The scaling limit of random walk and the intrinsic metric on planar critical percolation

TL;DR

This paper proves that the random walk and intrinsic metric on critical percolation clusters in 2D converge to continuous limits related to CLE6, establishing key exponents and relations.

Contribution

It establishes the scaling limits of random walk and intrinsic metric on critical percolation clusters, linking them to CLE6 and deriving fundamental exponents.

Findings

Random walk converges to CLE6 Brownian motion.

Intrinsic metric converges to CLE6 geodesic metric.

Chemical distance, resistance, and spectral dimension exponents are identified.

Abstract

We consider critical site percolation ($p=p_c=1/2$) on the triangular lattice $\mathbf{T}$ in two dimensions. We show that the simple random walk on the clusters of open vertices converges in the scaling limit to a continuous diffusion which lives in the gasket of a conformal loop ensemble with parameter $\kappa = 6$ $\big(\mathrm{CLE}_6\big)$, the so-called $\mathrm{CLE}_6$ Brownian motion. We also show that the intrinsic (i.e., chemical distance) metric converges in the scaling limit to the geodesic $\mathrm{CLE}_6$ metric. As a consequence, we deduce the existence of the chemical distance exponent, the resistance exponent, and the spectral dimension of the critical percolation clusters. Moreover, we show that the exponents satisfy the Einstein relations.