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

Jinjiong Yu

TL;DR

This work establishes sharp asymptotics for the $N$-point correlation functions of coalescing heavy-tailed random walks on ${\mathbb Z}$ with jumps in the domain of attraction of an $\alpha$-stable law for $0<\alpha\le 1$. The authors develop refined tools for heavy-tailed walks, including precise transition-probability differences, precise potential-kernel rates, and first-hitting-time tails, and they introduce a rigorous ODE- and time-reversal-based framework to connect the $N$-point function to the one-point density and a non-collision probability. The key contributions are (i) sharp density and transition estimates in the heavy-tailed setting, (ii) a detailed asymptotic for the non-collision probability $p_{\rm NC}(t)$ of $N$ independent walks, (iii) the derivation of sharp asymptotics for the density $\rho_1(t)$, and (iv) complete asymptotics for the $N$-point function $\rho_N({\bf x},t)$ with explicit error terms that depend on $\alpha$ and recurrence behavior. These results extend the understanding of long-range CRW to the heavy-tailed regime and pave the way for potential scaling-limit descriptions and dimension-theoretic interpretations in the $\alpha=1$ barely recurrent setting.

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 $α$-stable distribution with $α\in(0,1]$. 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.

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

TL;DR

This work establishes sharp asymptotics for the -point correlation functions of coalescing heavy-tailed random walks on with jumps in the domain of attraction of an -stable law for . The authors develop refined tools for heavy-tailed walks, including precise transition-probability differences, precise potential-kernel rates, and first-hitting-time tails, and they introduce a rigorous ODE- and time-reversal-based framework to connect the -point function to the one-point density and a non-collision probability. The key contributions are (i) sharp density and transition estimates in the heavy-tailed setting, (ii) a detailed asymptotic for the non-collision probability of independent walks, (iii) the derivation of sharp asymptotics for the density , and (iv) complete asymptotics for the -point function with explicit error terms that depend on and recurrence behavior. These results extend the understanding of long-range CRW to the heavy-tailed regime and pave the way for potential scaling-limit descriptions and dimension-theoretic interpretations in the barely recurrent setting.

Abstract

We study a system of coalescing continuous-time random walks starting from every site on , where the jump increments lie in the domain of attraction of an -stable distribution with . We establish sharp asymptotics for the -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 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.
Paper Structure (17 sections, 19 theorems, 172 equations)

This paper contains 17 sections, 19 theorems, 172 equations.

Table of Contents

  1. Introduction
  2. Model and main result
  3. Proof strategy
  4. Discussion
  5. Heavy-tailed random walks
  6. Transition probability
  7. Potential kernel
  8. First hitting time
  9. First collision time of random walks
  10. Proof of Theorem \ref{['T:noncoa']}
  11. Asymptotic independence
  12. Coalescing heavy-tailed random walks
  13. $N$-point correlation function
  14. Proof of Theorem \ref{['T:den']}
  15. Proof of Theorem \ref{['T:npf']}
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $(\xi_t)_{t\geq0}$ be the coalescing heavy-tailed random walks on ${\mathbb Z}$ with initial configuration $\xi_0\equiv1$. Assume the random walk has jump rate 1 and symmetric jump kernel $p(\cdot)$ of the form jump with $0<\alpha\leq 1$. In the recurrent case with $\alpha=1$ and $l(\infty)=\inf be the $N$-point correlation function. Recall $l(t),\ell(t)$ from svfl, and recall $K$ such that $\

Theorems & Definitions (25)

  • Theorem 1.1: $N$-point correlation function asymptotics
  • Theorem 1.2: Density asymptotics
  • Remark 1.3
  • Theorem 1.4: Non-collision probability asymptotics
  • Proposition 2.1: Transition probability difference
  • Remark 2.2
  • Proposition 2.3: Transition probability difference in time
  • Proposition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • ...and 15 more