A Universality Result for 2D Percolation Models

This abstract summarizes the talk concerning [1] that was delivered by one of us (L.C.) at the 2006 Oberwolfach meeting Spacial Random Processes and Statistical Mechanics. The starting point of the research – from a certain perspective – is the seminal result of S. Smirnov [4]. In this work, Smirnov showed that the so–called Carleson–Cardy functions, functions related to crossing probabilities in a (conformal) triangle, had a particular scaling limit: They are harmonic functions obeying certain boundary conditions on any triangle. These boundary conditions are asymptotically satisfied in the percolation problem defined on the discretization of any triangle and hence, if some version of harmoniticity or analyticity can be established at the discrete level, conformal invariance follows. Smirnov addressed this problem for the critical site percolation model on the triangular lattice. For this geometry, an approximate version of the Cauchy–Riemann equations among the triple of the Cardy– Carleson functions was demonstrated by exploiting the exact color symmetry of the random hexagon tiling realization of this problem. But the purported “power” of scaling theory for critical models is the notion that all models of a similar type should have the same scaling behavior – universality. Till now, there has been little substantive progress in this direction; the talk concerned a (modest) example where this form of universality was demonstrated to be the case. From our perspective, this work began a while ago with the investigation of bond percolation – and general q–state Potts and random cluster models – on the triangular bond lattice [2]. The relevant ingredient from [2] is a straightforward perspective on the somewhat mysterious duality relation for the bond problem on this lattice. The key is to abandon independence on any single triangle while keeping disjoint up–pointing triangles independent. Thus, as far as connectivity is concerned, there are only five (rather than eight) relevant configurations a single such triangle: full, empty, and three single bond events. Parametrizing the model in terms of the respective probabilities of these events a, e, and s, with a+e+3s = 1, it is not hard to see that the duality/criticality condition is simply a = e. (Much of this had been realized in the physics community, especially in the context of spin systems where the local correlations are represented by three–body interactions. See [5] and the references therein.) The desirable feature of this perspective is that the breakdown maps directly into a modified hexagon tiling of the plane. The “a” and “e” configurations correspond to yellow and blue hexagons while the three “s” configurations correspond to three half yellow and half blue configurations in which the hexagon is split at the midpoints of two opposing edges. It is noted that not all of the possible splits appear, ergo the model does not enjoy full color reversal symmetry. This and other features render the full bond problem too hard, at least for the present, and we have to make certain modifications. The model we treat will be described presently; first let us first fix some parlance. We