Bethe $M$-layer construction for the percolation problem

The major difference between percolation and other phase transition models is the absence of an Hamiltonian and of a partition function. For this reason it is not straightforward to identify the corresponding field theory to be used as starting point of Renormalization Group computations. Indeed, it could be identified with the field theory of $n+1$ states Potts model in the limit of $n \to 0$ only by means of the mapping discovered by Kasteleyn and Fortuin for bond percolation. In this paper we show that it is possible to recover the epsilon 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.