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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> states Potts model in the limit of n \to 0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . In this paper, we show that it is possible to recover the \epsilon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>ϵ</mml:mi> </mml:math> -expansion for critical exponents in finite dimension directly using the M <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> -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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>M</mml:mi> </mml:math> -layer method to systems where the underlying field theory is unknown or disputed.