The discrete Laplace asymptotic method and its application to the 3XOR satisfiability problem

Abstract

A standard way to calculate the asymptotic behavior of integrals of the form \int_Wg(x)e^{-nh(x)}dx is the (continuous) Laplace asymptotic method. However, also discrete sums like \sum_{x\in W\cap\Lambda_n}g_n(x)e^{-nh_n(x)} have similar behavior, when \Lambda_n is a discrete grid which becomes infinitely fine, and the functions g_n and h_n converge to g and h respectively. We go even further, and also derive the asymptotic formula for sums of the form \sum_{x\in W\cap\Lambda_n}S_n(x), where the summand S_n asymptotically behaves as g_ne^{-nh_n}. The motivation, and also an immediate application, will be filling in all details in the classical breakthrough paper of Dubois and Mandler from 2002, which gives the solvability (phase transition) threshold of the 3XOR-SAT problem using the second moment method. Various analytical arguments there were lightly described, but the appendix of this paper combines recent results to fill all of them in. We would expect our theorems on asymptotics to apply to other (especially combinatorial) problems as well. For example, they seem effective on 3XOR-GAME problems.