A simple ghost-field proof of the 1-arm exponent for percolation models in high dimensions

Date
2026/02/06 Fri 13:45 - 15:15
Room
3号館552号室
Speaker
Akira Sakai
Affiliation
Hokkaido University
Abstract

We discuss phase transitions and critical behavior for unoriented percolation on $\mathbb{Z}^d$ and oriented percolation (OP) on $\mathbb{Z}_+\times\mathbb{Z}^d$. Recently, significant attention has been focused on the 1-arm exponent $\rho$, which characterizes the power-law decay ($\asymp t^{-\rho}$) of the critical 1-arm probability (the probability of connection to distance/time $t$). Following initial bounds by Sakai (2004), establishing $\rho\le2$ for percolation and $\rho\le1$ for OP, matching lower bounds were proven in high dimensions by Kozma & Nachmias (2011) and van der Hofstad & Holmes (2013), respectively. Most recently, van Engelenburg et al. (2025) derived $\rho=2$ for spread-out percolation, using a remarkably elementary argument based on the entropic bound of Dewan & Muirhead (2023). We propose an alternative, strikingly simple approach to prove $\rho\ge2$ for percolation and $\rho\ge1$ for OP, both in sufficiently high dimensions. The key idea involves comparing the critical 1-arm probability against the connectivity to a "ghost site," with the ghost-field strength $h$ scaled as $1/t^4$ for percolation and $1/t^2$ for OP.