Sharp asymptotics for $N$-point correlation functions of coalescing heavy-tailed random walk

TL;DR

This paper derives precise asymptotic formulas for the $N$-point correlation functions of coalescing heavy-tailed random walks on $ Z$, revealing detailed probabilistic behavior of such systems with stable-like jumps.

Contribution

It provides the first sharp asymptotic estimates for correlation functions in coalescing heavy-tailed random walks, including tail estimates and non-collision probabilities.

Findings

Established sharp asymptotics for correlation functions.

Derived refined tail estimates for heavy-tailed random walks.

Analyzed non-collision probabilities for multiple independent walks.

Abstract

We study a system of coalescing continuous-time random walks starting from every site on $\mathbb{Z}$, where the jump increments lie in the domain of attraction of an $\alpha$-stable distribution with $\alpha\in(0,1]$. We establish sharp asymptotics for the $N$-point correlation function of the system. Our analysis relies on two precise tail estimates for the system density, as well as the non-collision probability of $N$ independent random walks with arbitrary fixed initial configurations. In addition, we derive refined estimates for heavy-tailed random walks, which may be of independent interest.