The paper by Morris Ang, Gefei Cai, Xin Sun, and Baojun Wu (Ang et al. 2021), first posted online in 2021, has now appeared electronically in the Journal of the American Mathematical Society. The paper proves several exact formulas for conformal loop ensembles. One particularly concrete consequence is a rigorous proof of a remarkable prediction from mathematical physics: for critical site percolation on the triangular grid, a suitably normalized probability that three points belong to the same cluster converges to the universal constant
In this post we discuss critical Bernoulli site percolation on the triangular grid. It is convenient to identify the Euclidean plane with the complex plane , where , and to write Two vertices of are joined by an edge if and only if their Euclidean distance is . For , let be the triangular grid with mesh size .
In Bernoulli site percolation, every vertex is independently declared open with some probability and closed with probability . An open path is a path all of whose vertices are open, and an open cluster is a connected component of the subgraph induced by the open vertices.
For site percolation on the triangular grid, the critical probability is We therefore declare every vertex of open or closed independently, each with probability . At this critical value there is almost surely no infinite open cluster, but large clusters occur with probabilities governed by power laws.
For each , choose a vertex nearest to . If there is more than one nearest vertex, choose one according to any fixed rule. Let be pairwise distinct. For , define and define
As , the lattice distance between any two of the vertices tends to infinity. Consequently, The question is whether these probabilities vanish in a related way.
The normalized three-point connectivity is The form of this normalization is not arbitrary. Heuristically, requiring a prescribed point to belong to a macroscopic cluster contributes one small “one-arm” factor. The probability contains three such factors. Each pairwise probability contains two, and hence the square root of the product of the three pairwise probabilities again contains a total of three. The local small-scale factors should therefore cancel in the quotient.
This heuristic can be made precise. Let be the one-arm probability from scale to scale . Results on the scaling limit of critical percolation imply that the normalized limits and exist and are conformally covariant.
Conformal covariance determines almost all of the dependence of these functions on the marked points. In fact, there are positive constants and such that and Here is the one-arm exponent for critical percolation.
It follows that so all dependence on cancels. This is also natural from the viewpoint of conformal geometry: unlike a configuration of four points, a configuration of three points has no conformally invariant cross-ratio.
Conformal covariance does not, however, determine the number In the language of conformal field theory, this number is a structure constant. Two-point functions determine the normalization of the relevant fields, while the normalized three-point function contains additional information about the theory.
In 2010, Delfino and Viti used methods from conformal field theory to predict an exact value for this structure constant. Their answer is expressed through the imaginary DOZZ formula, an explicit special-function formula related to the three-point functions of Liouville conformal field theory.
Let denote the imaginary DOZZ structure constant, and put The Delfino–Viti prediction is The full definition of involves the Barnes double gamma function and is too long to reproduce here; see (Ang et al. 2021) for the explicit formula.
Ang, Cai, Sun, and Wu proved that this prediction is correct.
Thus, after normalization by the three pairwise connection probabilities, all dependence on the size and shape of the triangle with vertices disappears. The same constant is obtained for an equilateral triangle, a very thin triangle, or three points separated by vastly different distances, provided that the points remain fixed and distinct as .
If the three-point probability satisfied the naive factorization then the limiting constant would be . The actual value is approximately larger. This small but nonzero correction measures a genuine three-way correlation that cannot be recovered from the three pairwise connection probabilities alone.
The scope of (Ang et al. 2021) goes well beyond the percolation constant. The same framework proves a conjecture of Ikhlef, Jacobsen, and Saleur concerning the nesting statistics of conformal loop ensembles and computes another three-point function associated with three points lying on the same loop. The paper therefore provides not only the value of one universal constant, but also a general method for deriving exact formulas for geometric observables in two-dimensional critical models.