A generalization of Cardy's and Schramm's formulae

TL;DR

The paper develops a discrete, parafermionic observable for critical site percolation on the triangular lattice, unifying Cardy’s crossing formula and Schramm’s surrounding probability through a single conformal-map framework. It proves that the difference between right- and left-facing interface probabilities for two marked points converges, in the scaling limit, to a conformal quantity $\frac{1}{\sqrt{3}}\mathrm{Im}\, g\left(\frac{\psi(u)+\overline{\psi(v)}}{\psi(u)-\overline{\psi(v)}}\right)$, where $\psi$ maps the domain to the upper half-plane and $g$ is an explicit Schwarz-type map to a rhombus. The results are obtained via discrete holomorphicity of the parafermionic observable, boundary-value analysis, and a continuum limit argument, with Cardy and Schramm as special cases. This provides a purely discrete approach to deriving conformally invariant scaling limits and suggests further generalizations to more complex disorder configurations. The work strengthens the connection between lattice observables and conformal mappings, offering new tools for exact computations in critical percolation and related models.

Abstract

We study critical site percolation on the triangular lattice. We find the difference of the probabilities of having a percolation interface to the right and to the left of two given points in the scaling limit. This generalizes both Cardy's and Schramm's formulae. The generalization involves a new interesting discrete analytic observable and an unexpected conformal mapping.

Figures (6)

• Figure 1: The percolation model; see Corollary \ref{['cor-Cardy']}
• Figure 2: Colorings with the interfaces $a^\delta b^\delta$ passing to the left and to the right from given points $u^\delta,v^\delta$, and the probabilities those colorings contribute to. The dashed paths demonstrate that $u^\delta$ and $v^\delta$ are in the same connected component of the union of black sides, which is one of our requirements.
• Figure 3: The conformal mapping $g(z)$ of the plane pierced by two slits onto a lozenge
• Figure 4: A coloring and a loop configuration. See the proof of Lemma \ref{['l-symmetric-difference']}.
• Figure 5: Link patterns and the probabilities they contribute to

Theorems & Definitions (35)

• Theorem 2.1
• Proposition 2.2
• Remark
• Remark
• Remark
• Corollary 2.3: Cardy's formula
• Corollary 2.4: Schramm's formula
• Remark
• Lemma 3.1
• Theorem 3.2: Continuum limit of the parafermionic observable