Uniform Lower Bound for Intersection Numbers of $\psi$-Classes

Uniform Lower Bound for Intersection Numbers
of $\boldsymbol{\psi}$-Classes††This paper is a contribution to the Special
Issue on Algebra, Topology, and Dynamics in Interaction in honor of Dmitry
Fuchs. The full collection is available at
https://www.emis.de/journals/SIGMA/Fuchs.html

Vincent DELECROIX ${}^{{\dagger}^{1}}$, Élise GOUJARD ${}^{{\dagger}^{2}}$,
Peter ZOGRAF ${}^{{\dagger}^{3}{\dagger}^{4}}$ and Anton ZORICH
${}^{{\dagger}^{5}{\dagger}^{6}}$

V. Delecroix, É. Goujard, P. Zograf and A. Zorich

${}^{{\dagger}^{1}}$ LaBRI, Domaine universitaire, 351 cours de la Libération,
33405 Talence, France<EMAIL_ADDRESS>
${}^{{\dagger}^{2}}$ Institut de Mathématiques de Bordeaux, Université de
Bordeaux,
${}^{{\dagger}^{2}}$ 351 cours de la Libération, 33405 Talence, France
<EMAIL_ADDRESS>
${}^{{\dagger}^{3}}$ Steklov Mathematical Institute, Fontanka 27, St.
Petersburg 191023, Russia<EMAIL_ADDRESS>
${}^{{\dagger}^{4}}$ Chebyshev Laboratory, St. Petersburg State University,
${}^{{\dagger}^{4}}$ 14th Line V.O. 29B, St. Petersburg, 199178, Russia

${}^{{\dagger}^{5}}$ Center for Advanced Studies, Skoltech, Russia
<EMAIL_ADDRESS>
${}^{{\dagger}^{6}}$ Institut de Mathématiques de Jussieu – Paris Rive Gauche,
Bâtiment Sophie Germain,
${}^{{\dagger}^{6}}$ Case 7012, 8 Place Aurélie Nemours, 75205 PARIS Cedex 13,
France

Received April 09, 2020, in final form August 21, 2020; Published online
August 26, 2020

We approximate intersection numbers
$\big{\langle}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\big{\rangle}_{g,n}$ on
Deligne–Mumford’s moduli space $\overline{\mathcal{M}}_{g,n}$ of genus $g$
stable complex curves with $n$ marked points by certain closed-form
expressions in $d_{1},\dots,d_{n}$. Conjecturally, these approximations become
asymptotically exact uniformly in $d_{i}$ when $g\to\infty$ and $n$ remains
bounded or grows slowly. In this note we prove a lower bound for the
intersection numbers in terms of the above-mentioned approximating expressions
multiplied by an explicit factor $\lambda(g,n)$, which tends to $1$ when
$g\to\infty$ and $d_{1}+\dots+d_{n-2}=o(g)$.

intersection numbers; $\psi$-classes; Witten–Kontsevich correlators; moduli
space of curves; large genus asymptotics

14C17; 14H70

Dedicated to D.B. Fuchs
on the occasion of his 80th birthday

##  1 Introduction

Let $\overline{\mathcal{M}}_{g,n}$ be the Deligne–Mumford moduli space of
genus $g$ complex stable algebraic curves (possibly with nodes), with $n>0$
distinct labeled marked points. Consider the tautological line bundles
$\mathcal{L}_{i}\to\overline{\mathcal{M}}_{g,n}$, $i=1,\dots,n$, defined
fiberwise by $\mathcal{L}_{i}|_{C,x_{1},\dots,x_{n}}\cong T^{*}_{x_{i}}C$,
where $C$ is a genus $g$ curve with marked points $x_{1},\dots,x_{n}$ (the
definition makes sense since the marked points are not allowed to coincide
with the nodes).

In the late 1980-ies E. Witten [21] introduced a theory of two-dimensional
topological gravity, where the classes $\psi_{i}=c_{1}(\mathcal{L}_{i})$,
$i=1,\dots,n,$ played the role of observables, and the intersection numbers

$\displaystyle\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n}=\big{\langle}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\big{\rangle}_{g,n}=\int_{\overline{\mathcal{M}}_{g,n}}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}},$

where $d_{1}+\dots+d_{n}=3g-3+n$, represented correlators of the theory.
Following a common convention, we will omit $g$ and $n$, or just $n$ when they
are clear from the context. On the basis of a low genus evidence and
considerations from physics, E. Witten conjectured that the generating
function

$\displaystyle F(t_{0},t_{1},\dots)=\sum_{\begin{subarray}{c}g\geq 0\\\ n\geq
1\end{subarray}}\frac{1}{n!}\sum_{\begin{subarray}{c}d_{1}+\cdots+d_{n}\\\
=3g-3+n\end{subarray}}\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle
t_{d_{1}}\cdots t_{d_{n}}$

(total free energy of Witten’s two-dimensional topological gravity) satisfies
the KdV (Korteveg–de Vries) hierarchy. An equivalent formulation of Witten’s
conjecture is that the partition function $e^{F}$ is a highest weight vector
of a Virasoro algebra representation, see papers [9, 10] of R. Dijkgraaf and
of R. Dijkgraaf, H. Verlinde, and E. Verlinde. Witten’s conjecture was first
proven by M. Kontsevich [16], and then several other proofs appeared,
including the proofs due to A. Okounkov and R. Pandharipande [20], to M.
Kazarian and S. Lando [15], to M. Mirzakhani [18, 19]. Note that the two-
dimensional topological gravity may be interpreted as the simplest instance of
the Gromov–Witten theory, where the tagret space is a point.

Computability of the intersection numbers
$\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle$ is an important open problem.
Besides the cases of small $g$ or $n$ no general explicit formulas for the
numbers $\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n}$ are known. In
applications it is often sufficient to know large genus behavior of these
intersection numbers, and here either not much was known in general before the
current paper and before even more recent beautiful work by A. Aggarwal [3].

In this note we prove a simple lower bound for the numbers that appears to be
asymptotically exact as $g\to\infty$. To begin with, we recall some basic
facts about the numbers $\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle$
(Witten’s correlators). They are uniquely defined by the initial data

$\big{\langle}\tau_{0}^{3}\big{\rangle}=1,\qquad\langle\tau_{1}\rangle=\frac{1}{24}$

via the recursive relations known as Virasoro constraints that we present
below.

Virasoro constraints (in Dijkgraaf–Verlinde–Verlinde form [9, 10]):

$\displaystyle\langle\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}=\frac{1}{(2k+3)!!}\Bigg{[}\sum_{j=1}^{n}\frac{(2k+2d_{j}+1)!!}{(2d_{j}-1)!!}\langle\tau_{d_{1}}\cdots\tau_{d_{j}+k}\cdots\tau_{d_{n}}\rangle_{g}$
$\displaystyle\hphantom{\langle\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}=}{}+\frac{1}{2}\sum_{\begin{subarray}{c}r+s=k-1\\\
r,s\geq
0\end{subarray}}(2r+1)!!(2s+1)!!\langle\tau_{r}\tau_{s}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g-1}$
$\displaystyle\hphantom{\langle\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}=}{}+\frac{1}{2}\sum_{\begin{subarray}{c}r+s=k-1\\\
r,s\geq 0\end{subarray}}(2r+1)!!(2s+1)!!$
$\displaystyle\hphantom{\langle\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}=}{}\times\sum_{\\{1,\dots,n\\}=I\coprod
J}\bigg{\langle}\tau_{r}\prod_{i\in
I}\tau_{d_{i}}\bigg{\rangle}_{g^{\prime}}\bigg{\langle}\tau_{s}\prod_{i\in
J}\tau_{d_{i}}\bigg{\rangle}_{g-g^{\prime}}\Bigg{]}.$ (1)

For $k=-1$ and $k=0$ the above relations have particularly simple form.

String equation ($\boldsymbol{k=-1}$):

$\displaystyle\langle\tau_{0}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n+1}=\langle\tau_{d_{1}-1}\cdots\tau_{d_{n}}\rangle_{g,n}+\dots+\langle\tau_{d_{1}}\cdots\tau_{d_{n}-1}\rangle_{g,n}.$
(2)

Dilaton equation ($\boldsymbol{k=0}$):

$\displaystyle\langle\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n+1}=(2g-2+n)\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n}.$

For any partition $\boldsymbol{d}$ of $3g-3+n$ into a sum of $n$ nonnegative
integers define $\varepsilon(\boldsymbol{d})$ through the following equation:

$\displaystyle\big{\langle}\psi_{1}^{d_{1}}\cdots\psi_{n}^{d_{n}}\big{\rangle}_{g,n}=\frac{(6g-5+2n)!!}{(2d_{1}+1)!!\cdots(2d_{n}+1)!!}\cdot\frac{1}{g!\cdot
24^{g}}\cdot\big{(}1+\varepsilon(\boldsymbol{d})\big{)}.$

We denote by $\Pi(m,n)$ the set of ordered partitions of an integer $m$ into a
sum of $n$ nonnegative integers.

######  Main Conjecture ([7]).

For any strictly positive constant $C$

$\lim_{g\to+\infty}\max_{1\leq n\leq
C\log(g)}\max_{\boldsymbol{d}\in\Pi(3g-3+n,n)}|\varepsilon(\boldsymbol{d})|=0.$

Corollary 6 below makes the first step towards a proof of the Main Conjecture.
It establishes an efficient uniform lower bound for
$\varepsilon(\boldsymbol{d})$ for those partitions $\boldsymbol{d}$ for which
the sum of the first $n-2$ entries is small with respect to the sum of the
remaining two entries.

######  Remark 1.

It follows from the definition of $\varepsilon(\boldsymbol{d})$ that
$\varepsilon(\boldsymbol{d})$ does not change under any permutation of the
entries of $\boldsymbol{d}$.

######  Remark 2.

It is plausible, that much stronger statement might be true, where the bound
$n<C\log(g)$ is replaced by the bound $n<g^{\alpha}$ with any fixed $\alpha$
satisfying $\alpha<\frac{1}{2}$. The reason why one cannot go beyond the bound
$n<\sqrt{g}$ is explained at the very end of Section 1.

######  Remark 3 (added in proofs).

The Main Conjecture was proved in a very strong form in the recent paper [3]
of A. Aggarwal.

Motivation. Certain universality phenomena in flat and hyperbolic geometry and
in dynamics of surfaces manifest themselves in large genera. The large genus
asymptotics of the Masur–Veech volumes of strata in moduli spaces of Abelian
differentials conjectured in [12] was successfully proved by independent
methods in [2] and in [6]. However, the analogous conjectures stated in [4] on
the large genus asymptotics of the Masur–Veech volumes of strata in moduli
spaces of quadratic differentials are open.

There are several approaches to evaluation of Masur–Veech volumes of such
strata. The original approach of A. Eskin and A. Okounkov discovered in [11]
uses characters of the symmetric group. Using modern computers for exact
computations based on this approach, É. Goujard evaluated in [13] the volumes
of all strata up to dimension 11. Currently it is not known how to obtain
volume asymptotics based on this approach.

The recent paper [5] of D. Chen, M. Möller and A. Sauvaget expressed the
Masur–Veech volume of the principal stratum through certain very special
linear Hodge integrals. The papers [14] of M. Kazarian and [22] of D. Yang, D.
Zagier and Y. Zhang provide extremely efficient recursive formulae for these
Hodge integrals, which allow to compute exact values of the volumes of the
principal strata up to genus 250 and more. However, currently it is not known
how to prove large genus asymptotic formulae for the Masur–Veech volume of the
principal stratum developing this approach.

One more approach to evaluation of Masur–Veech volumes is elaborated in our
paper [7] where we express the Masur–Veech volume of the principal stratum of
meromorphic quadratic differentials with at most simple poles through
intersection numbers of $\psi$-classes and suggest conjectures aimed to prove
large genus asymptotics of these volumes. This conjectural scheme involves the
Main Conjecture stated above as one of the key ingredients.

######  Remark 4 (added in proofs).

The asymptotic formula for the Masur–Veech volume conjectured in [7] was
recently proved in [3] by A. Aggarwal. Together with the Main Conjecture
proved by A. Aggarwal in the same paper, this allowed us to provide in [8] a
detailed description of the asymptotic geometry of random square-tiled
surfaces and of random simple closed multicurves on surfaces of large genus.

###  1.1 State of the art

Currently we have the following evidence towards the Main Conjecture. Direct
computation shows that $\varepsilon(0,0,0)=0$. It is known [21] that

$\langle\tau_{3g-2}\rangle_{g,1}=\frac{1}{24^{g}\cdot g!},$

so for all $1$-correlators we have

$\displaystyle\varepsilon(3g-2)=0.$ (3)

Applying the string equation recursively we get

$\displaystyle\big{\langle}\tau_{0}^{n-1}\tau_{3g-3+n}\big{\rangle}_{g,n}=\frac{1}{24^{g}\cdot
g!},$ (4)

so for all partitions with at most one nonzero entry we have

$\displaystyle\varepsilon\big{(}0^{n-1},3g-3+n\big{)}=0.$ (5)

For $2$-correlators the Main Conjecture is valid. Namely, by Remark 1, we have
$\varepsilon(d_{1},d_{2})=\varepsilon(d_{2},d_{1})$ for any
$(d_{1},d_{2})\in\Pi(3g-1,2)$. Thus, we may assume that $d_{1}<d_{2}$. We have
already seen that $\varepsilon(0,3g-1)=0$. For the remaining $2$-partitions we
have the following bounds:

######  Theorem ([7]).

For all $g\in\mathbb{N}$ and for all integer $k$ satisfying $2\leq
k\leq\frac{3g-1}{2}$ the following bounds are valid:

$\displaystyle-\frac{2}{6g-1}=\varepsilon(1,3g-2)<\varepsilon(k,3g-1-k)<0=\varepsilon(0,3g-1).$
(6)

We performed a detailed analysis of $\varepsilon(k,3g-1-k)$ in [7] based on
[23]. In particular, for large $g$ the error term $\varepsilon(k,3g-1-k)$
rapidly tends to $0$ when $k$ approaches $\frac{3g-1}{2}$, so the statement of
the above theorem can be seriously strengthened, if needed.

It is easy to compute $\varepsilon\big{(}\boldsymbol{d}\big{)}$ explicitly for
those partitions where all but one entries $d_{1},\dots,d_{n-1}$ are equal to
$0$ or $1$. Namely, we first apply recursively the dilaton equation
eliminating all those entries of the partition, which are equal to $1$, and
then apply (4). In particular,

$\big{\langle}\tau_{1}^{n-1}\tau_{3g-2}\big{\rangle}_{g,n}=(2g-3+n)(2g-4+n)\cdots(2g-1)\cdot\frac{1}{24^{g}\cdot
g!},$

so

$\displaystyle
1+\varepsilon((1,1,\dots,1,3g-2))=1+\varepsilon\big{(}\big{(}1^{n-1},3g-2\big{)}\big{)}$
$\displaystyle\hphantom{1+\varepsilon((1,1,\dots,1,3g-2))}{}=(2g-3+n)(2g-4+n)\cdots(2g-1)\cdot\frac{3^{n-1}\cdot(6g-3)!!}{(6g-5+2n)!!}$
$\displaystyle\hphantom{1+\varepsilon((1,1,\dots,1,3g-2))}{}=\frac{6g-3+3(n-2)}{6g-1+2(n-2)}\cdot\frac{6g-3+3(n-3)}{6g-1+2(n-3)}\cdots\frac{6g-3}{6g-1}.$

This implies that for any constant $\alpha$ satisfying $0<\alpha<\frac{1}{2}$
(respectively $\frac{1}{2}<\alpha$) we have

$\displaystyle\lim_{g\to+\infty}\max_{1\leq n\leq
g^{\alpha}}\big{|}\varepsilon\big{(}1^{n-1},3g-2\big{)}\big{|}=0,\quad\text{when
}0<\alpha<\frac{1}{2},$ $\displaystyle\lim_{g\to+\infty}\inf_{n\geq
g^{\alpha}}\varepsilon\big{(}1^{n-1},3g-2\big{)}=+\infty,\quad\text{when
}\frac{1}{2}<\alpha,$

which explains why the restriction $\alpha<\frac{1}{2}$ in Remark 2 cannot be
loosened.

##  2 Uniform lower bound

Given a real number $L$ and integers $g\geq 1$ and $n\geq 3$, denote by
$\Pi_{L}(3g-3+n,n)$ the following subset of ordered partitions:

$\Pi_{L}(3g-3+n,n)=\big{\\{}\boldsymbol{d}\in\Pi(3g-3+n,n)\,\big{|}\,d_{1}+\dots+d_{n-2}\leq
L\big{\\}}.$

For any nonnegative $L$ and any integer $g\geq 1$ we define
$\Pi_{L}(3g-2,1)=\Pi(3g-2,1)$ and $\Pi_{L}(3g-1,2)=\Pi(3g-1,2)$.

Define the following function of integer arguments $g$, $L$, satisfying
$g>L\geq 0$:

$\displaystyle\lambda(g,L)=\left(\prod_{i=0}^{L-1}\left(1-\frac{1}{6(g-i)+1}\right)\right)\cdot\left(1-\frac{2}{6(g-L)-1}\right),$
(7)

where, by convention,

$\displaystyle\lambda(g,0)=\left(1-\frac{2}{6g-1}\right).$ (8)

It follows from the definition of $\lambda(g,L)$ that $0<\lambda(g,L)<1$ for
any $g>L\geq 0$.

######  Theorem 5.

Let $g$, $L$ be nonnegative integers such that $g>L$. For any partition
$\boldsymbol{d}\in\Pi_{L}(3g-3+n,n)$ one has

$\displaystyle\varepsilon(\boldsymbol{d})\geq\lambda(g,L)-1.$ (9)

######  Corollary 6.

Let $L(g)$, where $g=1,2,\dots,$ be any sequence of nonnegative integers such
that $L(g)=o(g)$ as $g\to+\infty$. One has

$\displaystyle\lim_{g\to+\infty}\inf_{n\geq
1}\min_{\boldsymbol{d}\in\Pi_{L(g)}(3g-3+n,n)}\varepsilon(\boldsymbol{d})=0.$
(10)

###### Proof of Corollary 6.

Definition (7) of $\lambda(g,L)$ implies that for any sequence $L(g)$ of
nonnegative integers satisfying $L(g)=o(g)$ as $g\to+\infty$ one has

$\lim_{g\to+\infty}\lambda(g,L(g))=1.$

Now (10) follows from combination of (9) and (3). ∎

######  Remark 7.

Proposition 3.2 in [17] claims that for any triple $(n,K,M)$ of positive
integers one has

$\lim_{g\to+\infty}\max_{\boldsymbol{d}\in\Pi_{K}(3g-3+n,n)}|\varepsilon(\boldsymbol{d})|=0$

under the additional requirement that $d_{n-1}\leq M$.

We start by proving three Lemmas (corresponding to the string and the dilaton
equations, and to Virasoro constraints). It would be useful to introduce the
following notation. Given $\boldsymbol{d}\in\Pi(3g-3+n,n)$ let

$\displaystyle\lfloor\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n}=\frac{(6g-5+2n)!!}{(2d_{1}+1)!!\cdots(2d_{n}+1)!!}\cdot\frac{1}{g!\cdot
24^{g}}.$ (11)

By definition of $\varepsilon(\boldsymbol{d})$ we have

$\displaystyle\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n}=\lfloor\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n}\cdot\big{(}1+\varepsilon(\boldsymbol{d})\big{)}.$
(12)

From now on we suppose that $g\geq 1$.

######  Lemma 8.

Let $\boldsymbol{d}\in\Pi(3g-2+n,n-k)$ such that $d_{j}>0$ for
$j=1,\dots,n-k$. We assume that $k\geq 0$ and $n-k>0$. Define $\delta_{\rm
string}(0^{k+1},\boldsymbol{d})$ by equation

$\displaystyle\big{\lfloor}\tau_{0}^{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-k}}\big{\rfloor}_{g,n+1}\cdot\big{(}1+\delta_{\rm
string}\big{(}0^{k+1},\boldsymbol{d}\big{)}\big{)}$
$\displaystyle\qquad{}=\big{\lfloor}\tau_{0}^{k}\tau_{d_{1}-1}\cdots\tau_{d_{n-k}}\big{\rfloor}_{g,n}+\dots+\big{\lfloor}\tau_{0}^{k}\tau_{d_{1}}\cdots\tau_{d_{n-k}-1}\big{\rfloor}_{g,n}.$
(13)

Then

$\delta_{\rm
string}\big{(}0^{k+1},\boldsymbol{d}\big{)}=\frac{n-k-1}{6g-3+2n}.$

In particular, for any $\boldsymbol{d}$ as above and for any $k\geq 0$ we have

$\displaystyle\delta_{\rm string}\big{(}0^{k+1},\boldsymbol{d}\big{)}\geq 0.$
(14)

###### Proof.

Dividing both sides of equation (13) by
$\big{\lfloor}\tau_{0}^{k+1}\tau_{d_{1}}\dots\tau_{d_{n-k}}\big{\rfloor}_{g,n+1}$
and applying definition (11) to all terms involved in the right-hand side of
the resulting equation we get

$\displaystyle 1+\delta_{\rm
string}\big{(}0^{k+1},\boldsymbol{d}\big{)}=\frac{(2d_{1}+1)+\dots+(2d_{n-k}+1)}{6g-3+2n}$
$\displaystyle\hphantom{1+\delta_{\rm
string}\big{(}0^{k+1},\boldsymbol{d}\big{)}}{}=\frac{2(3g-2+n)+(n-k)}{6g-3+2n}=\frac{6g-4+3n-k}{6g-3+2n}=1+\frac{n-k-1}{6g-3+2n}.\\!\\!\\!$
∎

######  Corollary 9.

For any $(d_{1},d_{2})\in\Pi(3g-2+n,2)$ and for any $n\in\mathbb{N}$ one has

$\displaystyle\varepsilon\big{(}0^{n-1},d_{1},d_{2}\big{)}\geq-\frac{2}{6g-1}.$
(15)

###### Proof.

If one of $d_{1}$, $d_{2}$ is equal to zero, the statement for arbitrary $n$
follows from (5), so from now on we assume that both $d_{1}$, $d_{2}$ are
strictly positive. For $n=1$ the statement follows directly from (6). This
serves us as a base of induction in $n$. Suppose that for all $n=1,\dots,k$
the statement is true. Let us prove it for $n=k+1$:

$\displaystyle\big{\langle}\tau_{0}^{k+1}\tau_{d_{1}}\tau_{d_{2}}\big{\rangle}_{g,k+3}=\big{\langle}\tau_{0}^{k}\tau_{d_{1}-1}\tau_{d_{2}}\big{\rangle}_{g,k+2}+\big{\langle}\tau_{0}^{k}\tau_{d_{1}}\tau_{d_{2}-1}\big{\rangle}_{g,k+2}$
$\displaystyle\hphantom{\big{\langle}\tau_{0}^{k+1}\tau_{d_{1}}\tau_{d_{2}}\big{\rangle}_{g,k+3}}{}\geq\left(1-\frac{2}{6g-1}\right)\cdot\big{(}\big{\lfloor}\tau_{0}^{k}\tau_{d_{1}-1}\tau_{d_{2}}\big{\rfloor}_{g,k+2}+\big{\lfloor}\tau_{0}^{k}\tau_{d_{1}}\tau_{d_{2}-1}\big{\rfloor}_{g,k+2}\big{)}$
$\displaystyle\hphantom{\big{\langle}\tau_{0}^{k+1}\tau_{d_{1}}\tau_{d_{2}}\big{\rangle}_{g,k+3}}{}=\left(1-\frac{2}{6g-1}\right)\cdot\big{(}1+\delta_{\rm
string}\big{(}0^{k+1},d_{1},d_{2}\big{)}\big{)}\cdot\big{\lfloor}\tau_{0}^{k+1}\tau_{d_{1}}\tau_{d_{2}}\big{\rfloor}_{g,k+3}$
$\displaystyle\hphantom{\big{\langle}\tau_{0}^{k+1}\tau_{d_{1}}\tau_{d_{2}}\big{\rangle}_{g,k+3}}{}\geq\left(1-\frac{2}{6g-1}\right)\cdot\big{\lfloor}\tau_{0}^{k+1}\tau_{d_{1}}\tau_{d_{2}}\big{\rfloor}_{g,k+3},$

where the first equality is the string equation; inequality between the first
and the second lines is the assumption of the induction; the equality between
the second and the third line is equation (13); the inequality between the
third and the forth line is an implication of (14) and of the fact that all
the factors in both lines are positive. ∎

######  Corollary 10.

For any $g,n\in\mathbb{N}$ and for any $\boldsymbol{d}\in\Pi_{0}(3g-3+n,n)$
one has

$\displaystyle 1+\varepsilon(\boldsymbol{d})\geq\lambda(g,0).$ (16)

###### Proof.

Recalling convention (8) for $\lambda(g,0)$ we conclude that for $n=1$
inequality (16) follows from (3); for $n=2$ inequality (16) follows from (6);
for $n\geq 3$ inequality (16) corresponds to (15). ∎

######  Lemma 11.

Let $\boldsymbol{d}\in\Pi(3g-3+n,n)$. Define $\delta_{\rm
dilaton}(1,\boldsymbol{d})$ by equation

$\displaystyle\lfloor\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}\cdot\big{(}1+\delta_{\rm
dilaton}(1,\boldsymbol{d})\big{)}=(2g-2+n)\lfloor\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n}.$
(17)

Then

$\delta_{\rm dilaton}(1,\boldsymbol{d})=\frac{n-3}{6g-3+2n}.$

In particular,

$\displaystyle\delta_{\rm dilaton}(1,\boldsymbol{d})\begin{cases}\geq
0&\text{when }n\geq 3,\\\ =-\dfrac{1}{6g+1}&\text{when }n=2,\vspace{1mm}\\\
=-\dfrac{2}{6g-1}&\text{when }n=1.\end{cases}$ (18)

###### Proof.

Dividing both sides of equation (17) by
$\lfloor\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}$, applying
definition (11) and canceling common factors in the numerator and in the
denominator of the resulting expression we get

$1+\delta_{\rm
dilaton}(1,\boldsymbol{d})=\frac{3(2g-2+n)}{6g-3+2n}=\frac{6g-6+3n}{6g-3+2n}.$

######  Corollary 12.

For any partition $\boldsymbol{d}\in\Pi_{1}(3g-3+n,n)$ one has

$\displaystyle
1+\varepsilon(\boldsymbol{d})\geq\left(1-\frac{1}{6g+1}\right)\cdot\left(1-\frac{2}{6g-1}\right).$
(19)

###### Proof.

We have seen in (3) that for all $1$-correlators we have
$\varepsilon(3g-2)=0$, so for $n=1$ the statement is true. For $n=2$ the
statement is a direct implication of equation (6). Suppose that $n\geq 3$.

By Remark 1, the quantity $\varepsilon(\boldsymbol{d})$ does not change under
any permutation of the entries of $\boldsymbol{d}$. Thus, we can permute the
first $n-2$ elements of the partition without affecting the value of
$\varepsilon(\boldsymbol{d})$, in particular, we can place them in the growing
order. Since the sum of the first $n-2$ elements is less than or equal to $1$
either they are all equal to $0$ or they form the sequence $(0,\dots,0,1)$
after such reordering. If they all are equal to $0$, the statement follows
from equation (15) from Corollary (9).

It remains to consider the case when $n\geq 3$ and when the first $n-2$
elements form a sequence $(0,\dots,0,1)$. We prove first the desired
inequality for partitions of the form $(1,d_{1},d_{2})$:

$\displaystyle\langle\tau_{1}\tau_{d_{1}}\tau_{d_{2}}\rangle_{g,3}=(2g)\cdot\langle\tau_{d_{1}}\tau_{d_{2}}\rangle_{g,2}\geq\left(1-\frac{2}{6g-1}\right)\cdot(2g)\lfloor\tau_{d_{1}}\tau_{d_{2}}\rfloor_{g,2}$
$\displaystyle\hphantom{\langle\tau_{1}\tau_{d_{1}}\tau_{d_{2}}\rangle_{g,3}}{}=\left(1-\frac{2}{6g-1}\right)\cdot\lfloor\tau_{1}\tau_{d_{1}}\tau_{d_{2}}\rfloor_{g,3}\cdot\big{(}1+\delta_{\rm
dilaton}(1,d_{1},d_{2})\big{)}$
$\displaystyle\hphantom{\langle\tau_{1}\tau_{d_{1}}\tau_{d_{2}}\rangle_{g,3}}{}\geq\left(1-\frac{2}{6g-1}\right)\left(1-\frac{1}{6g+1}\right)\cdot\lfloor\tau_{1}\tau_{d_{1}}\tau_{d_{2}}\rfloor_{g,3}.$

Here the first equality is the dilaton equation; the inequality which follows
is equation (6); the equality between the first two lines is equation (17) and
the inequality between the second and the third line is based on equation
(18).

To complete the proof of Corollary 12 we prove it for partitions of the form
$\big{(}0^{k+1},1,d_{1},d_{2}\big{)}$ by induction in $k\geq 0$. The proof
follows line-by-line the proof of Corollary 9. ∎

######  Lemma 13.

Let $\boldsymbol{d}\in\Pi(3g-3+n-k,n)$, where $k,n\in\mathbb{N}$. Define
$\delta_{\rm Virasoro}(k+1,\boldsymbol{d})$ by equation

$\displaystyle\lfloor\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g}\big{(}1+\delta_{\rm
Virasoro}(k+1,\boldsymbol{d})\big{)}$
$\displaystyle\qquad{}=\frac{1}{(2k+3)!!}\Bigg{(}\sum_{j=1}^{n}\frac{(2k+2d_{j}+1)!!}{(2d_{j}-1)!!}\lfloor\tau_{d_{1}}\cdots\tau_{d_{j}+k}\cdots\tau_{d_{n}}\rfloor_{g}$
$\displaystyle\qquad\quad{}+\frac{1}{2}\sum_{\begin{subarray}{c}r+s=k-1\\\
r,s\geq
0\end{subarray}}(2r+1)!!(2s+1)!!\lfloor\tau_{r}\tau_{s}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g-1}\Bigg{)}.$
(20)

Then

$\displaystyle\delta_{\rm
Virasoro}(k+1,\boldsymbol{d})=\frac{n-3}{6g-3+2n}-\frac{2k(2n-5)}{(6g-3+2n)(6g-5+2n)}.$
(21)

###### Proof.

Dividing both sides of equation (20) by
$\lfloor\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}$, applying
definition (11) and canceling common factors in the numerator and in the
denominator of the resulting expression we get

$\displaystyle 1+\delta_{\rm Virasoro}(k+1,\boldsymbol{d})$
$\displaystyle\qquad{}=\frac{1}{6g-3+2n}\cdot\left(\big{(}(2d_{1}+1)+\dots+(2d_{n}+1)\big{)}+\frac{1}{2}\cdot
k\cdot\frac{24g}{6g-5+2n}\right)$
$\displaystyle\qquad{}=\frac{1}{6g-3+2n}\cdot\left(\big{(}6g-6+3n-2k\big{)}+2k-2k\cdot\frac{2n-5}{6g-5+2n}\right)$
$\displaystyle\qquad{}=1+\frac{n-3}{6g-3+2n}-\frac{2k(2n-5)}{(6g-3+2n)(6g-5+2n)}.$
∎

######  Remark 14.

In expression (20) we ignored the third term in the Virasoro constraints.
Since this third term is, clearly, positive, this is suitable for getting a
lower bound instead of exact asymptotics. It is widely believed that the third
term of Virasoro constraints becomes negligible in large genera. We expect
that technique from [1] might be useful for replacing the lower bound in (10)
by the exact asymptotics under strengthening restrictions on $\alpha$.

We shall need the following technical corollary of Lemma 13.

######  Corollary 15.

Let $k,n$ be integers satisfying $k\geq 0$, $n\geq 2$. Let $\boldsymbol{d}$ be
a partition $\boldsymbol{d}\in\Pi(3g-3+n-k,n)$, such that $k+1\leq d_{j}$ for
$j=1,\dots,n-2$, and $k+d_{1}+\dots+d_{n-2}\leq\frac{3}{2}g$. Then

$\displaystyle\delta_{\rm Virasoro}(k+1,\boldsymbol{d})\geq-\frac{1}{6g+1}.$
(22)

###### Proof.

Use expression (21) for $\delta_{\rm Virasoro}(k+1,\boldsymbol{d})$. When
$n=2$ we get

$\delta_{\rm
Virasoro}(k+1,\boldsymbol{d})=-\frac{1}{6g+1}+\frac{2k}{(6g+1)(6g-1)}\geq-\frac{1}{6g+1}.$

Let $n\geq 3$. By assumption, $(k+1)\leq d_{j}$ for $j=1,\dots,d_{n-2}$, so
$(k+1)(n-2)\leq\frac{3}{2}g$, and hence $2k(2n-5)<6g$, which implies that

$\displaystyle\delta_{\rm
Virasoro}((k+1,\boldsymbol{d}))=\frac{n-3}{6g-3+2n}-\frac{2k(2n-5)}{(6g-3+2n)(6g-5+2n)}$
$\displaystyle\hphantom{\delta_{\rm
Virasoro}((k+1,\boldsymbol{d}))}{}\geq-\frac{2k(2n-5)}{(6g-3+2n)(6g-5+2n)}\geq-\frac{6g}{(6g+3)(6g+1)}>-\frac{1}{6g+1}.\\!\\!\\!$
∎

Before passing to the step of induction, we recapitulate the properties of the
function $\lambda(g,L)$ defined in (7). Recall that the arguments $g$, $L$ of
$\lambda(g,L)$ are nonnegative integers satisfying $g>L$. Inequalities
(23)–(25) below follow from the definition of $\lambda(g,L)$. Each inequality
is applied to those ordered pairs $g$, $L$ for which the argument of $\lambda$
on both sides of the inequality belongs to the domain of definition of
$\lambda$. We have

$\displaystyle 1>\lambda(g+1,L)>\lambda(g,L)>\lambda(g,L+1)>0,$ (23)

and

$\displaystyle\left(1-\frac{1}{6g+1}\right)\cdot\lambda(g-1,L-1)=\lambda(g,L).$
(24)

Combining the latter two relations we get

$\displaystyle\left(1-\frac{1}{6g+1}\right)\cdot\lambda(g,L)>\left(1-\frac{1}{6(g-1)+1}\right)\cdot\lambda(g,L)$
$\displaystyle\hphantom{\left(1-\frac{1}{6g+1}\right)\cdot\lambda(g,L)}{}>\left(1-\frac{1}{6(g-1)+1}\right)\cdot\lambda(g-1,L)=\lambda(g,L+1).$
(25)

######  Proposition 16 (step of induction).

Suppose that for some nonnegative integer $L_{0}$ the following uniform bound
is valid: for all integers $g$, $L$ satisfying $g>L_{0}$, $0\leq L\leq L_{0}$,
for all partitions $\boldsymbol{d}\in\Pi_{L}(3g-2+n,n+1)$, where $n\geq 0$,
one has

$1+\varepsilon(\boldsymbol{d})\geq\lambda(g,L).$

Then for all integers $g$, $L$ satisfying $g>L_{0}+1$, $0\leq L\leq L_{0}+1$,
for all partitions $\boldsymbol{d}\in\Pi_{L}(3g-2+n,n+1)$, where $n\geq 0$,
one also has

$1+\varepsilon(\boldsymbol{d})\geq\lambda(g,L).$

###### Proof.

We warn the reader that the total number of elements of the partition is
denoted in Proposition 16 by $n+1$ and not by $n$ as in Theorem 5. This allows
us to use formulae from the key Lemmas 11 and 13 without adjustments.

By convention $\Pi_{L}(3g-2,1)=\Pi(3g-2,1)$ and $\Pi_{L}(3g-1,2)=\Pi(3g-1,2)$
for any $L\in\mathbb{Z}_{\geq 0}$. We have seen in (3) that for all
$1$-correlators we have $\varepsilon(3g-2)=0$, so for $n=0$ the statement is
trivially true. For $n=1$ the statement is a direct implication of inequality
(6):

$1+\varepsilon(\boldsymbol{d})\geq\left(1-\frac{2}{6g-1}\right)=\lambda(g,0)\geq\lambda(g,L).$

Thus, from now on we can assume that $n\geq 2$. Let
$\boldsymbol{d}\in\Pi_{L}(3g-2+n,n+1)$. If $L\leq L_{0}$, then the statement
makes part of the induction assumption. Hence, from now on we can assume that

$\boldsymbol{d}\in\Pi_{L_{0}+1}(3g-2+n,n+1)\setminus\Pi_{L_{0}}(3g-2+n,n+1),$

where $n\geq 2$ and $g>L_{0}+1$. This implies that
$d_{1}+\dots+d_{n-1}=L_{0}+1$ and, hence,

$\displaystyle d_{n}+d_{n+1}=3g-2+n-(L_{0}+1)>2L_{0}+2.$

By Remark 1, the quantity $\varepsilon(\boldsymbol{d})$ does not change under
any permutation of the entries of $\boldsymbol{d}$. Place to the leftmost
position the smallest strictly positive element among the first $n-1$
elements. This operation does not change the last two elements of the
partition and does not change the sum of its first $n-1$ elements. Denote the
resulting partition by $(k+1,d_{1},\dots,d_{n})$. To prove the proposition we
have to prove the inequality

$\displaystyle 1+\varepsilon(k+1,d_{1},\dots,d_{n})\geq\lambda(g,L_{0}+1),$
(26)

where

$\displaystyle k+d_{1}+d_{2}+\dots+d_{n-2}=L_{0},\qquad n\geq 2,\qquad k\geq
0,$ $\displaystyle k+1\leq\min_{\begin{subarray}{c}1\leq i\leq n-2\\\
d_{i}>0\end{subarray}}d_{i},\qquad g>L_{0}+1\geq 1.$ (27)

We consider the special case $k=0$ separately. In this special case, when
$n=2$ the partition $(k+1,d_{1},\dots,d_{n})$ becomes $(1,d_{1},d_{2})$ and
the desired inequality is proved in (19). Assume that $k=0$ and $n\geq 3$. By
(18) we have

$\displaystyle\delta_{\rm dilaton}(1,d_{1},\dots,d_{n})\geq 0,\qquad\text{for
}n\geq 3.$ (28)

Thus, for any $g>L_{0}+1$ we have

$\displaystyle\big{(}1+\varepsilon(1,d_{1},\dots,d_{n})\big{)}\cdot\lfloor\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}=\langle\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n+1}$
$\displaystyle\qquad{}=(2g-2+n)\langle\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g,n}\geq(2g-2+n)\cdot\big{(}\lambda(g,L_{0})\cdot\lfloor\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n}\big{)}$
$\displaystyle\qquad{}=\lambda(g,L_{0})\cdot\lfloor\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}\cdot\big{(}1+\delta_{\rm
dilaton}(1,\boldsymbol{d})\big{)}$
$\displaystyle\qquad{}\geq\lambda(g,L_{0})\cdot\lfloor\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}>\lambda(g,L_{0}+1)\cdot\lfloor\tau_{1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g,n+1}.$

Here the first equality is definition (12) of
$\varepsilon(1,d_{1},\dots,d_{n})$; the second equality is the string equation
(2); the inequality in the middle of the second line is the induction
assumption; the equality in the beginning of the third line is definition (17)
of $(1+\delta_{\rm dilaton}(1,\boldsymbol{d}))$; the inequality in the
beginning of the last line is a direct implication of (28); the last
inequality is an implication of (23).

Suppose now that $k\geq 1$. We first prove the desired inequality (26) in the
special case when

$\displaystyle d_{j}\geq 1,\qquad\text{for }j=1,\dots,n-2$ (29)

and then prove it in the most general situation when some of $d_{j}$ (possibly
all of them) are equal to $0$.

Note that by assumption, $k+1$ is less than or equal to any strictly positive
element among $d_{1},\dots,d_{n-2}$, so inequalities (29), actually, imply
that

$\displaystyle d_{j}\geq k+1,\qquad\text{for }j=1,\dots,n-2.$

Also, from (27) we get

$k+d_{1}+d_{2}+\dots+d_{n-2}=L_{0}<\frac{3g}{2}.$

Thus, the partition $(k+1,d_{1},\dots,d_{n})$ satisfies assumptions of
Corollary 15.

Let $g>L_{0}+1$. Under the above assumptions we have

$\displaystyle\big{(}1+\varepsilon(k+1,d_{1},\dots,d_{n})\big{)}\cdot\lfloor\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g}=\langle\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g}$
$\displaystyle\qquad{}\geq\frac{1}{(2k+3)!!}\Bigg{(}\sum_{j=1}^{n}\frac{(2k+2d_{j}+1)!!}{(2d_{j}-1)!!}\langle\tau_{d_{1}}\cdots\tau_{d_{j}+k}\cdots\tau_{d_{n}}\rangle_{g}$
$\displaystyle\qquad\quad{}+\frac{1}{2}\sum_{\begin{subarray}{c}r+s=k-1\\\
r,s\geq
0\end{subarray}}(2r+1)!!(2s+1)!!\langle\tau_{r}\tau_{s}\tau_{d_{1}}\cdots\tau_{d_{n}}\rangle_{g-1}\Bigg{)}$
$\displaystyle\qquad{}\geq\frac{1}{(2k+3)!!}\Bigg{(}\sum_{j=1}^{n-2}\frac{(2k+2d_{j}+1)!!}{(2d_{j}-1)!!}\cdot\lambda(g,L_{0})\cdot\lfloor\tau_{d_{1}}\cdots\tau_{d_{j}+k}\cdots\tau_{d_{n}}\rfloor_{g}$
$\displaystyle\qquad\quad{}+\frac{(2k+2d_{n-1}+1)!!}{(2d_{n-1}-1)!!}\cdot\lambda(g,L_{0}-k)\cdot\lfloor\tau_{d_{1}}\cdots\tau_{d_{n-2}}\tau_{d_{n-1}+k}\tau_{d_{n}}\rfloor_{g}$
$\displaystyle\qquad\quad{}+\frac{(2k+2d_{n}+1)!!}{(2d_{n}-1)!!}\cdot\lambda(g,L_{0}-k)\cdot\lfloor\tau_{d_{1}}\cdots\tau_{d_{n-1}}\tau_{d_{n}+k}\rfloor_{g}$
$\displaystyle\qquad\quad{}+\frac{1}{2}\sum_{\begin{subarray}{c}r+s=k-1\\\
r,s\geq
0\end{subarray}}(2r+1)!!(2s+1)!!\cdot\lambda(g-1,L_{0}-1)\cdot\lfloor\tau_{r}\tau_{s}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g-1}\Bigg{)}$
$\displaystyle\qquad{}\geq\lambda(g,L_{0})\cdot\lfloor\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g}\big{(}1+\delta_{\rm
Virasoro}(k+1,\boldsymbol{d})\big{)}$
$\displaystyle\qquad{}\geq\left(1-\frac{1}{6g+1}\right)\cdot\lambda(g,L_{0})\cdot\lfloor\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g}$
$\displaystyle\qquad{}\geq\lambda(g,L_{0}+1)\cdot\lfloor\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n}}\rfloor_{g}.$

Here the first equality is the definition (12) of
$\varepsilon(1,d_{1},\dots,d_{n})$; the first inequality is an instant
corollary of the Virasoro constraints in which we omitted the terms in the
third line of (1). The second inequality is the induction assumption. The
third inequality combines the inequality $\lambda(g,L_{0}-k)>\lambda(g,L_{0})$
which follows from (23), the inequality
$\lambda(g-1,L_{0}-1)>\lambda(g,L_{0})$ which follows from (24), and the
definition (20) of $\delta_{\rm Virasoro}(k+1,\boldsymbol{d})$. The inequality
$\big{(}1+\delta_{\rm
Virasoro}(k+1,\boldsymbol{d})\big{)}\geq\big{(}1-\frac{1}{6g+1}\big{)}$ is
justified by (22). The last inequality is justified in (25).

It remains to prove inequality (26), without extra assumptions (29). In other
words, we have to prove the inequality

$1+\varepsilon\big{(}0^{s},k+1,d_{1},\dots,d_{n-s}\big{)}\geq\lambda(g,L_{0}+1).$

The case $n-s=0$ follows from (5). For $n-s=1$ inequality (15) implies

$1+\varepsilon\big{(}0^{s},k+1,d_{1}\big{)}\geq
1-\frac{2}{6g-1}=\lambda(g,0)\geq\lambda(g,L_{0}+1).$

Thus, we may assume that $n-s\geq 2$ and that the following inequalities are
valid:

$\begin{cases}k+d_{1}+d_{2}+\dots+d_{n-s-2}=L_{0},\\\ n\geq s\geq 0,\\\ k\geq
1,\\\ d_{j}>k\quad\text{for }j=1,\dots,n-s-2.\end{cases}$

We proceed by induction in $s$. For $s=0$, which serves us as a base of
induction, the statement is already proved. We perform a step of induction as
follows

$\displaystyle\big{(}1+\varepsilon\big{(}0^{s+1},k+1,d_{1},\dots,d_{n-s}\big{)}\big{)}\cdot\big{\lfloor}\tau_{0}^{s+1}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}=\big{\langle}\tau_{0}^{s+1}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rangle}_{g}$
$\displaystyle\qquad{}=\big{\langle}\tau_{0}^{s}\tau_{k}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rangle}_{g}+\big{\langle}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}-1}\cdots\tau_{d_{n-s}}\big{\rangle}_{g}+\dots+\big{\langle}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}-1}\big{\rangle}_{g}$
$\displaystyle\qquad{}=\big{(}1+\varepsilon\big{(}0^{s},k,d_{1},\dots,d_{n-s}\big{)}\big{)}\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}$
$\displaystyle\qquad\quad{}+\big{(}1+\varepsilon\big{(}0^{s},k+1,d_{1}-1,\dots,d_{n-s}\big{)}\big{)}\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}-1}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}+\cdots$
$\displaystyle\qquad\quad{}+\big{(}1+\varepsilon\big{(}0^{s},k+1,d_{1},\dots,d_{n-s}-1\big{)}\big{)}\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}-1}\big{\rfloor}_{g}$
$\displaystyle\qquad{}\geq\lambda(g,L_{0})\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}+\lambda(g,L_{0})\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}-1}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}+\cdots$
$\displaystyle\qquad\quad{}+\lambda(g,L_{0})\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s-2}-1}\tau_{d_{n-s-1}}\tau_{d_{n-s}}\big{\rfloor}_{g}$
$\displaystyle\qquad\quad{}+\lambda(g,L_{0}+1)\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s-2}}\tau_{d_{n-s-1}-1}\tau_{d_{n-s}}\big{\rfloor}_{g}$
$\displaystyle\qquad\quad{}+\lambda(g,L_{0}+1)\cdot\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s-2}}\tau_{d_{n-s-1}}\tau_{d_{n-s}-1}\big{\rfloor}_{g}$
$\displaystyle\qquad{}\geq\lambda(g,L_{0}+1)\big{(}\big{\lfloor}\tau_{0}^{s}\tau_{k}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}+\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}-1}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}+\cdots$
$\displaystyle\qquad\quad{}+\big{\lfloor}\tau_{0}^{s}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}-1}\big{\rfloor}_{g}\big{)}$
$\displaystyle\qquad{}=\lambda(g,L_{0}+1)\cdot\big{\lfloor}\tau_{0}^{s+1}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}\cdot\big{(}1+\delta_{\rm
string}\big{(}0^{s+1},k+1,d_{1},\dots,d_{n-s}\big{)}\big{)}$
$\displaystyle\qquad{}\geq\lambda(g,L_{0}+1)\cdot\big{\lfloor}\tau_{0}^{s+1}\tau_{k+1}\tau_{d_{1}}\cdots\tau_{d_{n-s}}\big{\rfloor}_{g}.$

Here the first equality is the definition (12) of
$\varepsilon\big{(}0^{s+1},k+1,d_{1},\dots,d_{n-s}\big{)}$. The second
equality is the string equation (2). (Recall that by convention, if one of
$d_{n-s-1}$ or $d_{n-s}$ is equal to zero, the term, containing the negative
index $d_{n-s-1}-1$ or $d_{n-s}-1$ respectively, is missing in the string
equation and below.) The equality which follows, is equation (12) applied to
every term of the resulting expression. The inequality, where $\lambda$
appears on the left-hand side for the first time, is the induction assumption
applied to each term. The next inequality follows from the inequality
$\lambda(g,L_{0})>\lambda(g,L_{0}+1)$, see (23). The equality which follows is
the definition (13) of $\delta_{\rm
string}\big{(}0^{s+1},k+1,d_{1},\dots,d_{n-s}\big{)}$. The last inequality is
justified by (14). ∎

###### Proof of Theorem 5.

For $L=0$ Theorem 5 follows from Corollary 10. For $L=1$ it follows from (19)
combined with the fact that
$\big{(}1-\frac{1}{6g+1}\big{)}\cdot\big{(}1-\frac{2}{6g-1}\big{)}>\lambda(g,1)$.
For $L>1$ we apply recursively Proposition 16. ∎

### Acknowledgements

We thank A. Aggarwal for precious comments on the preliminary version of this
text, which allowed to correct several misprints and improve the presentation.
We are grateful to N. Anantharaman, A. Borodin, D. Chen, M. Kazarian, S.
Lando, M. Möller, A. Okounkov, M. Shapiro for stimulating discussions. We
thank MPIM in Bonn and MSRI in Berkeley for providing us with excellent
working environment. We thank anonymous referees for their generous reports
and for helpful suggestions which allowed to improve the presentation. The
research of the second author was partially supported by PEPS. The results of
Section 1 were obtained at Saint Petersburg State University under support of
RSF grant 19-71-30002. This material is based upon work supported by the
ANR-19-CE40-0021 grant. It was also supported by the NSF Grant DMS-1440140
while part of the authors were in residence at the MSRI during the Fall 2019
semester.

## References

  * [1] Aggarwal A., Large genus asymptotics for Siegel–Veech constants, Geom. Funct. Anal. 29 (2019), 1295–1324, arXiv:1810.05227.
  * [2] Aggarwal A., Large genus asymptotics for volumes of strata of abelian differentials, J. Amer. Math. Soc., to appear, arXiv:1804.05431.
  * [3] Aggarwal A., Large genus asymptotics for intersection numbers and principal strata volumes of quadratic differentials, arXiv:2004.05042.
  * [4] Aggarwal A., Delecroix V., Goujard É., Zograf P., Zorich A., Conjectural large genus asymptotics of Masur–Veech volumes and of area Siegel–Veech constants of strata of quadratic differentials, Arnold Math. J. 6 (2020), 149–161, arXiv:1912.11702.
  * [5] Chen D., Möller M., Sauvaget A., Masur–Veech volumes and intersection theory: the principal strata of quadratic differentials (with an appendix by G. Borot, A. Giacchetto, D. Lewanski), arXiv:1912.02267.
  * [6] Chen D., Möller M., Sauvaget A., Zagier D., Masur–Veech volumes and intersection theory on moduli spaces of abelian differentials, Invent. Math., to appear, arXiv:1901.01785.
  * [7] Delecroix V., Goujard É., Zograf P., Zorich A., Masur–Veech volumes, frequencies of simple closed geodesics and intersection numbers of moduli spaces of curves, arXiv:1908.08611.
  * [8] Delecroix V., Goujard É., Zograf P., Zorich A., Large genus asymptotic geometry of random square-tiled surfaces and of random multicurves, arXiv:2007.04740.
  * [9] Dijkgraaf R., Intersection theory, integrable hierarchies and topological field theory, in New Symmetry Principles in Quantum Field Theory (Cargèse, 1991), NATO Adv. Sci. Inst. Ser. B Phys., Vol. 295, Plenum, New York, 1992, 95–158, arXiv:hep-th/9201003.
  * [10] Dijkgraaf R., Verlinde H., Verlinde E., Topological strings in $d<1$, Nuclear Phys. B 352 (1991), 59–86.
  * [11] Eskin A., Okounkov A., Pillowcases and quasimodular forms, in Algebraic Geometry and Number Theory, Progr. Math., Vol. 253, Birkhäuser Boston, Boston, MA, 2006, 1–25, arXiv:math.DS/0505545.
  * [12] Eskin A., Zorich A., Volumes of strata of Abelian differentials and Siegel–Veech constants in large genera, Arnold Math. J. 1 (2015), 481–488, arXiv:1507.05296.
  * [13] Goujard É., Volumes of strata of moduli spaces of quadratic differentials: getting explicit values, Ann. Inst. Fourier (Grenoble) 66 (2016), 2203–2251, arXiv:1501.01611.
  * [14] Kazarian M., Recursion for Masur–Veech volumes of moduli spaces of quadratic differentials, J. Inst. Math. Jussieu, to appear, arXiv:1912.10422.
  * [15] Kazarian M., Lando S., An algebro-geometric proof of Witten’s conjecture, J. Amer. Math. Soc. 20 (2007), 1079–1089, arXiv:math.AG/0601760.
  * [16] Kontsevich M., Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147 (1992), 1–23.
  * [17] Liu K., Xu H., A remark on Mirzakhani’s asymptotic formulae, Asian J. Math. 18 (2014), 29–52, arXiv:1103.5136.
  * [18] Mirzakhani M., Simple geodesics and Weil–Petersson volumes of moduli spaces of bordered Riemann surfaces, Invent. Math. 167 (2007), 179–222.
  * [19] Mirzakhani M., Weil–Petersson volumes and intersection theory on the moduli space of curves, J. Amer. Math. Soc. 20 (2007), 1–23.
  * [20] Okounkov A., Pandharipande R., Gromov–Witten theory, Hurwitz numbers, and matrix models, in Algebraic Geometry – Seattle 2005, Part 1, Proc. Sympos. Pure Math., Vol. 80, Amer. Math. Soc., Providence, RI, 2009, 325–414, arXiv:math.AG/0101147.
  * [21] Witten E., Two-dimensional gravity and intersection theory on moduli space, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh University, Bethlehem, PA, 1991, 243–310.
  * [22] Yang D., Zagier D., Zhang Y., Masur–Veech volumes of quadratic differentials and their asymptotics, arXiv:2005.02275.
  * [23] Zograf P.G., An explicit formula for Witten’s 2-correlators, J. Math. Sci. 240 (2019), 535–538.