Conformally covariant probabilities, operator product expansions, and logarithmic correlations in two-dimensional critical percolation

TL;DR

This paper rigorously demonstrates that two-dimensional critical percolation exhibits conformal covariance and logarithmic correlations, providing proofs of logarithmic singularities and operator product expansions in the scaling limit.

Contribution

It offers the first rigorous proof of logarithmic singularities and conformal covariance in the scaling limit of 2D critical percolation, connecting geometric events to CFT features.

Findings

Logarithmic singularities in connection probabilities are proven.

Conformally covariant scaling limits of connectivity events are established.

Asymptotic expansions resembling CFT operator product expansions are derived.

Abstract

The large-scale behavior of two-dimensional critical percolation is expected to be described by a conformal field theory (CFT). Moreover, this putative CFT is believed to be of the logarithmic type, exhibiting logarithmic corrections to the most commonly encountered behavior of CFT correlations. While constructing a full-fledged percolation CFT is still an open problem, in this paper we prove various CFT features of the scaling limit of two-dimensional critical percolation. In particular, we provide the first rigorous proof of the emergence of logarithmic singularities in the scaling limit of connection probabilities. More precisely, we study several connectivity events, including arm-events and the events that a vertex is pivotal or belongs to the percolation backbone, whose probabilities have conformally covariant scaling limits and can be interpreted as CFT correlation functions. For some of these probabilities, we prove asymptotic expansions that can be regarded as CFT operator product expansions (OPEs). Our analysis identifies various logarithmic singularities and explains the geometric mechanism that produces them. In follow-up work, the results of this paper are used to define a percolation energy field and its logarithmic partner.