
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Large genus asymptotics}
\label{sec:asy}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section we study the large genus asymptotics of
Masur-Veech volumes and area Siegel-Veech constants and prove the
conjectures of Eskin and Zorich in \cite{EZVol} by using our previous results.

%%%%%%%%%%%%%%%%%%%%%
\subsection{Volume asymptotics}
\label{sec:asymin}
%%%%%%%%%%%%%%%%%%%%%
We recall from \cite[Theorems~12.1 and~19.1]{cmz} and
\cite[Theorem~1.9]{SauvagetMinimal} that the asymptotic expansions of $v(2g-2)$
and $v(1,\ldots,1)$ can be computed using the mechanisms of very rapidly
divergent series (\cite[Appendix]{cmz} as 
\begin{eqnarray*} v(1^{2g-2})  \;&\sim&\; {4} \,
\Bigl(1\,-\, \frac{\pi^2}{24g} \,-\, 
\frac{60\pi^2 - \pi^4}{1152g^2}\,-\, \cdots\Bigr)\,,  \\
v(2g-2)  \;&\sim&\; {4} \,
\Bigl(1\,-\, \frac{\pi^2}{12g} \,-\, 
\frac{24\pi^2 - \pi^4}{288g^2} \,-\, \cdots\Bigr)\,.
\end{eqnarray*} 
\par
Let $\mu=(m_1,\ldots,m_n)$ be a partition of $2g-2$ into $n$ positive integers
with $n\geq 2$. We write $\mu'=(m_1+m_2, m_3,\ldots,m_n)$ and
$\mu''=(m_1+m_2-2, m_3,\ldots,m_n)$. We use Theorem~\ref{intro:VolRec} and the
two obvious backbone graphs, the one with a single top level component of
genus~$g$ (i.e.\ $k=1$) and the one with two top level components (i.e.\ $k=2$)
of genus~$1$ and~$g-1$ respectively, to deduce the inequality 
\be \label{eq:vmu2terms}
v(\mu)\geq v(\mu') + \frac{\pi^2 (2g-5+n)!}{6(2g-3+n)!} v(\mu'')\,,
\ee
where we use $h_{\proj^1}((m_1,m_2), (m_1+m_2+2))=h_{\proj^1}((m_1,m_2), (m_1+m_2,1))=1$
for $m_1,m_2 >0$ and $v(0)=\pi^2/6$.
In particular this inequality implies that $v(\mu)\geq v(\mu')$ and thus 
$$
v(2g-2) \,\leq\, v(\mu) \,\leq\, v(1,\ldots,1)
$$
for all $\mu$.  Consequently, there exists a constant $C>0$ such that for all $\mu$ we have the inequality $|v(\mu)-4|<C/g$. Now we introduce the notation
$$\widehat{v}(\mu):= v(\mu)-4+\frac{2\pi^2}{3(2g-3+n)}\,.$$
Then the inequality~\eqref{eq:vmu2terms} implies that 
\begin{eqnarray*}
\widehat{v}(\mu)- \widehat{v}(\mu')&\geq& - \frac{2\pi^2}{3(2g-3+n)(2g-4+n)}+ \frac{\pi^2 (2g-5+n)!}{6(2g-3+n)!} v(\mu'')\\ 
&\geq &  - \frac{C \pi^2 }{6g(2g-3+n)(2g-4+n)}\,.
\end{eqnarray*}
In particular for all $\mu$ we have 
$$
\widehat{v}(2g-2)- \frac{C \pi^2 (n-1)}{6g(2g-1)^2}  \leq \widehat{v}(\mu)\leq \widehat{v}(1,\ldots,1) +  \frac{C \pi^2 (2g-2-n) }{6g  (2g-1)^2}\,.
$$
Since $n$ is bounded by $2g-2$, there exists a constant $C'$ such that $|\widehat{v}(\mu)|\leq C'/g^2$ for all $\mu$. Thus the first part of Theorem~\ref{cor:EZconj} holds. 

%%%%%%%%%%%%%%%
\subsection{Asymptotics of Siegel-Veech constants}
%%%%%%%%%%%%%%%%
We apply the same strategy to control the asymptotic behavior of area
Siegel-Veech constants. We denote by 
$$
 \widetilde{d}(\mu)\,:=\, c_{\rm area}(\mu)v(\mu)
 \ = \frac{1}{4\pi^2} \frac{2(2\pi){^2g}}{(2g-3+n)!}
 \left( \prod_{i=1}^n (m_i+1) \right) \cdot  |d(\mu)| 
$$
where $d(\mu)$ is defined in~\eqref{eq:defd} and where the second equality
stems from~\eqref{eq:IntF2reform}. For $\mu = (m_1,\ldots, m_n)$ with $n \geq 2$, we write 
$\mu'=(m_1+m_2, m_3,\ldots,m_n)$ and $\mu''=(m_1+m_2-2, m_3,\ldots,m_n)$ as before. Then from
Proposition~\ref{prop:indintSV} we have $\widetilde{d}(0)=1$ and we obtain the inequality
$$
\widetilde{d}(\mu) \,\geq\, \widetilde{d}(\mu') \+  \frac{\pi^2 (2g-5+n)!}{6(2g-3+n)!}
\widetilde{d}(\mu'') \+ \frac{(2g-5+n)!}{2(2g-3+n)!} v(\mu'')\,.
$$
In particular this inequality implies that $\widetilde{d}(\mu) \geq
\widetilde{d}(\mu')$. Moreover, we know the asymptotic expansions
\bes
\widetilde{d}(2g-2) \,\sim\, 2 - \frac{3+\pi^2}{6g} +\cdots \quad
\text{and} \quad 
\widetilde{d}(1,\ldots,1)\,\sim\, 2 - \frac{3+\pi^2}{12g}+\cdots \, 
\ees
from\cite[Theorem~19.4]{cmz} and~\cite[Theorem~1.9]{SauvagetMinimal}.
Consequently, there exists a constant $C$ such that $|\widetilde{d}(\mu)-2|< C/g$ for all $\mu$. Then by the same argument as above we can show that there exists a constant $C'$ such that
$$
\Big|\widetilde{d}(\mu)-2+ \frac{3+\pi^2}{3(2g-3+n)} \Big| \,< \, C'/g^2\,.
$$
Therefore, using the fact that $c_{\rm area}(\mu)=\widetilde{d}(\mu)/v(\mu)$ we thus deduce the second part of Theorem~\ref{cor:EZconj}.

%%%%%%%%%%%%%%%
\subsection{Spin asymptotics}
%%%%%%%%%%%%%%%%

The goal here is to prove that the volumes of the odd and even spin
components are asymptotically equal. This is the content of
Theorem~\ref{thm:EZconj3} in the introduction that refines the conjecture
of Eskin and Zorich (\cite[Conjecture~2]{EZVol}).
Recall that we defined in Section~\ref{sec:spinD2}
\be \label{eq:PZredef}
\bfP_Z(u) \= \exp\Bigl(\sum_{ j \geq 1} \Bigl(\frac12\Bigr)^{j/2}
\frac{\zeta(-j)}{2} u^{j+1}\Bigr)\,.
\ee
\par
\begin{Prop} \label{prop:spinminasy}
The difference $v(2g-2)^{\Delta} = v(2g-2)^{\rm even}
  -v(2g-2)^{\rm odd}$ can be computed as the coefficient extraction  
\be \label{eq:deltamin}
v(2g-2)^{\Delta}  \=  \frac{2 (2\pi i)^{2g}}{(2g-1)!}
\bigl[u^{2g-1}\bigr] \frac{1}{(u/\bfP_Z)^{-1}}\,.
\ee
Moreover, it has the asymptotics
\be \label{eq:deltaAsyMin}
v(2g-2)^{\Delta} \;\sim\; \Bigl(\frac{-1}{2}\Bigr)^{g-2} \,
\Bigl(1\,+\, \frac{2\pi^2}{3g} \,+\, 
\frac{12\pi^2 + \pi^4}{18g^2} \,+\, \cdots\Bigr)\,
\ee
as $g \to \infty$.
\end{Prop}
\par
For the reader's convenience we give a table for low genus values: 
\begin{figure}[H]
$$ \begin{array}{|c|c|c|c|c|c|}
\hline
g & 1 & 2 & 3 & 4 & 5 \\
\hline &&&&& \\ [-\halfbls]
v(2g-2)^{\Delta}  & \,\,\frac{-1}{3} \,\,&  \frac{1}{40}
& \frac{-143}{108864}  & \frac{15697}{279936000} & \frac{-2561}{1103872000}  \\
      [-\halfbls] &&&&&\\
\hline
\end{array}
$$
\end{figure}
\par

\par
\begin{proof}[Proof of Proposition~\ref{prop:spinminasy}]
The first statement is a reformulation of a special case of
Corollary~\ref{cor:eodiff}.
\par
The power series $\bfP_Z(u)$ is a very rapidly divergent series, just as $P_B(u)$ is,
since the coefficients $\ell!b_\ell$ and 
$\zeta(-\ell)/2 =  \ell!b_\ell \cdot  \sqrt{2}(2^{\ell}-2^{-\ell})$
differ by a factor that grows only geometrically. The asymptotic statement
thus follows from the method of very rapidly divergent series. 
\end{proof}
\par
\begin{proof}[Proof of Theorem~\ref{thm:EZconj3}]
Proposition~\ref{prop:spinminasy} together with Theorem~\ref{cor:EZconj}
implies that there exists a constant $C'$ such that for all $g\geq 1$
$$
|v(2g-2)^{\rm odd}-v(2g-2)^{\rm even}|\, \leq \, C'/g\,.
$$
Repeated application of Theorem~\ref{cor:EZconj} implies that 
$v(\mu)^\bullet\geq v(2g-2)^{\bullet}$ for all $\mu$ with $|\mu| = 2g-2$.
Theorem~\ref{cor:EZconj} moreover implies that there exists a
constant $C''$ such that for all $\mu$ with $|\mu| = 2g-2$ the  inequality 
$$
|v(\mu)-4| \, \leq\,  C''/g
$$
holds. Thus for all $\mu$ with $|\mu| = 2g-2$ we have 
\begin{eqnarray*}
v(2g-2)^{\rm odd}\leq v(\mu)^{\rm odd} &=& v(\mu)-v(\mu)^{\rm even}
\,\,\, \leq \,\,\, v(\mu)- v(2g-2)^{\rm even}\\ 
& \leq & 2 + (C'+3C'')/g\,.
\end{eqnarray*} 
It follows that $|v(\mu)^{\rm odd}-2|\leq (C'+3C'')/g$ and the same holds for
$v(\mu)^{\rm even}$. This implies the claim for the volume comparison. 
\par
For the comparison of area Siegel-Veech constants, since we have shown that the two spin components have the same volume asymptotic, combining with \cite[Propositions 4.3 and 4.4]{Agg2} it suffices to compare the odd and even one-cylinder contributions in the part of the principal boundary where the corresponding configurations have either a figure-eight boundary or a pair of holes boundary. Using again the same volume asymptotics of the spin components in genus $g-1$
and combining with \cite[Lemmas 14.2 and 14.4]{emz}, we see that the odd and even contributions are comparable for those two parts of the principal boundary,
thus implying the claim.
\end{proof}

