conformal invariance and the SLE connection
/ Smirnov: smeer-NOFF; Schramm-Loewner: shram-LERV-ner; Cardy: CAR-dee /
At criticality, two-dimensional percolation has no characteristic scale, and physicists long conjectured something stronger than scale invariance: full conformal invariance — invariance not just under rescaling but under all angle-preserving maps of the plane. If true, the continuum limit of critical interfaces should be a canonical random curve determined by conformal symmetry alone. Schramm's insight turned this physical dream into a precise, classifiable mathematical object, and Smirnov's theorem proved the conjecture for one model.
Schramm (2000) observed that any conformally invariant random curve with a natural Markov-type property (the conformal Markov property: the law of the future of the curve, given its past, is the conformal image of the original law) must be the Schramm-Loewner evolution SLE_kappa — a one-parameter family of random curves obtained by driving the Loewner differential equation with a Brownian motion of variance kappa t. The single parameter kappa selects the model. Smirnov (2001) proved that critical SITE percolation on the triangular lattice is conformally invariant in the scaling limit: crossing probabilities of conformal rectangles converge to Cardy's formula (a hypergeometric expression in the conformal cross-ratio), and the exploration-process interface between open and closed converges to SLE_6. From SLE_6 one rigorously derives the 2D critical exponents — beta = 5/36, nu = 4/3, and the rest — because SLE has computable dimensions and multifractal spectra.
The honesty here is sharp and important. Smirnov's theorem is proved ONLY for site percolation on the triangular lattice; conformal invariance and Cardy's formula are still CONJECTURAL for bond percolation on Z^2 and for general lattices, because the proof exploited a special combinatorial identity (Cardy-Carleson contour integrals) that is not known to hold elsewhere — universality of the scaling limit in 2D percolation remains open. Conformal invariance is also strictly a two-dimensional phenomenon: in d = 3 and above there is no conformal-mapping group rich enough to pin down the limit, and SLE has no higher-dimensional analogue. So this beautiful, exact picture is a triumph confined to the plane and, rigorously, to one lattice.
Take a conformal rectangle (a Jordan domain with four marked boundary arcs). Cardy's formula predicts the limiting probability of an open crossing between two opposite arcs as a hypergeometric function 2F1 of the conformal modulus, independent of how the lattice is laid down inside. Smirnov verified this exactly for triangular-lattice site percolation, confirming both the formula and its conformal covariance.
Cardy's formula: a conformally invariant crossing probability, proved for triangular-lattice site percolation.
Conformal invariance and the SLE_6 limit are proved ONLY for site percolation on the triangular lattice; for Z^2 bond percolation they remain conjectural. And the whole picture is intrinsically two-dimensional — there is no SLE or conformal program in d >= 3.