%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The viewpoint of Hurwitz spaces}
\label{sec:NewHurwitz}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

This section is a digression on how to interpret the volume
recursion and the saddle connection Siegel-Veech constant
from the viewpoint of Hurwitz spaces. The results in this
section are not needed for proving any of the theorems stated in the
introduction. We will rather explain and motivate
\begin{itemize}
\item why the homologous count of saddle connections is more natural
  than the physical count from the viewpoint of intersection theory,
\item how to heuristically deduce the value of the saddle connection
Siegel-Veech constant in Theorem~\ref{intro:SVproduct} from an equidistribution
of cycles in Hurwitz tuples, and
\item why backbone graphs correspond to configurations.
\end{itemize}
\par
As usual $\mu = (m_1,\ldots,m_n)$ is a partition of $2g-2$ and the
ramification profile~$\Hmu$ consists of~$n$ cycles $\mu^{(i)}$ of
length $m_i +1$ unless stated otherwise. Let  $r(\mu) = 2g + n(\mu) - 1$
be the dimension of the (unprojectivized) stratum $\omoduli[g](\mu)$. 
\par
\begin{Thm} \label{thm:H2}
There exists a constant $M(\mu)$ such that the Hurwitz numbers~$N_d^\circ(\Hmu)$
for connected torus covers of profile~$\Hmu$ can be approximated as
\begin{flalign} 
&\phantom{\=} M(\mu)\,N_d^\circ(\Hmu) \label{eq:H2} \\
& \=  \sum_{\Gamma \in \BB^\star_{1,2}} \frac{d}{k!}\,h_{\PP^1}((m_1,m_2),\bfp)
\sum_{d_1+\cdots +d_k = d}
\prod_{i=1}^k p_i N^\circ_{d_i}(\Hmu_i,(p_i)) +  o(d^{r(\mu)-1})\,, \nonumber
\end{flalign}
where $\Hmu \setminus \{\mu^{(1)}, \mu^{(2)}\}
\= \Hmu_1 \sqcup \cdots \sqcup \Hmu_k$ is the
decomposition of the profile according to the leg assignment in~$\Gamma$
and where $\bfp = (p_1,\ldots p_k)$ is the unique 
twist compatible with~$\Gamma$.
\end{Thm}
\par
At the end of the section we will show by combining Theorem~\ref{intro:VolRec}
together with Theorem~\ref{thm:H2} the following result.
 \par
\begin{Prop}
\label{prop:M-value}
The constant $M(\mu)$ in Theorem~\ref{thm:H2} is $(m_1+1)(m_2+1)$.
In particular, $M(\mu)$ depends only on the first two zero orders of $\mu$. 
\end{Prop}
\par
Indeed an independent proof of Proposition~\ref{prop:M-value} would provide
an alternative proof of Theorem~\ref{intro:VolRec} (and hence
Theorem~\ref{intro:IntFormula}) that would  bypass the complicated
combinatorics in Sections~\ref{sec:D2rec} and~\ref{sec:D2VR}.
\par
The strategy to prove Theorem~\ref{thm:H2} consists of comparing the Hurwitz
number~$N_d^\circ(\Hmu)$, that is the fiber cardinality of the forgetful
map $f_T\colon H_d(\Hmu) \to \moduli[1,n]$ to the target curve with the
fiber cardinality of the extension of $f_T$ to the space of admissible
covers $\BH_d(\Hmu) = \BH_{d,g,1}(\Hmu)$ over degenerate targets of the
following type. 
\par
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Admissible torus covers}
\label{sec:NotHurwitz}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Let $\ES$ be the stable curve of genus one consisting of a
$\PP^1$-component carrying precisely the first two marked points and of
an elliptic curve~$E$ carrying the remaining marked points, joint
at a node~$q_E$. If $p\colon X \to \ES$ is an admissible cover, we denote by~$X_0$
and $X_{-1}$ the (possibly reducible) curves mapping to~$E$ and to $\PP^1$
respectively, both deprived of their unramified~$\PP^1$-components.
See Figure~\ref{fig:exadmcov} for examples of such admissible covers.
\par
The admissible covers of $\ES$ come in two types. One possibility is
that the first two branch points are in the same (hence the unique)
component of $X_{-1}$. The stable dual graph of the cover is thus
a graph $\Gamma \in \ABB^\star_{1,2}$ (recall the definition of ABB graphs
in Section~\ref{ssec:boundary})  and we denote by $N^{\circ}_d(\Pi,\ES,\Gamma)$
the number of such covers. The second possibility is that each of the two
branch points is on its own component~$X_{-1}^{(i)}$ for $i=1,2$. Consequently,
the map $p|_{X_{-1}^{(i)}}$ is a cyclic cover of degree~$(m_i+1)$. By 
contracting the components over $\PP^1$ we see that such covers
are in bijective correspondence (up to the automorphism group of size
$|\Aut(X_{-1}/\PP^1)|  = (m_1+1)(m_2+1)$)  with
covers of~$E$ with the profile $\Hmu_{(12)} = ((\mu^{(1)}, \mu^{(2)}),\mu^{(3)},
\ldots,\mu^{(n)})$, where the first two ramification points are piled over the same branch point. 
\par
\begin{Prop}\label{prop:admissible}
For $\Gamma \in \ABB^\star_{1,2}$ the bound $N^{\circ}_d(\Pi,\ES,\Gamma)  
\= O(d^{r(\mu)-2})$ holds as $d\to \infty$. The upper bound
is attained, i.e.\ there exists $C>0$ such that 
$N^{\circ}_d(\Pi,\ES,\Gamma)  \,\geq \,C d^{r(\mu)-2}$, 
if and only if $\Gamma \in \BB^\star_{1,2}$. Moreover in this case 
\be \label{eq:hur-adm}
N^{\circ}_d(\Pi,\ES,\Gamma) \=  \frac{1}{k!} h_{\PP^1}((m_1,m_2),\bfp)
\sum_{d=d_1+\cdots +d_k} \prod_{i=1}^k N^\circ_{d_i}(\Hmu_i,(p_i)) \,,
\ee
where $\Hmu_i$ and $\bfp$ are associated with~$\Gamma$ as in 
Theorem~\ref{thm:H2}. 
\end{Prop}
\par
\begin{proof} Recall from \cite{eo} or the proof of~\cite[Proposition~9.4]{cmz} that if $\Hmu$ is the profile for a cover $\pi$ with $\pi^* \omega \in
\omoduli[g,n](\mu)$, then there exist $C_1,C_2 \neq 0$ such that
$C_1\cdot d^{r(\mu)-1} \leq N^{\circ}_d(\Hmu) \leq C_2\cdot d^{r(\mu)-1}$
as $d \to \infty$.
 Suppose that $\Gamma$ has $k$ components
on level $0$, each of genus $g_i$ and with $n_i$ marked points or nodes.
Let $g_0$ be the genus of the component on level $-1$, and let
$b = h^1(\Gamma)$. Then 
$$b + \sum_{i=0}^k g_i = g \quad \mbox{and}\quad \sum_{i=1}^k n_i
\= n-2 + k + b\,.$$
The cover of the~$\PP^1$-component has finitely many choices independent of $d$. Over the
elliptic component of~$\ES$, the number of choices of covers has asymptotic
growth given by $B \sum_{d_1+\cdots+d_k = d} d_i^{2g_i -2 +n_i}$ 
for some constant~$B$ independent of $d$. This quantity is a polynomial
of degree 
$$ \Bigl( \sum_{i=1}^k (2g_i - 2+ n_i) \Bigr) + (k-1)
\= 2g - 2g_0 - b + n - 3\,. $$
We thus conclude that the total number of admissible covers
$N^0_d(\Pi,\ES,\Gamma)$ has asymptotic growth  given by a polynomial of
degree $r(\mu)-2-b - 2g_0 \leq r(\mu)-2$, with equality attained if and only if 
$b = g_0 = 0$, i.e. if and only if $\Gamma\in \BB^\star_{1,2}$.
\par
To justify equation~\eqref{eq:hur-adm} we refer to the computation of the
Hurwitz numbers in Proposition~\ref{prop:intpsiPP1}
and divide by $k!$
to account for our auxiliary labeling of the $k$ components of~$X_0$.
\end{proof}
\par
We remark that 
Proposition~\ref{prop:admissible} is not used in any other proofs in the paper, 
but it reveals the geometric reason for the homologous count of
saddle connections behind the recursions in Sections~\ref{sec:RelMZInt}
to~\ref{sec:D2VR}. The factor $h_{\PP^1}((m_1, m_2),\bfp)$ (possibly
with $1/k!$ if all branches are labeled) appears in the direct count
of admissible covers and in the count of configurations. There is no
extra factor~$k$ in~\eqref{eq:hur-adm}, which corresponds to our setting of 
the coefficient $m^{\hom}(\mathcal C) = 1$ (instead of $k$) in~\eqref{eq:SVviaboundary} 
for homologous count of saddle connections (instead of physical count).  
\par
\begin{proof}[Proof of Theorem~\ref{thm:H2}] We first show that
\ba \label{eq:H2app}
&\phantom{\=} N_d^\circ(\Hmu) -  N_d^\circ(\Hmu_{(12)}) \\
& \=  \sum_{\Gamma \in \BB^\star_{1,2}}  \frac{1}{k!} h_{\PP^1}((m_1, m_2),\bfp)
\sum_{d_1+\cdots +d_k = d}
\prod_{i=1}^k p_i N^\circ_{d_i}(\Hmu_i,(p_i)) +  O(d^{r(\mu)-2})\,. 
\ea
To see this, note that the ramification order of $f_T$ over $\ES$
at the branch through a cover~$\pi\colon X \to E $ is equal to the product of
ramification orders at the nodes of ~$X$.
This results in the factors $p_i$ inside the product of the right-hand
side and cancels the factor $1/|\Aut(X_{-1}/\PP^1)|$ when counting $\ES$-covers
instead of counting $N_d^\circ(\Hmu_{(12)})$.
\par
On the other hand, since the volume of the stratum can be approximated 
by counting covers of profile $\Hmu_{(12)}$ and since the generating function
of counting these covers is a quasimodular form, arguments as in
\cite[Proposition~9.4]{cmz} imply the existence of a constant
$M(\mu)$ such that
\ba\label{eq:hurwitz12}
 \frac{N_d^\circ(\Hmu_{(12)})}{N_d^{\circ}(\Hmu)} \=
\frac{d -  M(\mu)}{d} \,+ \,o (1/d )\,. 
\ea
The combination of equations~\eqref{eq:H2app} and~\eqref{eq:hurwitz12} thus implies the desired formula~\eqref{eq:H2}.
\end{proof}
\par
We now address the equidistribution heuristics for saddle connection
Siegel-Veech constants. Recall that $N^\circ(\Hmu)$ is the number
(weighted by $|\Aut(p)|$) of transitive Hurwitz tuples
$(\alpha,\beta,(\gamma_i)_{i=1}^n) \in S_d^{n+2}$
with $[\alpha,\beta] = \prod_{i=1}^n \gamma_i$ and $\gamma_i$ of type $\mu^{(i)}$.
\par
\begin{Prop} If the pairs $(\gamma_1,\gamma_2)$ appearing in the Hurwitz tuples
of profile $\Hmu$ equidistribute among pairs of $(m_i+1)$-cycles in $S_d^2$
as $d \to \infty$, then $M(\mu) = (m_1+1)(m_2+1)$.
\end{Prop}
\par
\begin{proof} If the non-trivial cycles in $\gamma_1$ and $\gamma_2$
have no letter in common, then taking $(\alpha,\beta,\gamma_1 \circ \gamma_2,
\gamma_3,\ldots,\gamma_n)$ is a Hurwitz tuple of profile $\Hmu_{(12)}$.
Assuming equidistribution and comparing to the total number of Hurwitz tuples, 
the number of Hurwitz tuples with
$\gamma_1$ and $\gamma_2$ having two letters in common is negligible,
while the ratio of those having one letter in common is $(m_1+1)(m_2+1)/d + o(1/d)$.
\end{proof}
\par
\begin{Ex} 
For the reader's convenience we illustrate the contributions to the
right-hand side of~\eqref{eq:H2app} for the stratum $\omoduli[2](1,1)$
explicitly in Figure~\ref{fig:exadmcov}.
\input{pic_adm_cover}
The picture on the left gives stable graphs in $\BB^\star_{1,2}$, while
the pictures in the middle and on the right give graphs in
$\ABB^\star_{1,2} \setminus \BB^\star_{1,2}$. The preimages
of~$E$ in the middle and on the right are unramified and thus again are elliptic
curves, while on the left the preimage of $E$ is a curve of genus two.
\par
The saddle connection Siegel-Veech counting in this case was carried out
in \cite{ems} in a similar way as summarized in Theorem~\ref{thm:H2},
despite that  only primitive torus covers were considered.
\par
\end{Ex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The principal boundary of Hurwitz spaces}
\label{sec:Pb}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We focus on saddle connections joining the first
two marked zeros and determine a full set of configurations and
the corresponding principal boundary of the Hurwitz spaces. 
We say that $\Gamma \in {\rm ABB}^\star_{1,2}$ is {\em realizable} in $\BH_d(\Hmu)$
if there is an admissible cover  $p\colon X \to \ES$ whose stable graph
is~$\Gamma$ and such that the vertices with $\ell(v)=0$ correspond
bijectively to the components of $X_0$. Recall also the definition of ribbon 
graphs associated to configurations in Section~\ref{ssec:configurations}.  
\par
\begin{Prop} \label{prop:ConfandGamma}
Associating with a configuration~$\cCC$ the boundary
curves of the ribbon graph~$R(\cCC)$ induces a map $\varphi\colon \cCC
\to \Gamma(\cCC)$ from a full set of saddle connection configurations
onto the subset of $\ABB^\star_{1,2}$  that is realizable in $\BH_d(\Hmu)$.
The image of~$\varphi$ is independent of~$d$ for~$d$ large enough. 
The fibers of~$\varphi$ are finite with cardinality bounded
independently of~$d$.
\par
Moreover if a graph~$\Gamma \in \BB^\star_{1,2}$ is realizable, then 
the configurations in $\varphi^{-1}(\Gamma)$ are in bijection with cyclic 
orderings of the components at level $0$.
\end{Prop}
\par
\begin{proof}  To define $\varphi$, we pinch the boundary curves 
of~$R(\cCC)$ to obtain a pointed nodal curve. The configuration $\cCC$ of 
saddle connections remains in one component of the curve that contains $z_1$ 
and $z_2$. We provide the dual graph of the curve with the level structure 
such that the component containing~$z_1$ and~$z_2$ is the unique one at 
level~$-1$ and all the other components are on level $0$.  
This way we thus obtain a graph $\varphi(\cCC) \in \ABB^\star_{1,2}$. 
We leave the straightforward verification of the other statements to 
the reader.
\end{proof}
\par
An application of the Riemann-Hurwitz formula shows that any configuration in
$\varphi^{-1}(\Gamma)$ has multiplicity $|E(\Gamma)|+2g(X_{-1})$.
In the special case $\Gamma \in \BB^\star_{1,2}$ (i.e.\ if $g(X_{-1}) = 0$
and $\Gamma$ is of compact type), the configuration consists of 
$k=|E(\Gamma)| $ pairwise homologous arcs. However, the cover on the 
right-hand side of Figure~\ref{fig:exadmcov} shows that graphs
in $\Gamma \in \ABB^\star_{1,2} \setminus \BB^\star_{1,2}$ also contribute.
It is not hard to give an example that the fiber cardinality of~$\varphi$
over a target graph with $g(X_{-1})>0$ can indeed be larger than one,
and we leave it to the reader since it is irrelevant to our applications.  
\par
\medskip
Finally we address that Theorem~\ref{thm:H2} and an a priori knowledge
that $M(\mu) = (m_1+1)(m_2+1)$ would give an alternative proof of Theorem~\ref{intro:VolRec}.
In terms of our volume normalization, \cite[Proposition~9.4]{cmz} says that  
\ba \label{eq:vol-hur}
\sum_{d=1}^D N_d^\circ(\Hmu) 
&\= \frac{v(\mu)}{2r\prod_{i=1}^n (m_i+1)} \,D^{r} + O(D^{r-1} \log D)
\ea
as $D\to \infty$, where $r = 2g + n -1$. To sum the right-hand side of~\eqref{eq:H2} we let
$S_D(\Pi_i) = \sum_{d=1}^D N_{d}^\circ(\Pi_i)$. The Euler integral
$\int_{0}^1 t^{a-1}(1-t)^{b-1}dt = \frac{(a-1)!(b-1)!}{(a+b-1)!}$ used recursively
implies the following result.  
\par
\begin{Lemma} \label{lm:eulerintegral}
Suppose that $S_D(\Pi_i) = v_i D^{r_i} + O(D^{r_i-1} \log D)$
as $D \to \infty$ and that there exists a constant $C$ depending
on $\Pi_i$ only such that $N_{d}^\circ(\Pi_i) < C d^{r_i-1}$
for $i=1,\ldots,k$. Then 
\be
\lim_{D \to \infty}  \sum_{d_1+\cdots + d_k\leq D}
\frac{(d_1+\cdots + d_k)\,N_{d_1}^\circ(\Pi_1)\cdots N_{d_k}^\circ(\Pi_k)}
{D^{r_1 + \cdots +r_k+1}}
\= \frac{\prod_{i=1}^k (r_i! v_k)\sum_{i=1}^k r_i}{(r_{1} +\cdots + r_k+1)!}\,.
\ee
\end{Lemma}
\par
\begin{proof}[Alternative proof of Theorem~\ref{thm:H2} (assuming 
$M(\mu) = (m_1+1)(m_2+1)$)]
With the ab\-bre\-viation
$r_i = 2g_i + n(\mu_i)$ we obtain from~\eqref{eq:vol-hur} that  
$$ \sum_{d_i=1}^{D} N_{d_i}^{\circ} (\Hmu_i, (p_i)) \=
\frac{ v(\mu_i, p_i-1)}{2r_ip_i \prod_{m_i\in \mu_i} (m_i+1)} D^{r_i}
+ O(D^{r_i-1} \log D)\,. $$
Since $r_1+\cdots + r_k = 2g+n-2$, Lemma~\ref{lm:eulerintegral} implies that 
\ba \label{eq:5.2RHS}
& \phantom{\=}\, \sum_{d_1+\cdots +d_n\leq D} (d_1+\cdots +d_n) \prod_{i=1}^k
p_i\,N^\circ_{d_i}(\Hmu_i,(p_i)) \\ 
& \=  D^{2g+n-1} \frac{ \prod_{i=1}^k  (2g_i+n(\mu_i)-1)! v(\mu_i, p_i-1)}
{2^k(2g+n-1)(2g+n-3)!\prod_{i=3}^n (m_i+1)}\,.
\ea
Summing over all backbone graphs and taking the limit after dividing
by $D^{2g+n-1}$ thus implies the desired formula~\eqref{eq:volintro}.
\end{proof}
\par
\begin{proof}[Proof of Proposition~\ref{prop:M-value}]
  Conversely, the above argument shows that the mere knowledge of
  Theorem~\ref{thm:H2}
gives the recursion in Theorem~\ref{intro:VolRec} with $M(\mu)$ on the
left-hand side that replaces $(m_1+1)(m_2+1)$, and hence the two theorems taken
together thus determine the value  $M(\mu) = (m_1+1)(m_2+1)$ as claimed
in Proposition~\ref{prop:M-value}. 
\end{proof}

