
%% if you are submitting an initial manuscript then you should have "submission" as an option here
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\documentclass[revision]{FPSAC2019}
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\articlenumber{72}
\addbibresource{72.bib}

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\usepackage{lipsum}
\usepackage{enumerate}
\usepackage{subcaption}

\newtheorem{thm}{Theorem}[section]
\newtheorem*{thm*}{Theorem}
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\newtheorem{innercustomthm}{Theorem}
\newenvironment{customthm}[1]
  {\renewcommand\theinnercustomthm{#1}\innercustomthm}
  {\endinnercustomthm}
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\newtheorem{prop}[thm]{Proposition}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}

\theoremstyle{definition}
\newtheorem{dfn}[thm]{Definition}
\newtheorem{prob}[thm]{Problem}
\newtheorem{eg}[thm]{Example}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{conj}[thm]{Conjecture}
\newtheorem{ques}[thm]{Question}

\newcommand{\CC}{\mathbb C}
\newcommand{\RR}{\mathbb R}
\newcommand{\ZZ}{\mathbb Z}
\newcommand{\PP}{\mathbb P}

\DeclareMathOperator{\val}{val}
\DeclareMathOperator{\trop}{trop}
\DeclareMathOperator{\rspan}{row.span}
\DeclareMathOperator{\Sym}{Sym}
\DeclareMathOperator{\Div}{Div}
\DeclareMathOperator{\Eff}{Eff}
\DeclareMathOperator{\Supp}{Supp}
\DeclareMathOperator{\indeg}{indeg}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\Arg}{Arg}
%\DeclareMathOperator{\ker}{ker}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\Capacity}{\textsc{cap}}
\DeclareMathOperator{\coamoeba}{co\mathcal{A}}
\DeclareMathOperator{\PL}{PL}
\DeclareMathOperator{\rS}{S}
%%% use either \Delta or Div %%%
\DeclareMathOperator{\Divisor}{\Delta}
\DeclareMathOperator{\zspan}{span}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\zeros}{\Divisor^{+}}%{zeros}
\DeclareMathOperator{\poles}{\Divisor^{--}}%{poles}
\DeclareMathOperator{\vol}{vol}
\DeclareMathOperator{\len}{len}

%%%%%%%%%%%%%%%%%%%%%%%%%%
% reduced divisor macro
% #1 = point, #2 = divisor
\newcommand{\reddiv}[2]{\red_{#1} [#2]}
%%%%%%%%%%%%%%%%%%%%%%%%%%
% voltage / potential function macro
% #1 = sink, #2 = source
\newcommand{\potent}[2]{j_{#1}^{#2}}
\newcommand{\potential}[3]{j_{#1}({#2},{#3})}
%%%%%%%%%%%%%%%%%%%%%%%
% divisorial preimage of a PL function
\newcommand{\divfiber}[1]{{#1}^{-1}_{\Divisor}}


%% define your title in the usual way
\title[Tropical Weierstrass points]{The distribution of Weierstrass points on a tropical curve}

%% define your authors in the usual way
%% use \addressmark{1}, \addressmark{2} etc for the institutions, and use \thanks{} for contact details
\author{David Harry Richman\thanks{\href{mailto:hrichman@umich.edu}{hrichman@umich.edu}. 
This work was partially supported by NSF grant DMS-1600223 and a Rackham Predoctoral Fellowship.}}
% \and you\addressmark{2}}

%% then use \addressmark to match authors to institutions here
\address{Department of Mathematics, University of Michigan, USA }
% \\ \addressmark{2}School of Something, University of 
% Something Else, Somewhere}

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\received{\today}

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%\revised{}

%% put your English abstract here, or comment this out if you don't have one yet
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\abstract{The set of (higher) Weierstrass points on a metric graph of genus $g > 1$ 
is an analogue of the set of $N$-torsion points on a circle. 
As $N$ grows, the torsion points "distribute evenly" over a circle. 
This makes it natural to ask how Weierstrass points distribute on a graph, 
as the degree of the corresponding divisor grows. 
We study how Weierstrass points behave on tropical curves (i.e. finite metric graphs)
in analogy with complex algebraic curves (i.e. Riemann surfaces), 
and explain how their distribution can be described in terms of electrical networks. 
This is a tropical analogue of a result of Neeman, 
for the distribution of Weierstrass points on a compact Riemann surface, 
and extends previous work of Zhang and Amini on the non-Archimedean case.
}

%% 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{tropical curve, metric graph, Weierstrass point, electrical network}

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%\usepackage[backend=bibtex]{biblatex}
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\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}
On a circle equipped with a length metric, 
for any positive integer $N$ there is a natural notion of $N$-torsion points.
This is a %finite set of points 
set of $N$ points which by definition are ``evenly spaced''
around the circle.
A specific choice of $N$-torsion points requires the additional data
 of choosing a basepoint 
on the underlying circle.
\begin{figure}[h]
\centering
\includegraphics[width=0.2\textwidth]{circle-7tors}
\caption{$7$-torsion points on a circle.}
\label{fig:plot}
\end{figure}

What if we replace the circle with an arbitrary metric graph, 
i.e. a graph equipped with edge lengths
and the path metric?

There is no longer an obvious notion of how to choose a finite set of points 
to be ``evenly spaced'' on a graph, 
especially if we want a choice that reflects global combinatorial or topological properties
of the underlying graph.

By applying the notions of tropical rank and reduced divisors of Baker--Norine~\cite{BN}
we define the {\em Weierstrass points} of a divisor on a metric graph,
and propose that this should be considered a good analogue 
of $N$-torsion points on a metric circle.
(The dependence on a divisor corresponds to the 
dependence on a basepoint
of $N$-torsion on a circle;
the parameter $N$ corresponds to the {\em degree} of the divisor.)
This viewpoint follows the suggestion of Mumford~\cite{Mum} for Weierstrass points
on an %complex 
algebraic curve.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{graph-Nweier}
\caption{Weierstrass points on a genus $4$ metric graph, for a degree $15$ divisor.}
\label{fig:plot}
\end{figure}


In the full version of this paper~\cite{R} we study the set of   
Weierstrass points 
associated to a divisor on an arbitrary metric graph.
%on an arbitrary metric graph,
%which is associated to an (arbitrary) %given %(effective) 
%divisor class on the graph.
In particular, we ask 
\begin{ques}
When is the set of Weierstrass points finite?
%\quad and
\end{ques}
\noindent and
\begin{ques}
How are these points distributed as the degree approaches infinity?
\end{ques}
%\[ \text{(A) When is the set of Weierstrass points finite?}\]
%and
%\[ \text{(B) How are these points distributed as the degree approaches infinity?}\]
We show that for any metric graph $\Gamma$, 
the Weierstrass locus is finite for a generic divisor class, 
and we prove that for any degree-increasing sequence of such generic divisors the 
Weierstrass points  become distributed according to 
the %Arakelov--
Zhang canonical measure on $\Gamma$~\cite{Z}.
This measure can be described via interpreting $\Gamma$ as an electrical network of resistors,
following Baker--Faber~\cite{BF}.


\subsection{Previous work}
The canonical Weierstrass points on a complex algebraic curve of genus $g\geq 2$ 
were studied classically. % (see e.g. \cite{Mir}).
This is a set of $g^3 - g$ points (counted with multiplicity) on $X$
which are intrinsic to $X$ as an abstract curve.
% without reference to any (non-canonical) embedding of $X$ into an ambient space.
They form a useful tool, e.g. for proving that the automorphism group of such a curve is finite.
This notion naturally extends to (higher) Weierstrass points, 
which are a finite set of points on $X$
associated to a choice of divisor $D$ on $X$.
%(With multiplicity, the number of points associated to $(X,[D])$ is $g(r+1)^2$ where $r+1 = h^0(D)$.)
The number of Weierstrass points of $D$ (counted with multiplicity) 
grows quadratically as a function of the degree of $D$.

Mumford suggests in~\cite{Mum} that
the Weierstrass points associated to a divisor of degree $N$
form a higher-genus analogue of the set of $N$-torsion points on an elliptic curve.
%(Just as choosing a different origin for the group law on a genus $1$ curve 
%leads to a different set of torsion points,
%choosing different degree $N$ divisors will give you different sets of Weierstrass points.)
The fact that $N$-torsion points on a complex elliptic curve become ``evenly distributed'' as $N$ grows large
leads one to ask whether a similar phenomenon holds for Weierstrass points generally. 
%(on a curve of higher genus).
%
This was answered %in the affirmative 
in 1984 by Neeman~\cite{N} (a student of Mumford):
for a complex algebraic curve $X$ of genus $g\geq 2$,
the Weierstrass points of a degree $N$ divisor
become equidistributed according to the Bergman measure on $X$
as $N\to \infty$ .

If the ground field $\CC$ is replaced with a non-Archimedean field, %with non-Archimedean valuation,
one may consider the same question of how Weierstrass points are distributed
inside the Berkovich analytification $X^{\text{an}}$,
%or how they are distributed 
say after retracting to a (compact) skeleton $\Gamma$.
This was addressed by Amini in \cite{A}:
%Here the answer %after retracting to a skeleton 
%is that 
the Weierstrass points are equidistributed
according to the ``Arakelov--Zhang canonical admissible measure'' $\mu = \mu_\Gamma$,
constructed by Zhang in \cite{Z};
%This measure does not have support on bridge edges, so it is independent of the choice of skeleton.
%Zhang's construction was motivated 
% by Arakelov's pairing for divisors on a Riemann surface %\cite{Ar},
%on an arithmetic surface, 
%for the purpose of answering questions in arithmetic geometry.
we make use of a definition of $\mu$ along more elementary lines 
following Baker--Faber~\cite{BF}, 
%(and Shokrieh \cite{BS}),  
using the notions of current flow and electric potential in a (1-dimensional) network of resistors.

In his preprint, Amini raises the question of whether 
the distribution of Weierstrass points is possibly intrinsic to the metric graph $\Gamma$,
without needing to identify $\Gamma$ with the skeleton of some Berkovich curve $X^{\text{an}}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{berk-to-circle}
\caption{Weierstrass points on a genus $1$ Berkovich curve, 
and retraction to a skeleton (adapted from an illustration of Matt Baker).}
\end{figure}
One major obstacle to this idea is that on a metric graph, 
the  Weierstrass locus for a divisor may fail to be a finite set of points.
We sidestep this issue % entirely 
by showing that finiteness does hold 
for a generic choice of divisor class (\cref{thm:w-finite}).
With this assumption of genericity, we are able to show that 
the distribution of Weierstrass points in large degree 
is indeed intrinsic to $\Gamma$ (\cref{thm:equidistribution}).

%(Technical note: our %abstract 
%tropical curves $\Gamma$ have no ``hidden genus'' and no infinite legs,
%i.e. we restrict our attention to $X^{\text{an}}$ with totally degenerate reduction 
%and no punctures.)


%\subsection{Statement of results}
%Given a compact, connected metric graph $\Gamma$ and a divisor $D$ of rank $r = r(D)$,
%we define the Weierstrass locus $W(D) = W_{\Gamma,D}$ as 
%\[ W(D) = \{ x\in \Gamma : D \sim (r+1)x + E \text{ for some }E \geq 0\},\]
%where $\sim$ denotes linear equivalence and $r(D)$ is the Baker-Norine rank.
%The set $W(D)$ may fail to be finite; in some cases it contains all of $\Gamma$
%(see Example \ref{eg:3loop-wedge}).
%Our first result addresses when the Weierstrass locus is finite.
%Here ``generic'' means %the condition holds 
%on a dense open subset of the space of divisor classes.
%
%\begin{customthm}{A}
%\label{thm:w-finite}
%Let $\Gamma$ be a compact, connected metric graph. 
%For a generic divisor class $[D]$ on $\Gamma$, 
%the associated Weierstrass locus $W(D)$ is finite.
%%{ Additionally: When $\geq 2g-1$,
%%the associated Weierstrass locus $W(D)$ is a discrete set of $g(N-g+1)$ points.}
%\end{customthm}
%The next theorem is the main result of our paper, 
%which describes the equidistribution of tropical Weierstrass points.
%\begin{customthm}{B}
%\label{thm:equidistribution}
%Let $\Gamma$ be a metric graph of genus $g$,
%and let $\{ [D_N] : N\geq 1\} $ be a sequence of generic divisor classes on $\Gamma$ with $\deg D_N = N$.
%Let $W_N$ denote the Weierstrass locus of the divisor $D_N$, and let
%\[ \delta_N  = \frac1{N} \sum_{x\in W_N} \delta_x\]
%denote the normalized discrete measure on $\Gamma$ associated to $W_N$
%(where $\delta_x$ is the Dirac measure at $x$). 
%Then as $N\to \infty$, the measures $\delta_N$ converge weakly 
%to the Zhang canonical measure $\mu$ on $\Gamma$.
%\end{customthm}
%(Warning: we use a different normalization for $\mu$ than previous authors, 
%namely we have $\mu(\Gamma) = g$ rather than $\mu(\Gamma) = 1$.)
%We also obtain a quantitative version of this result which specifies 
%a bound on the  rate of convergence.
%\begin{customthm}{C}
%\label{thm:quant-equidistribution}
%Let $\Gamma$ be a metric graph of genus $g$,
%let $[D_N]$ be a generic degree $N$ divisor class, % on $\Gamma$.
%and let $W_N$ denote the  Weierstrass locus of $D_N$.
%Let $\mu$ denote the Arakelov-Zhang canonical measure on $\Gamma$.
%\begin{enumerate}[(a)]
%\item For any segment $e$ in $\Gamma$,
%\begin{equation*}
%N \mu(e) - 3g - 1 \leq \# (W_N\cap e) \leq N \mu(e) + g + 2.
%\end{equation*}
%%or,
%%\begin{equation*}
%% \mu(e) - \frac{3g + 1}{N} \leq \frac{\# (W_N\cap e)}{N} \leq  \mu(e) + \frac{g + 1}{N}.
%%\end{equation*}
%%where $W_N$ is the set of $D_N$-Weierstrass points on $\Gamma$ and $\mu$ is the canonical measure.
%
%\item As $N\to \infty$,
%\begin{equation*}
%\frac{\#(W_N \cap e)}{ N} = \mu(e) + O\left( \frac1{N} \right).
%\end{equation*}
%
%\item[(b')] For a fixed continuous function $f: \Gamma \to \RR$,
%\begin{equation*}
%\frac1{N}\sum_{x \in W_N} f(x) = \int_\Gamma f(x) \mu(dx) + O\left(\frac1{N}\right).
%\end{equation*}
%(The big-$O$ constant may depend on $f$.)
%
%\item If $e$ is a segment of $\Gamma$ with  $\mu(e) > \frac{3g+1}{N}$,
%then $e$ contains at least one $D_N$-Weierstrass point.
%
%\end{enumerate}
%\end{customthm}
%It is likely that the bounds in part (a) are not sharp.
%%(For the technical definition of a ``segment'' of $\Gamma$ see Section \ref{subsec:metric-graphs}.)

\section{Tropical curves}
%In this section we define metric graphs and linear equivalence of divisors on metric graphs. 
We use the terms ``metric graph'' and ``abstract tropical curve'' interchangeably.
%We recall the Baker-Norine rank of a divisor, and state the Riemann-Roch theorem which is satisfied by this rank function.

A {\em metric graph} is a compact, connected metric space 
which comes from assigning positive real edge lengths to a finite connected combinatorial graph.
%Namely, we construct a metric graph $\Gamma$  by taking 
%a finite set of edges $E = \{e_i\}$ each isometric to a real interval $e_i = [0,L_i]$ of length $L_i>0$,
%gluing their endpoints to a finite set of vertices $V$,
%and imposing the path metric.
The underlying combinatorial graph $G = (E,V)$ is called a {\em combinatorial model} for $\Gamma$;
%A metric graph usually has infinitely many possible combinatorial models.
we allow loops and parallel edges in $G$.
We say $e$ is a {\em segment} of $\Gamma$ if it is an edge in some combinatorial model. 

The {\em valence} $\val(x)$ of a point $x$ on a metric graph $\Gamma$ is  the number 
on connected components of a sufficiently small punctured neighborhood of $x$.
Points in the interior of a segment %of $\Gamma$ 
always have valence 2.
All points $x$ with $\val(x) \neq 2$ are contained in the vertex set of any combinatorial model.

The {\em genus} of a metric graph $\Gamma$ is its first Betti number as a topological space,
\[ g(\Gamma) = b_1(\Gamma) = \dim_\RR H_1(\Gamma,\RR).\]
%A metric graph of genus $g$ is homotopy equivalent to a wedge of $g$ loops, 
%and has Euler characteristic $\chi(\Gamm a) = 1 - g$.
If $G$ is a combinatorial model for $\Gamma$, the genus is equal to
$ g(\Gamma) = \# E(G) - \# V(G) + 1$.

\begin{eg}
The metric graph in \cref{fig:genus0} has genus 0.
A minimal combinatorial model has $8$ vertices and $7$ edges.
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{genus0}
\caption{Metric graph of genus $0$.}
\label{fig:genus0}
\end{figure}
\end{eg}

\begin{eg}
The metric graph in \cref{fig:genus2} has genus $2$.
A minimal combinatorial model has $2$ vertices and $3$ edges.
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{genus2theta}
\caption{Metric graph of genus $2$.}
\label{fig:genus2}
\end{figure}
\end{eg}

A {\em divisor} on a metric graph $\Gamma$ is a finite formal sum of  
points of $\Gamma$ with integer coefficients.
%We express a divisor $D$ by
%\[ D = \sum_{x\in \Gamma} a_x x \qquad \text{where } a_x \in \ZZ,  
%\text{ and }a_x = 0 \text{ for almost all } x. \]
The {\em degree} of a divisor is the sum of its coefficients; 
i.e.
for the divisor $D = \sum_{x\in \Gamma} a_x x $,
we have
${ \deg( D ) = \sum_{x\in\Gamma} a_x }$.
We let $\Div(\Gamma)$ denote the abelian group of all divisors on $\Gamma$,
and let $\Div^d(\Gamma)$ denote the divisors of degree $d$.
We say a divisor is {\em effective} if all its coefficients are nonnegative;
we write $D\geq 0$ %as a shorthand 
to indicate that $D$ is effective.
We let $\Sym^d(\Gamma)$  denote the set of effective divisors of degree $d$ on $\Gamma$.
%Since $\Sym^d(\Gamma)$ may be identified with the $d$-fold symmetric power of $\Gamma$,
$\Sym^d (\Gamma)$ inherits from $\Gamma$ the structure of a polyhedral cell complex of dimension $d$.

%Since an effective divisor of degree $d$ is naturally equivalent to an unordered $d$-tuple of points of $\Gamma$
%(not necessarily distinct), 
%we can identify $\Sym^d\Gamma$ with the $d$-fold symmetric product $\Sym^d \Gamma$;
%since $\Gamma$   is a 1-dimensional cell complex,
%$\Sym^d \Gamma$ inherits from $\Gamma$ the structure of a polyhedral cell complex of dimension $d$.

We let $\Div_\RR(\Gamma)$ denote the set of divisors on $\Gamma$ with coefficients in $\RR$.
In other words, $\Div_\RR(\Gamma) = \Div(\Gamma)\otimes_\ZZ \RR$.

\subsection{Linear equivalence}

We define linear equivalence for divisors on metric graphs, 
following Gathmann--\linebreak Kerber~\cite{GK}. %and Mikhalkin--Zharkov \cite{GK,MZ}.
This notion is analogous to linear equivalence of divisors on an algebraic curve,
where rational functions are replaced with piecewise $\ZZ$-linear functions. 
%[Originally due to Baker--Norine + Mikhalkin--Zharkov + ???]
%The analogue of the set of rational functions on an algebraic curve
% is the set of piecewise $\ZZ$-linear functions on a metric graph, which we define below.

A {\em piecewise linear function} on $\Gamma$ is a continuous function $f: \Gamma \to \RR$ 
such that there is 
% some decomposition of $\Gamma$ into a union of finitely many segments 
% where $f$ restricted to each segment is a linear (affine) function, 
some combinatorial model for $\Gamma$ such that 
$f$ restricted to each edge is an affine linear function,
i.e. a function of the form
\[ f(x) = ax + b, \qquad a,b\in \RR\]
%after a length-preserving parametrization 
where $x$ is a length-preserving parameter on the edge.
We let $\PL_\RR(\Gamma)$ denote the set of all piecewise linear functions on $\Gamma$.
%Note that $\PL_\RR(\Gamma)$ naturally has the structure of an $\RR$-vector space.
%(It is moreover a tropical $\RR$-module, under operations addition and minimum.)
%
A {\em piecewise $\ZZ$-linear function} on $\Gamma$ is a piecewise linear function 
such that all its slopes are integers, 
i.e. $f$ restricted to each edge (for some combinatorial model) has the form
\[ f(x) = ax + b, \qquad a\in\ZZ,\, b\in \RR .\]
We let $\PL_\ZZ(\Gamma)$ denote the set of all piecewise $\ZZ$-linear functions on $\Gamma$.
%The functions $\PL_\ZZ(\Gamma)$ are closed under the operations of addition,
%multiplication by $\ZZ$,
%and taking pairwise $\max$ and $\min$.

We let $UT_x\Gamma$  denote the 
unit tangent fan of $\Gamma$ at $x$,
which is the set of ``directions going away from $x$'' on $\Gamma$.
For $v \in UT_x\Gamma$,  
the symbol $\epsilon v$ for sufficiently small $\epsilon \geq 0$
means the point in $\Gamma$
at distance $\epsilon$ away from $x$ in the direction $v$.
%which is a finite set of size $\val(x)$.
%An element $v \in UT_x\Gamma$ is a formal symbol naturally identified with a component $e_v$
%of a sufficiently small punctured $\epsilon$-neighborhood of $x$ in $\Gamma$
%(such that this punctured neighborhood has $\val(x)$ components); 
%then $\delta v$  for $0 \leq \delta \leq \epsilon$ means the point of the closed segment $\text{cl}(e_v)$ 
%that is a distance $\delta$ away from $x$.
For  $v\in UT_x\Gamma$
and a function $f: \Gamma \to \RR$   we let 
\[D_vf(x) = \lim_{\epsilon\to 0^+} \frac{f(x+\epsilon v)-f(x)}{\epsilon}\]
denote the slope of $f$ while travelling away from $x$ in the direction $v$.

Given %a piecewise $\ZZ$-linear function 
$f \in \PL_\ZZ(\Gamma)$, 
we define the {\em principal divisor} $\Divisor(f) \in \Div^0(\Gamma)$ %(or $\Div(f)$?) defined 
by
\[ \Divisor(f) = \sum_{x\in \Gamma} \mathrm{ord}_x(f) x 
\qquad\text{where}\qquad 
\mathrm{ord}_x(f) = \sum_{v\in UT_x \Gamma} D_v f(x). \]
In words, the coefficient in $\Divisor(f)$ at $x$ is equal to 
the sum of the outgoing slopes.
This divisor is supported on the finite set of points at which $f$ is not linear, 
 called the {\em break locus} of $f$.

If $\Divisor(f) = D - E$ where $D, E$ are effective divisors with disjoint support, 
then we call $D = \zeros(f)$ the {\em divisor of zeros of $f$} 
and $E = \poles(f)$ the {\em divisor of poles} of $f$.
%
We say two divisors $D, E$ are {\em linearly equivalent}, denoted $D\sim E$, if there exists a piecewise $\ZZ$-linear function $f$ such that
\[ \Divisor(f) = D - E.\]
Note that linear equivalence preserves degree.
We let $[D]$ denote the linear equivalence class of divisor $D$, i.e.
\[ [D] = \{ E \in \Div(\Gamma) : E \sim D\} 
= \{D + \Divisor(f) : f\in \PL_\ZZ(\Gamma)\} .\]
%We say a divisor class $[D]$ is {\em effective}, or write $[D]\geq 0$, 
%if there is an effective representative ${ E\sim D },\, E\geq 0$ in the equivalence class.
%
We let $|D|$ denote the {\em complete linear system} of $D$, 
which is the set of effective divisors linearly equivalent to $D$.
Namely, we have
\begin{align*}
 |D| &= \{ E \in \Sym^d(\Gamma) : E \sim D\} 
 = \{D + \Divisor(f) : f\in \PL_\ZZ(\Gamma),\, \Divisor(f) \geq -D\} .
\end{align*}
%Unlike $[D]$, the linear system $|D|$ is naturally a compact polyhedral complex, 
%with topology induced by the inclusion
The topology on $|D|$ is induced by the inclusion
$|D| \subset \Sym^d(\Gamma)$.

\begin{rmk}[Linear interpolation along $f$]
Given a function $f \in \PL_\ZZ(\Gamma)$,
we may associate to $f$  
a 1-parameter family of effective divisors 
which ``linearly interpolate'' between the zeros $\zeros(f)$ and poles $\poles(f)$.
(We can think of this construction as specifying a unique ``geodesic path'' 
between any two points in the complete linear system $|D|$.)


Namely, for $\lambda \in \RR$ we let $\lambda \in \PL_\ZZ(\Gamma)$ also denote the constant function on $\Gamma$
by abuse of notation,
and we define the effective divisor $\divfiber{f}(\lambda)$ 
by
\[ \divfiber{f}(\lambda) = \poles(f) + \Divisor(\max\{f ,\lambda\}). \] 
% = \zeros(f) - \Divisor(f \wedge \lambda).\]
% ($\vee = \max$, $\wedge = \min$).
See \cref{fig:interpolation} for an illustration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{pl-function-preimage}
\caption{Linear interpolation showing the divisor $\divfiber{f}(\lambda)$
in dark blue.}
\label{fig:interpolation}
\end{figure}

Note that according to this definition,
${ \divfiber{f}(\lambda) = \poles(f) }$ for $\lambda$ sufficiently large and 
${ \divfiber{f}(\lambda) = \zeros(f) }$ for $\lambda$ sufficiently small.
%(i.e. $\lambda \geq \max_\Gamma(f)$ and $\lambda \leq \min_\Gamma(f)$ respectively).
%
%(We also have the more symmetric, but more complicated, expression
%\[ f^{-1}(\lambda) := D(f,\lambda) =  \zeros(f \vee \lambda) + \poles(f \wedge \lambda) 
%- \left( \cap \right) .\]
It is clear from definition that for any $\lambda$, 
$\divfiber{f}(\lambda)$ is %an (effective) divisor which is 
linearly equivalent to $\zeros(f)$ and to $\poles(f)$.
\end{rmk}

\subsection{Reduced divisors}
A divisor class $[D]$ is typically very large, 
so it is convenient to have a method of choosing a somewhat-canonical representative divisor inside $[D]$.
We can do so after fixing a basepoint $q$ on our metric graph $\Gamma$.
The {\em reduced divisor} $\reddiv{q}{D}$ is the unique divisor in $[D]$ which is effective away from $q$, and which minimizes a certain ``energy function'' among such 
representatives~\cite[Appendix A]{BS}.
Intuitively, $\reddiv{q}{D}$ is the divisor in $[D]$ whose effective part is ``as close as possible'' to the basepoint $q$.
%We defer giving the full definition until Section \ref{subsec:red-div},
%following  [Baker--Shokrieh, Appendix A]. %(unpublished??)].
We state these important properties of the reduced divisor:
\begin{enumerate}
%\item $\reddiv{q}{D}$ is a unique divisor inside $[D]$
%\item if $D$ has degree $d$, then $\reddiv{q}{D}$ has degree at least $d-g$ at $q$ 
%and at most $g$ away from $q$
%\item $\reddiv{q}{D}$ is effective away from $q$
\item[(RD1)] $[D] \geq 0$ if and only if $\reddiv{q}{D} \geq 0$
\item[(RD2)] the degree of $\reddiv{q}{D}$ away from $q$ is at most $g$
\item[(RD3)] for a fixed divisor $D$, the map $\Gamma \to |D|$ sending $q \mapsto \reddiv{q}{D}$ is continuous 
%(due to  Amini \cite[Theorem 3]{A-reddiv})
\end{enumerate}

\section{Weierstrass points}
On an algebraic curve $X$ of genus $g$, 
the canonical Weierstrass points are defined as follows.
The canonical divisor $K$ on $X$ determines a map to projective space 
$\varphi_K : X \to \PP^{g-1}$.
Generically a point on $\varphi_K(X)$ will have an osculating hyperplane in $\PP^{g-1}$ 
which intersects $\varphi_K(X)$ with multiplicity $g-1$.
For finitely many ``exceptional'' points on $\varphi_K(X)$, 
the osculating hyperplane will intersect the curve with higher multiplicity; 
the preimages of these exceptional points are the {\em canonical Weierstrass points} of $X$.
(These  are also known as the {\em flex points} of the embedded curve $\varphi_K(X) \subset \PP^{g-1}$)

This notion is generalized 
%to higher Weierstrass points 
by replacing $K$ 
with an arbitrary divisor.
Given a (basepoint-free) divisor $D$ on $X$, 
%of degree at least $2g-1$, 
there is an associated map to projective space 
$\varphi_D : X \to \PP^r$.
%for some nonnegative $r$.
If the degree of $D$ is at least $2g+1$ then $r = \deg D - g$ and this map is an embedding.
The set of flex points of the embedded curve $\varphi_D(X)$, 
where the osculating hyperplane intersects the curve with multiplicity greater than $r$, 
are the {\em (higher) Weierstrass points} associated to the divisor $D$.


%The existence of an osculating hyperplane of multiplicity greater than $r$, at the point $\varphi_D (x)\in \varphi_D (X)$, is equivalent to the existence of a non-zero global section of the line bundle  $\mathcal L(X,D-(r+1)x)$,
%%the existence of a nonconstant meromorphic function on $X$ ...
%i.e. to having $h^0(X,D-(r+1)x) \geq 1$.


\subsection{Tropical Weierstrass points}

We define the Weierstrass points of a divisor on a metric graph $\Gamma$, 
using the Baker--Norine rank function and
the notion of reduced divisors~\cite{BN,BS}.
%which is the analogue of $h^0(D)-1$.

For a divisor on a metric graph,
the {\em Baker--Norine rank} $r(D)$ is defined by
\[ r(D) = \max \{ r\geq 0 : |D - E| \neq \emptyset 
          \text{ for all }E \in \Sym^r(\Gamma)\} .\]
The rank satisfies Riemann's inequality
$r(D) \geq \deg(D) - g$.
When the degree is sufficiently large, 
which is the case of interest in \cref{thm:equidistribution},
%i.e. $\deg(D) \geq 2g-1$, 
this is an equality;
\[ r(D) = \deg(D) - g \quad\text{when}\quad \deg(D) \geq 2g-1 .\]

\begin{dfn}
Let $D$ be a divisor on a metric graph $\Gamma$, with rank $r = r(D)$.
A point $x\in \Gamma$ is a {\em Weierstrass point for $D$} %, or a {\em $D$-Weierstrass point}, 
if 
% r(D - (r+1)x) \geq 0 %%%% <-- NO GOOD, "r" is overloaded
%\[ [D-(r+1)x] \geq 0.\]
\[ \reddiv{x}{D} \geq (r+1)x .\]
The {\em Weierstrass locus} $W(D) \subset \Gamma$ of $D$ is the set of its Weierstrass points.

%A {\em canonical Weierstrass point}  is a Weierstrass point for the canonical divisor $K$.
%Namely, $x\in \Gamma$ is a Weierstrass point if
%\[ r(K - gx) \geq 0 \quad\Leftrightarrow\quad 
%\[ [K-gx] \geq 0.\]
\end{dfn}
Note that the Weierstrass locus of $D$ depends only on the divisor class $[D]$.

%\begin{rmk}
%If the divisor class $[D]$ is not effective, i.e. $r(D) = -1$, 
%then the set of $D$-Weierstrass points is empty.
%Thus we may restrict our attention to studying  
%Weierstrass points for effective divisor classes. 
%\end{rmk}

\begin{eg}
Suppose $\Gamma$ is a genus $1$ graph and $D$ is a divisor of degree $6$,
indicated by the black dots in \cref{fig:gen0-w} with multiplicities.
This divisor has rank $r = 5$ since it is in the ``non-special range'' of Riemann-Roch.
The Weierstrass locus of $D$ consists of $6$ points evenly spaced around $\Gamma$,
indicated in red.
\begin{figure}[h]
\centering
\includegraphics[scale=0.4]{genus1-weierstrass2}
\caption{Weierstrass points of a degree $6$ divisor on a genus $1$ metric graph.}
\label{fig:gen0-w}
\end{figure}
\end{eg}
\begin{eg}
Suppose $\Gamma$ is 
%a complete graph on $4$ vertices, which has
the  genus $3$ metric graph in \cref{fig:gen3-w}.
The canonical divisor $K$ on $\Gamma$ is supported on the four trivalent vertices.
This divisor has rank $r=2$.
Assuming generic  edge lengths, the Weierstrass locus of $K$ 
consists of $8$ distinct points on $\Gamma$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.3]{genus3-weierstrass}
\caption{Metric graph with Weierstrass locus $W(K)$ finite.}
\label{fig:gen3-w}
\end{figure}
\end{eg}


%\begin{eg}
%Suppose $\Gamma$ is a wedge of $g$ circles,
%and let $x_0$ denote the point of $\Gamma$ lying on all $g$ circles.
%For a generic divisor class $[D_N]$ of degree $N$,
%% (generic inside of $\Pic^N( \Gamma)$), 
%the $x_0$-reduced representative of $[D_N]$ consists of $N-g$ chips at $x_0$ and one chip in the interior of each circle.
%Thus the Weierstrass locus of $D_N$
%will contain $N-g+1$ evenly-spaced points on each circle of $\Gamma$, for a total of $g(N-g+1)$ points.
%\end{eg}



\begin{eg}[Failure of $W(D)$ to be finite]
\label{eg:3loop-wedge}
Suppose $\Gamma$ is the genus $2$ graph shown in \cref{fig:w-degenerate},
and $D$ is a degree $4$ divisor supported on the bridge edge in \cref{fig:deg1}.
This divisor has rank $r=2$.
Every point of $\Gamma$ is in the Weierstrass locus of $D$.
%
%Consider the genus $3$ graph shown in Figure~\ref{fig:w-degenerate}.
%Suppose $D$ is a degree $4$ divisor supported on one of the bridge edges as shown.
%(Note that $D \sim K$.)
%This divisor has rank $r \leq 2$,
%since we cannot move the chips in $D$ to lie on three distinct loops freely.
%However, for any point $x$, the reduced divisor $\reddiv{x}{D}$ has at least $3$ chips at $x$.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.4\textwidth}
	\centering
	\includegraphics[scale=0.4]{w-degenerate}
	\caption{Nongeneric divisor class $[D]$}
	\label{fig:deg1}	
\end{subfigure}
\begin{subfigure}[b]{0.4\textwidth}
	\centering
	\includegraphics[scale=0.4]{w-generic}
	\caption{Generic divisor class $[D']$}
	\label{fig:deg2}
\end{subfigure}
\caption{Metric graph with $W(D) = \Gamma$ 
and  a nearby divisor with $W(D')$  finite.}
\label{fig:w-degenerate}
\end{figure}

%\begin{figure}[h]
%\centering
%\includegraphics[scale=0.4]{w-generic}
%\caption{Metric graph with finite Weierstrass locus $W(D)$.}
%\label{fig:w-generic}
%\end{figure}
\noindent When $[D]$ is deformed to a nearby divisor class $[D']$, 
%the rank does not change and
the Weierstrass locus becomes a finite set of $6$ points
as shown in \cref{fig:deg2}.
The rank does not change between $D$ and $D'$.
\end{eg}

%\begin{eg}[Failure of $W(D)$ to be finite, v2]
%Consider the genus $3$ graph shown in Figure~\ref{fig:w-degenerate2}.
%Suppose $D=K$ is the canonical divisor on $\Gamma$. 
%By Riemann-Roch $K$ has rank $r = 2$.
%Since it is possible to move all 4 chips to lie on the middle loop, 
%any point in the middle loop is a Weierstrass point for $K$.
%\begin{figure}[h]
%%\includegraphics[scale=0.4]{w-degenerate2}
%%$\qquad$ 
%\centering
%\includegraphics[scale=0.4]{w-degenerate2red}
%\caption{Metric graph with Weierstrass locus $W(K)$ not finite.}
%\label{fig:w-degenerate2}
%\end{figure}
%\end{eg}

%\begin{eg}[Failure of $W(D)$ to be finite, v3]
%Suppose $\Gamma$ is a metric graph constructed by adding a bridge edge 
%connecting two disjoint metric graphs $\Gamma_1$, $\Gamma_2$
%of positive genus $g_1$, $g_2$ respectively.
%Let $K = K_\Gamma$ denote the canonical divisor on $\Gamma$; it is clear that
%\[ K = K_{\Gamma_1} + K_{\Gamma_2} + y_1 + y_2,\]
%where $y_1$ and $y_2$ are the endpoints of the bridge edge.
%
%By considering $\PL$ functions which vary on $\Gamma_1$, we have
%$K_{\Gamma_1} \sim_{\Gamma} (g_1-1)y_1 + E_1$
%for some effective divisor $E_1$, 
%and similarly 
%$K_{\Gamma_2} \sim_{\Gamma} (g_2-1)y_2 + E_2$.
%Thus on $\Gamma$ there is a linear equivalence
%$$ K \sim g_1 y_1 + g_2 y_2 + E_1 + E_2 .$$
%Since chips may move freely on a bridge edge, 
%$g_1 y_1 + g_2  y_2 \sim (g_1+g_2) y$
%for any $y \in e$.
%This shows that $e$ is contained in the Weierstrass locus since 
%$$r(K) = g-1 = g_1 + g_2 -1 .$$
%\end{eg}

Our first result addresses when the Weierstrass locus is finite.
Here ``generic'' means %the condition holds 
on a dense open subset of the space of divisor classes.

\begin{customthm}{A}
%\begin{thm}
\label{thm:w-finite}
Let $\Gamma$ be a compact, connected metric graph. 
For a generic divisor class $[D]$ on $\Gamma$, 
the associated Weierstrass locus $W(D)$ is finite.
%{ Additionally: When $\geq 2g-1$,
%the associated Weierstrass locus $W(D)$ is a discrete set of $g(N-g+1)$ points.}
%\end{thm}
\end{customthm}
The next theorem 
describes the distribution of tropical Weierstrass points;
this is the main result of our paper.
\begin{customthm}{B}
\label{thm:equidistribution}
Let $\Gamma$ be a metric graph of genus $g$,
and let $\{ [D_N] : N\geq 1\} $ be a sequence of generic divisor classes on $\Gamma$ with $\deg D_N = N$.
Let $W_N$ denote the Weierstrass locus of the divisor $D_N$, and let
\[ \delta_N  = \frac1{N} \sum_{x\in W_N} \delta_x\]
denote the normalized discrete measure on $\Gamma$ associated to $W_N$,
where $\delta_x$ is the Dirac measure at $x$. 
Then as $N\to \infty$, the measures $\delta_N$ converge weakly 
to the Zhang canonical measure $\mu$ on $\Gamma$.

In other words, for any segment $e$ of $\Gamma$, as $N\to \infty$ we have
\begin{equation*}
\frac{\#(W_N \cap e)}{ N} \to \mu(e) .
\end{equation*}
\end{customthm}
%(Warning: we use a different normalization for $\mu$ than previous authors, 
%namely we have $\mu(\Gamma) = g$ rather than $\mu(\Gamma) = 1$.)
We also obtain a quantitative version of this result which specifies 
a bound on the  rate of convergence.
\begin{customthm}{C}
\label{thm:quant-equidistribution}
Let $\Gamma$ be a metric graph of genus $g$,
let $[D_N]$ be a generic degree $N$ divisor class, % on $\Gamma$.
and let $W_N$ denote the  Weierstrass locus of $D_N$.
Let $\mu$ denote the Zhang canonical measure on $\Gamma$.
\begin{enumerate}[(a)]
\item For any segment $e$ in $\Gamma$,
\begin{equation*}
N \mu(e) - 3g - 1 \leq \# (W_N\cap e) \leq N \mu(e) + g + 2.
\end{equation*}
%or,
%\begin{equation*}
% \mu(e) - \frac{3g + 1}{N} \leq \frac{\# (W_N\cap e)}{N} \leq  \mu(e) + \frac{g + 1}{N}.
%\end{equation*}
%where $W_N$ is the set of $D_N$-Weierstrass points on $\Gamma$ and $\mu$ is the canonical measure.

%\item As $N\to \infty$,
%\begin{equation*}
%\frac{\#(W_N \cap e)}{ N} = \mu(e) + O\left( \frac1{N} \right).
%\end{equation*}

\item For a fixed continuous function $f: \Gamma \to \RR$,
\begin{equation*}
\frac1{N}\sum_{x \in W_N} f(x) = \int_\Gamma f(x) \mu(dx) + O\left(\frac1{N}\right).
\end{equation*}
(The big-$O$ constant may depend on $f$.)

\item If $e$ is a segment of $\Gamma$ with  $\mu(e) > \frac{3g+1}{N}$,
then $e$ contains at least one $D_N$-Weierstrass point.

\end{enumerate}
\end{customthm}

\section{Canonical measure}
In this section we describe 
%the Arakelov--Zhang canonical measure 
the limiting measure of Weierstrass points
on a metric graph (defined by Zhang~\cite{Z}) 
via the perspective of resistor networks following Baker--Faber~\cite{BF}. 
%[cite Zhang, Baker-Faber \cite{BF}, ..]
We may view this construction as a one-dimensional  analogue 
of Gaussian curvature on a closed two-dimensional surface.

\subsection{Electrical networks} %, voltage, current}
We  view a metric graph $\Gamma$ as a resistor network by interpreting an edge of length $L$ as a resistor of resistance $L$.
%Note that this is well-defined on a metric graph due to the series rule for combining resistances, 
%so we have compatibility with subdividing an edge into edges of shorter length.
%This interpretation is not only mathematically convenient, but physically 
%honest---the electrical resistance of a wire is directly proportional to its length, 
%a fact known as Pouillet's law.

On a resistor network we may send current from one point to another.
By Ohm's law,
the voltage drop across a segment is equal to its resistance (i.e. length) 
multiplied by the amount of current passing through. 
%the segment---this is %known %in physics as 
%Ohm's law.
Under an externally-applied current, % flow, % or voltage differential,
the flow of current within the network is determined by Kirchoff's circuit laws: 
the current law says that the sum of directed currents out of any point is equal to zero 
(accounting for external currents),
and the voltage law says that the sum of directed voltage differences around any closed loop is equal to zero.
It is a well-known empirical fact that Kirchoff's circuit laws can be solved uniquely 
for any externally-applied current flow which satisfies conservation of current.
%(i.e. internal current flows are unique). %, voltage function is unique up to additive constant) 
(To some, it is a well-known mathematical result.)

Our convention is that current flows from higher voltage to lower voltage.

\begin{eg}
\label{eg:current}
We illustrate in \cref{fig:current} the induced current flow
from sending $1$ unit of current
from $y$ to $z$, where the metric graph has unit edge-lengths.
Assuming $\Gamma$ is grounded at $z$, the voltages at the trivalent vertices are
$0$, $\frac{2}{12}$, $\frac{3}{12}$, $\frac{3}{12}$, and $\frac{5}{12}$.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{current_flow}
\caption{Current flow induced by external current from $y$ to $z$.}
\label{fig:current}
\end{figure}
\end{eg}

\begin{rmk}
We may interpret any piecewise linear function $f\in \PL_\RR(\Gamma)$
as a voltage function on $\Gamma$,
which results from the externally applied current $\Delta(f)\in \Div_\RR(\Gamma)$.
In other words, 
$f$ gives a voltage resulting from sending current from $\poles(f)$ to $\zeros(f)$
in $\Gamma$.
%
%The existence of $\potent{z}{y} \in \PL_\RR(\Gamma)$ for any $y,z\in \Gamma$ implies that 
%the principal divisor map
%$ \Divisor : \PL_\RR(\Gamma) \to \Div_\RR^0(\Gamma)$
%is surjective.
%This verifies the claim made in Remark~\ref{rmk:real-div} concerning the exactness of the sequence
%\[ 0 \to \RR \xrightarrow{\text{const}} \PL_\RR(\Gamma)\xrightarrow{\Divisor} \Div_\RR(\Gamma) \xrightarrow{\text{deg}} \RR \to 0 .\]
\end{rmk}

\begin{dfn}
Let $r : \Gamma \times \Gamma \to \RR$ denote the  resistance function on the metric graph $\Gamma$. 
Namely, viewing $\Gamma$ as a resistor network
%we view $\Gamma$ as a resistor network where 
%each edge is a resistor whose resistance is equal to its length, and
\begin{align*}
r(x,y) &= \text{effective resistance between $x$ and $y$} \\
&= \text{total voltage drop when sending 1 unit of current from $x$ to $y$}
\end{align*}
%In terms of the voltage function from Section \ref{subsec:voltage}, $r(x,y) = \potent{y}{x}(x)$.
\end{dfn}

It is straightforward to verify that the resistance function satisfies the following properties:
%\begin{enumerate}
%\item 
(1) $r(x,x) = 0$,
%\item 
(2) $r(x,y) > 0$ when $x \neq y$,
%\item 
(3) $r(x,y)$ is continuous with respect to $x$ and $y$, and
%\item 
(4) $r(x,y) = r(y,x)$.
%\item for fixed $y$, the function $x \mapsto r(x,y)$ is in $\PL_\RR(\Gamma)$.
%\end{enumerate}
\begin{eg}
\label{eg:circle-resistance}
Let $\Gamma$ be a circle of circumference $L$. 
%(we use $R$ for ``resistance''). 
By choosing a basepoint which we denote as $0$, we may parametrize $\Gamma$ with the interval $[0,L]$.
Identifying points in this way, we have
\begin{align*}
r(x,0) &= \text{parallel combination of resistances $x$ and $L-x$} \\
&= \frac{x(L-x)}{x + (L-x)} = x - \frac1 L x^2.
\end{align*}
The effective resistance is maximized when $x = \frac12 L$, with maximum value %$r(\frac12 L,0) = 
$ \frac14 L$.
The effective resistance is minimized when $x= 0$ or $x = L$, with effective resistance $0$.
\end{eg}

\begin{dfn}
The {\em canonical measure} $\mu = \mu_\Gamma$ on a metric graph $\Gamma$ is 
the continuous measure defined by
\[ \mu  = -\frac12 \frac{d^2}{dx^2} r(x,y_0) \,dx.\]
where $x$ is a length-preserving parameter on a segment,
$dx$ is the Lebesgue measure,
and $y_0$ is a fixed point in $\Gamma$.
This defines $\mu$ on the open dense subset of $\Gamma$ where the second derivative exists;
%at the finite set of points where $r(-,y_0)$ is not differentiable, 
%or where the valence of $x$ differs from 2, 
otherwise we let $\mu_\Gamma= 0$.
\end{dfn}
\cref{eg:circle-resistance} shows that 
on a circle of circumference $L$ 
the canonical measure 
is $\frac1L dx$.
For an arbitrary metric graph,
the canonical measure on an open segment is
a constant multiple of the Lebesgue measure.
%
The following characterization of the canonical measure is used to 
prove Theorem B on the distribution of Weierstrass points.
\begin{prop}[Baker--Faber \cite{BF}]
\label{prop:current}
Let $e$ be a segment on a metric graph with endpoints $e^-$, $e^+$
and length $R(e)$.
Then the canonical measure on $e$ is
\begin{align*}
 \mu(e) &= 1 - (\text{current through $e$ when $1$ unit is sent from $e^-$ to $e^+$}) \\
&= 1 - \frac{r(e^-,e^+)}{R(e)}.
\end{align*}
\end{prop}

\begin{eg}
In \cref{fig:canonical} we show the canonical measure of each edge 
on a metric graph with unit edge-lengths.
The value $\mu(e) = \frac{7}{12}$ follows from the current shown in \cref{fig:current}
and \cref{prop:current}.
\begin{figure}[h]
\centering
\includegraphics[width=0.4\textwidth]{canonical_measure}
\caption{Canonical measure on edges of a unit-length metric graph.}
\label{fig:canonical}
\end{figure}
\end{eg}

On a metric graph with unit edge-lengths, the canonical measure has the following
combinatorial interpretation.

\begin{prop}[Classical \cite{P}]
Let $G$ be a metric graph with unit edge-lengths,
and let $\mu$ denote the canonical measure on $G$.
Then for each (unit-length) edge $e$,
\[ \mu(e) = \frac{\kappa (G \backslash e)}{\kappa(G)}\]
where $\kappa(G)$ denotes the number of spanning trees and 
$G \backslash e$ denotes the edge-deleted subgraph.
\end{prop}

\acknowledgements{
The author thanks David Speyer for introducing him to tropical geometry and guiding his interest in the field, and thanks Matt Baker for permission to use a modified version of his illustration of a Berkovich curve.}


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