High-order behavior inφ3field theories and the percolation problem

The percolation problem is normally described in field theory by the $n\to 0$ limit of the $n$-component Potts model. This model has trilinear interactions which give rise to an $\epsilon$ expansion in $6-\epsilon$ dimensions. In contrast to positive values of $n$, the $n=0\mathrm{}$ case is shown to have oscillatory growth at high orders in $\epsilon$ which permits Padé-Borel resummation. A new formulation of the $n=0\mathrm{}$ limit is required to obtain this result. An improved estimate is given for the critical exponent $\omega$ which describes corrections to scaling.