Sharp asymptotics for arm probabilities in critical planar percolation

TL;DR

This paper provides precise asymptotic estimates for arm event probabilities in critical planar percolation, significantly improving previous results and addressing longstanding questions in the field.

Contribution

It derives sharp asymptotics for arm probabilities in critical planar percolation, advancing understanding of critical phenomena on the triangular lattice.

Findings

Sharp estimates for half-plane arm events

Improved asymptotic bounds for planar arm events

Answers to open questions by Schramm (2006)

Abstract

In this work, we consider critical planar site percolation on the triangular lattice and derive sharp estimates on the asymptotics of the probability of half-plane $j$-arm events for $j \geq 1$ and planar (polychromatic) $j$-arm events for $j>1$. These estimates greatly improve previous results and in particular answer (a large part of) a question of Schramm (ICM Proc., 2006).