Abstract
We study four-point functions of critical percolation in two dimensions, and more generally of the Potts model. We propose an exact ansatz for the spectrum: an infinite, discrete and non-diagonal combination of representations of the Virasoro algebra. Based on this ansatz, we compute four-point functions using a numerical conformal bootstrap approach. The results agree with Monte-Carlo computations of connectivities of random clusters.
Highlights
Let us consider the random cluster formulation of the Potts model [6]
What do we know on the spectrums S(k) that should correspond to four-point functions such as the connectivities Pσ? First of all, in the limit z1 → z2, the connectivity P0 must reduce to the probability that z2, z3, z4 are in the same cluster
An additional motivation for both ansatze is that for q = 4, these spectrums are known to occur in fourpoint functions of the type of Pσ
Summary
In the limit z1 → z2, the connectivity P0 must reduce to the probability that z2, z3, z4 are in the same cluster It follows that the leading state of the corresponding spectrum, i.e. the state with the lowest total dimension ∆ + ∆ ̄ , again has conformal. We point out that there is no reason to assume that the values of c or of conformal dimensions are real. An additional motivation for both ansatze is that for q = 4, these spectrums are known to occur in fourpoint functions of the type of Pσ Such four-point functions have been computed in the Ashkin–Teller model, of which the four-state Potts model is a special case [15]. 1 2 appear in the partition functions discussed in [19]
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