau_\Reg(q):=p$. Clearly, $\tau_\Reg$ depends only on $\Reg$ and not on $\Tiling$. %\todof


Let $\fel,\fis:[2\N]\to[2\N]$ be defined by $\fel(p):=\tau_\Reg(p)+1$ (modulo $2\N$) and $\fis(p):=\tau_\Reg(p)$. One can uniquely extract the square roots to obtain the tuple $\bth_\Reg=(\th_1,\th_2,\dots,\th_{2\N})$ that is both $\fel$- and $\fis$-admissible, such that $\t_p=\exp(2i\th_p)$ for all $p\in[2\N]$, and such that $\th_{\tau_\Reg(p)}=\th_p+\pi/2$ whenever $p<\tau_\Reg(p)$. 
% The condition of $\bth_\Reg$ being non-degenerate translates into the condition that $\Reg$ is \emph{non-alternating}~\cite{ising_crit}, that is, there exist no $1\leq a<b<c<d\leq 2\N$ such that $\t_a=-\t_b=\t_c=-\t_d$.

Lam~\cite{Lam} has constructed an embedding $\pel$ of the space of $\N\times \N$ electrical response matrices into $\Grtnn(\N+1,2\N)$.\footnote{More precisely, Lam's embedding lands in $\Grtnn(\N-1,2\N)$. To get an element of $\Grtnn(N+1,2\N)$, one needs to take the orthogonal complement and then change the sign of every second column.} Together with Pylyavskyy~\cite{GP}, we constructed an embedding $\pis$ of the space of $\N\times \N$ Ising boundary correlation matrices into $\Grtnn(\N,2\N)$.
\begin{theorem}
For any region $\Reg$, we have
\begin{equation*}%\label{eq:*}
  \pel(\La^\Reg)=\Meas(\fel,\bth_\Reg) \quad\text{and}\quad \pis(\M^\Reg)=\Meas(\fis,\bth_\Reg)
\end{equation*}
\end{theorem}

The matrices $\La^\Reg$ and $\M^\Reg$ can be easily recovered from their respective images inside the Grassmannian. Let us explain the Ising model case in detail, the case of electrical networks being similar.  Introduce a $2\N\times \N$ matrix $K_\N$ defined as
\begin{equation*}\label{eq:K_n}
K_\N=\frac12 \begin{pmatrix}
1 & 0 & \dots & 0\\
1 & 0 & \dots & 0\\
0 & 1 & \dots & 0\\
0 & 1 & \dots & 0\\
\vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & 1\\
0 & 0 & \dots & 1
\end{pmatrix}.
\end{equation*}

\begin{corollary}\label{cor:main}
Let $\Reg$ be a non-alternating region. Choose an $\N\times 2\N$ matrix $A$ whose row span equals $\Span(\curve_\Reg)$.  Let $B=(b_{p,q})$ be the $\N\times 2\N$ matrix given by
\begin{equation*}\label{eq:AKA}
B:=(AK_\N)^{-1}A.
\end{equation*}
Then, up to a sign, the entries of $B$ are the boundary correlations: we have
\begin{equation*}\label{eq:corr_formula_B}
  \<\sigma_p\sigma_q\>_\Reg=\<\sigma_p\sigma_q\>_\Reg=|b_{p,2q-1}|=(-1)^{q-p+1}b_{p,2q-1} \quad\text{for all $1\leq p< q\leq \N$.}
\end{equation*}
\end{corollary}
\noindent Part of the content of \cref{cor:main} is that the matrix $AK_\N$ is always invertible.

\begin{remark}
Our formula involves inverting an $\N\times\N$ matrix $AK_\N$. There are other celebrated matrix formulas for the dimer and Ising models, e.g., the Kasteleyn matrix or the Kac--Ward matrix~\cite{KacWard,Kasteleyn}. However, these matrices are indexed by the vertices of $G_\Tiling$, and therefore have size $O(\N^2\times \N^2)$. By contrast, the matrix $AK_\N$ has size $\N\times \N$.
\end{remark}

% \TODO{cyc symm pt and \cite{GKL}}

% \TODO{extended abstract}

% \TODO{future dir}


\acknowledgements{I am indebted to Pasha Pylyavskyy for his numerous contributions at various stages of the development of~\cite{ising_crit}, where the boundary measurement formula was first discovered in the context of the Ising model. The generalization to the Grassmannian level was inspired by the results of~\cite{CLR,KLRR}, presented by Marianna Russkikh at the ``Dimers in Combinatorics and Cluster Algebras'' conference at the University of Michigan. I thank Marianna for bringing these results to my attention, and also thank the organizers of the conference (Sebastian Franco, Gregg Musiker, Richard Kenyon, David Speyer, and Lauren Williams) for making such an interaction possible.}



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