Large Deviations and Moments for the Euler Characteristic of a Random Surface

TL;DR

This paper investigates the probabilistic properties of the Euler characteristic of random surfaces formed by gluing polygons, deriving large deviations and moment estimates that support conjectures in combinatorics and topology.

Contribution

It provides new large deviations bounds and moment estimates for the number of cycles in related random permutations, connecting surface topology with permutation group theory.

Findings

Established large deviations bounds for cycle counts

Derived sharp moment estimates for the number of cycles

Confirmed specific conjectures by Pippenger and Schleich

Abstract

We study random surfaces constructed by glueing together $N/k$ filled $k$-gons along their edges, with all $(N-1)!! = (N-1)(N-3)...3\cdot 1$ pairings of the edges being equally likely. (We assume that lcm $\{2,k\}$ divides $N$.) The Euler characteristic of the resulting surface is related to the number of cycles in a certain random permutation of $\{1, ..., N\}$. Gamburd has shown that when 2 lcm $\{2,k\}$ divides $N$, the distribution of this random permutation converges to that of the uniform distribution on the alternating group $A_N$ in the total-variation distance as $N\to\infty$. We obtain large-deviations bounds for the number of cycles that, together with Gamburd's result, allow us to derive sharp estimates for the moments of the number of cycles. These estimates allow us to confirm certain cases of conjectures made by Pippenger and Schleich.