In the long-range low-dimensional regime of percolation, the cluster volume tail and k-point functions are determined up to constants, yielding the hyperscaling identities delta=(d+alpha)/(d-alpha) and d_f=(d+alpha)/2.
The percolation energy field and its logarithmic partner
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abstract
For site percolation on the triangular lattice, we define two lattice fields that form a logarithmic pair in the sense of conformal field theory. We show that, at the critical point, their two- and three-point correlation functions have well-defined scaling limits, whose structure agrees with that predicted for logarithmic field theories. One of the two fields can be identified with the percolation analog of the Ising energy field, while the other is related to the percolation four-arm event.
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Critical long-range percolation II: Low effective dimension
In the long-range low-dimensional regime of percolation, the cluster volume tail and k-point functions are determined up to constants, yielding the hyperscaling identities delta=(d+alpha)/(d-alpha) and d_f=(d+alpha)/2.