Bethe $M$-layer construction for the percolation problem

TL;DR

This paper introduces an $M$-layer construction method for percolation, enabling direct epsilon expansion of critical exponents without analytical continuation, and confirms the equivalence of site and bond percolation exponents.

Contribution

It demonstrates the use of the $M$-layer expansion to compute critical exponents in percolation directly in finite dimensions, bypassing the need for a known field theory.

Findings

Successful derivation of epsilon expansion for critical exponents

Explicit proof that site and bond percolation share the same exponents

Provides a new reference method for systems with unknown or disputed field theories

Abstract

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.