Conformal invariance in two-dimensional percolation

Abstract

The immediate purpose of the paper was neither to review the basic definitions of percolation theory nor to rehearse the general physical notions of universality and renormalization (an important technique to be described in Part Two). It was rather to describe as concretely as possible, although in hypothetical form, the geometric aspects of universality, especially conformal invariance, in the context of percolation, and to present the numerical results that support the hypotheses. On the other hand, one ulterior purpose is to draw the attention of mathematicians to the mathematical problems posed by the physical notions. Some precise basic definitions are necessary simply to orient the reader. Moreover a brief description of scaling and universality on the one hand and of renormalization on the other is also essential in order to establish their physical importance and to clarify their mathematical content.