Crossover exponents, fractal dimensions and logarithms in Landau–Potts field theories

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 ( $$q\rightarrow 0$$ ), and bond percolations ( $$q\rightarrow 1$$ ). 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.