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Bethe $M$-layer construction for the percolation problem

by Maria Chiara Angelini, Saverio Palazzi, Tommaso Rizzo, Marco Tarzia

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Authors (as registered SciPost users):

Saverio Palazzi

Abstract

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 $n \to 0$. 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.

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