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

We approximate intersection numbers $\big\langle \psi_1^{d_1}\cdots \psi_n^{d_n}\big\rangle_{g,n}$onDeligne-Mumfor{d}^{\mathrm{\prime }}smodulispace$\overline{\mathcal M}_{g,n}$ofgenus$g$stablecomplexcurveswith$n$markedpointsbycertainclosed-formexpressionsin$d_1,\dots,d_n$. Conjecturally, these approximations become asymptotically exact uniformly in $d_i$ when $g\to \mathrm{\infty }$ 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 approximatingexpressions multiplied by an explicit factor $\lambda (g,n)$, which tends to $1$ when $g\to \mathrm{\infty }$ and $d_1+\cdots +d_{n-2}=o(g)$.