The study of random surfaces of large genus has grown to be a pivotal area in modern geometry and mathematical physics, offering profound insights into the statistical properties of geometric structures. A cornerstone of this field is the seminal work of Maryam Mirzakhani, who investigated the asymptotic behavior of Weil–Petersson volumes and explored geometric properties of random hyperbolic surfaces as the genus grows. In particular, Mirzakhani analyzed the lengths of simple closed geodesics and established foundational results on the expected values of geometric invariants such as the diameter, injectivity radius, and Cheeger constant for hyperbolic surfaces of large genus.

Building upon Mirzakhani’s framework, subsequent studies have delved deeper into the statistical properties of hyperbolic surfaces. Notably, Mirzakhani and Petri examined the distribution of lengths of closed geodesics on random surfaces of large genus, providing Poisson approximation results for the length spectrum and computing the large-genus limit of the expected systole. Further advances include the very recent fundamental work by Anantharaman and Monk, who proved that a random hyperbolic surface of large genus has an almost optimal spectral gap with probability which tends to one as genus grows. Parallel results are obtained by Hide, Magee, and Naud for random covers. Major advances in the study of diameter and Cheeger constant of a random hyperbolic surface of large genus have recently been obtained by Budzinski, Curien, and Petri.

While hyperbolic surfaces have been extensively studied, there is a burgeoning interest in extending these investigations to the realm of flat (translation) surfaces.