This paper studies how certain numbers that arise from integrating basic geometric classes over the moduli space of complex curves behave when the genus (the number of holes in the surface) becomes large. The authors introduce and study a particular normalization C(d) of Witten’s psi‑class intersection numbers and prove that, for a wide class of inputs, these normalized numbers converge uniformly to 1/π as the genus grows. They also give a more detailed expansion that allows some marked points to carry zero insertions, and they apply the results to a formal solution of the Painlevé I equation. A complementary result gives a new proof of a polynomiality statement about the coefficients that appear in large‑genus expansions.

Witten’s intersection numbers are integrals of certain natural cohomology classes (psi classes) on the Deligne–Mumford moduli space Mg,n of stable algebraic curves with n marked points. These numbers connect to several important topics in geometry and mathematical physics, including Witten’s conjecture (now a theorem) linking the intersection numbers to the Korteweg–de Vries integrable hierarchy, and to counting problems such as Weil–Petersson and Masur–Veech volumes. To study their large‑genus behavior the authors use a specific normalization C(d) (defined by a product of double factorials and factorials) that isolates the main growth and makes a clean limit visible.

Technically, the paper builds on earlier ideas and formulas: an explicit generating‑series formula for n‑point intersection numbers, and the Dijkgraaf–Verlinde–Verlinde (DVV) recursion, which relates intersection numbers with different arguments. Using these tools the authors prove Theorem 1: for vectors d with positive integer entries and genus g(d)→∞, C(d) = 1/π + O(1/g(d)) uniformly in the allowed inputs. They then refine this in Theorem 2 to give a more precise multiplicative correction depending on the multiplicities of the entries of d, again with an error O(1/g). A concrete corollary describes the family with many zeros: for k up to order √g, C(0^k, 2/3 g − 3 + k) is asymptotically (1/π) exp(−k^2/(30 g)), while if k grows faster than √g the same normalized quantity tends to zero.