Bethe M-layer construction for the percolation problem

One way to perform field theory computations for the bond percolation problem is through the Kasteleyn and Fortuin mapping to the n + 1 states Potts model in the limit of n → 0. In this paper, we show that it is possible to recover the ε-expansion for critical exponents in finite dimension directly using the M-layer expansion, without the need to perform any analytical continuation. Moreover, we also show explicitly that the critical exponents for site and bond percolation are the same. This computation provides a reference for applications of the M-layer method to systems where the underlying field theory is unknown or disputed.