Continuum Limits for Critical Percolation and Other Stochastic Geometric Models

TL;DR

This paper discusses the scaling limits of critical percolation and related stochastic geometric models, focusing on their construction, properties, and potential conformal invariance, with open questions about measure uniqueness and universality.

Contribution

It introduces a framework for the continuum limit of critical percolation models using a random Web measure, highlighting key regularity conditions and open problems in measure uniqueness.

Findings

Proposes a continuum limit construction for critical percolation models.

Identifies a curve-regularity condition essential for the limit's construction.

Highlights open problems regarding measure uniqueness and universality.

Abstract

The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the scaling limit we keep track of features seen on the macroscopic scale, in situations where the short--distance scale at which the system's basic variables are defined is taken to zero. Among the challenging questions are the construction of the limit, and the explanation of some of the emergent properties, in particular the behavior under conformal maps as discussed in [LPS 94]. A descriptive account of the project, and some related open problems, is found in ref. [A] and in [AB] (joint work with A. Burchard) where tools are developed for establishing a curve--regularity condition which plays a key role in the construction of the limit. The formulation of the scaling limit as a random Web measure permits to formulate the question of uniqueness of measure(s) describing systems of random curves satisfying the conditions of independence, Euclidean invariance, and regularity. The uniqueness question remains open; progress on it could shed light on the purported universality of critical behavior and the apparent conformal invariance of the critical measures. The random Web yields also another perspective on some of the equations of conformal field theory which have appeared in this context, such as the equation proposed by J. Cardy [C].