Finite size scaling of random XORSAT

TL;DR

This paper analyzes the phase transition in random XORSAT problems, showing that the transition window's width scales as the inverse square root of the system size and providing the exact scaling function.

Contribution

It establishes the finite size scaling behavior of the SAT-UNSAT transition in random XORSAT, including the precise scaling function and transition width.

Findings

Transition window width is Θ(n^{-1/2})

Exact scaling function derived for the transition

Sharp threshold at m/n = ρ_k for satisfiability

Abstract

We consider a "configuration model" for random XORSAT which is a random system of $n$ equations over $m$ variables in $\mathbb F_2$. Each equation is of the form $y_1 + y_2 + \cdots + y_k = b$ where $k \geq 3$ is fixed, $y_1, y_2, \cdots$ are variables (not necessarily distinct) and $b \in \mathbb F_2$. The equations are chosen independently and uniformly at random with replacement. It is known \cite{Dubois02, Dietzfelbinger10, pittel2016} that there exists $\rho_k$ such that $m / n = \rho_k$ is a sharp threshold for the satisfiability of this system. In this note we show that for the configuration model, the width of SAT-UNSAT transition window for random $k$-XORSAT is $\Theta(n^{-1/2})$ and also derive the exact scaling function.