Critical site percolation on the triangular lattice: From integrability to conformal partition functions

Abstract

Critical site percolation on the triangular lattice is described by the Yang-Baxter solvable dilute $A_2^{(2)}$ loop model with crossing parameter specialized to $\lambda =\frac{\pi }{3}$, corresponding to the contractible loop fugacity $\beta =-2\cos 4\lambda =1$. We study the functional relations satisfied by the commuting transfer matrices of this model and the associated Bethe ansatz equations. The single and double row transfer matrices are respectively endowed with strip and periodic boundary conditions, and are elements of the ordinary and periodic dilute Temperley-Lieb algebras. The standard modules for these algebras are labeled by the number of defects $d$ and, in the latter case, also by the twist $e^{i\gamma }$. Nonlinear integral equation techniques are used to analytically solve the Bethe ansatz functional equations in the scaling limit for the central charge $c=0$ and conformal weights $\Delta ,\bar{\Delta }$. For the groundstates, we find $\Delta ={\Delta }_{1,d+1}$ for strip boundary conditions and $(\Delta ,\bar{\Delta })=({\Delta }_{\gamma /\pi ,d/2},{\Delta }_{\gamma /\pi ,-d/2})$ for periodic boundary conditions, where ${\Delta }_{r,s}=\frac{1}{24}((3r-2s{)}^2-1)$. We give explicit conjectures for the scaling limit of the trace of the transfer matrix in each standard module. For $d\le 8$, these conjectures are supported by numerical solutions of the logarithmic form of the Bethe ansatz equations for the leading $20$ or more conformal eigenenergies. With these conjectures, we apply the Markov traces to obtain the conformal partition functions on the cylinder and torus. These precisely coincide with our previous results for critical bond percolation on the square lattice described by the dense $A_1^{(1)}$ loop model with $\lambda =\frac{\pi }{3}$. The concurrence of all this conformal data provides compelling evidence supporting a strong form of universality between these two stochastic models as logarithmic CFTs.

arXiv commentv2: 84 pages, with a new section 4.4

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