Uniform lower bound for intersection numbers of psi-classes

We approximate intersection numbers $\langle \psi_1^{d_1} \dots \psi_n^{d_n}\rangle_{g,n}$ 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 $d_1,\ldots,d_n$. 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 $d_1+\ldots+d_{n-2}=o(g)$.

Following a common convention, we omit g and n, or just n when they are clear from the context. All correlators are uniquely defined by the initial data τ 3 0 = 1, τ 1 = 1 24 via the recursive relations known as Virasoro constraints that we present below.
Date: April 6, 2020. 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.
For k = −1 and k = 0 the above relations have particularly simple form.
We denote by Π(m, n) the set of ordered partitions of an integer m into a sum of n nonnegative integers.
Theorem 1 below makes the first step towards a proof of the Main Conjecture. It establishes an efficient uniform lower bound for ε(d) for those partitions 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 ε(d) that ε(d) does not change under any permutation of the entries of 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 α with any fixed α satisfying α < 1 2 .
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 [EZo] was successfully proved by independent methods in [Ag1] and in [CMöSZa]. However, the analogous conjectures stated in [DGZZ] and in [ADGZZ] on the large genus asymptotics of the Masur-Veech volumes of strata in moduli spaces of quadratic differentials are open. Assuming validity of the Main Conjecture stated above, the Conjecture on large genus asymptotics of the Masur-Veech volume of the principal stratum of holomorphic quadratic differentials, and the Conjecture on uniform convergence of certain expression in multivariate harmonic sum, we provide in [DGZZ] a detailed description of the asymptotic geometry of random square-tiled surfaces and of random simple closed multicurves on surfaces of large genus.
It is easy to compute ε d explicitly for those partitions where all but one entries d 1 , . . . , 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 (6). In particular, This implies that for any constant α satisfying 0 < α < 1 2 (respectively 1 2 < α) we have which explains why the restriction α < 1 2 in Remark 2 cannot be loosened.
Define the following function of integer arguments g, L, satisfying g > L ≥ 0: where, by convention, It follows from the definition of λ(g, L) that 0 < λ(g, L) < 1 for any g > L ≥ 0.
Remark 3. Proposition 2.2 in [LX] claims that for any triple (n, K, M ) of positive integers one has lim under the additional requirement that d n−1 ≤ 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 d ∈ Π(3g − 3 + n, n) let .
From now on we suppose that g ≥ 1.
In particular, for any d as above and for any k ≥ 0 we have Proof. Dividing both sides of equation (15) and applying definition (13) to all terms involved in the right-hand side of the resulting equation we get Corollary 2. For any (d 1 , d 2 ) ∈ Π(3g − 2 + n, 2) and for any n ∈ N one has Proof. If one of d 1 , d 2 is equal to zero, the statement for arbitrary n follows from (7), so from now on we assume that both d 1 , d 2 are strictly positive. For n = 1 the statement follows directly from (8). This serves us as a base of induction in n.
Suppose that for all n = 1, . . . , k the statement is true. Let us prove it for n = k +1.
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 (15); the inequality between the third and the forth line is an implication of (16) and of the fact that all the factors in both lines are positive.
Corollary 3. For any g, n ∈ N and for any d ∈ Π 0 (3g − 3 + n, n) one has Proof. Recalling convention (10) for λ(g, 0) we conclude that for n = 1 inequality (18) follows from (5); for n = 2 inequality (18) follows from (8); for n ≥ 3 inequality (18) corresponds to (17). Then In particular, Proof. Dividing both sides of equation (19) by ⌊τ 1 τ d1 . . . τ dn ⌋ g,n+1 , applying definition (13) and canceling common factors in the numerator and in the denominator of the resulting expression we get Corollary 4. For any partition d ∈ Π 1 (3g − 3 + n, n) one has Proof. We have seen in (5) that for all 1-correlators we have ε(3g − 2) = 0, so for n = 1 the statement is true. For n = 2 the statement is a direct implication of equation (8). Suppose that n ≥ 3. By Remark 1, the quantity ε(d) does not change under any permutation of the entries of d. Thus, we can permute the first n − 2 elements of the partition without affecting the value of ε(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, . . . , 0, 1) after such reordering. If they all are equal to 0, the statement follows from equation (17) from Corollary (2).
It remains to consider the case when n ≥ 3 and when the first n − 2 elements form a sequence (0, . . . , 0, 1). We prove first the desired inequality for partitions of the form (1, d 1 , d 2 ).
Here the first equality is the dilaton equation; the inequality which follows is equation (8); the equality between the first two lines is equation (19) and the inequality between the second and the third line is based on equation (20).
To complete the proof of Corollary 4 we prove it for partitions of the form (0 k+1 , 1, d 1 , d 2 ) by induction in k ≥ 0. The proof follows line-by-line the proof of Corollary 2.
We shall need the following technical Corollary of Lemma 3.
Then for all integers g, L satisfying g ≥ g 0 + 1, 0 ≤ L ≤ L 0 + 1, for all partitions d ∈ Π L (3g − 2 + n, n + 1), where n ≥ 0, one also has: Proof. The total number of elements of the partition is denoted by n + 1. By convention Π L (3g − 2, 1) = Π(3g − 2, 1) and Π L (3g − 1, 2) = Π(3g − 1, 2) for any L ∈ Z ≥0 . We have seen in (5) that for all 1-correlators we have ε(3g − 2) = 0, so for n = 0 the statement is trivially true. For n = 1 the statement is a direct implication of inequality (8): Thus, from now on we can assume that n ≥ 2. Let d ∈ Π L (3g − 2 + n, n + 1). If L ≤ L 0 , then the statement makes part of the induction assumption. Hence, from now on we can assume that where n ≥ 2 and g ≥ g 0 + 1. This implies that d 1 + · · · + d n−1 = L 0 + 1 and, hence, By Remark 1, the quantity ε(d) does not change under any permutation of the entries of 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 , . . . , d n ). To prove the Proposition we have to prove the inequality We consider the special case k = 0 separately. In this special case, when n = 2 the partition (k + 1, d 1 , . . . , d n ) becomes (1, d 1 , d 2 ) and the desired inequality is proved in (21). Assume that k = 0 and n ≥ 3. By (20) we have Thus, for any g ≥ g 0 we have Here the first equality is definition (14) of ε(1, d 1 , . . . , 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 (19) of (1 + δ dilaton (1, d)); the inequality in the beginning of the last line is a direct implication of (31); the last inequality is an implication of (25).
The case n − s = 0 follows from (7). For n − s = 1 inequality (17) implies Thus, we may assume that n − s ≥ 2 and that the following inequalities are valid: 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.
Here the first equality is the definition (14) of ε(0 s+1 , k + 1, d 1 , . . . , d n−s ). 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 (14) applied to every term of the resulting expression. The inequality, where λ 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 λ(g, L 0 ) > λ(g, L 0 + 1), see (25). The equality which follows is the definition (15) of δ string (0 s+1 , k + 1, d 1 , . . . , d n−s ). The last inequality is justified by (16).