  We compute the crossover exponents of all quadratic and cubic deformations of
critical field theories with permutation symmetry $S_q$ in $d=6-\epsilon$
(Landau-Potts field theories) and $d=4-\epsilon$ (hypertetrahedral models) up
to three loops.We use our results to determine the $\epsilon$-expansion of the
fractal dimension of critical clusters in the most interesting cases, which
include spanning trees and forests ($q\to0$), and bond percolations ($q\to1$).
We also explicitly verify several expected degeneracies in the spectrum of
relevant operators for natural values of $q$ upon analytic continuation, which
are linked to logarithmic corrections of CFT correlators, and use the
$\epsilon$-expansion to determine the universal coefficients of such
logarithms.
