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Using the symmetric group $S_Q$ symmetry of the $Q$-state Potts model, weclassify the (scalar) operator content of its underlying field theory inarbitrary dimension. In addition to the usual identity, energy andmagnetization operators, we find fields that generalize the $N$-clusteroperators well-known in two dimensions, together with their subleadingcounterparts. We give the explicit form of all these operators -- up tonon-universal constants -- both on the lattice and in the continuum limit forthe Landau theory. We compute exactly their two- and three-point correlationfunctions on an arbitrary graph in terms of simple probabilities, and give thegeneral form of these correlation functions in the continuum limit at thecritical point. Specializing to integer values of the parameter $Q$, we arguethat the analytic continuation of the $S_Q$ symmetry yields logarithmiccorrelations at the critical point in arbitrary dimension, thus implying amixing of some scaling fields by the scale transformation generator. All theselogarithmic correlation functions are given a clear geometrical meaning, whichcan be checked in numerical simulations. Several physical examples arediscussed, including bond percolation, spanning trees and forests, resistornetworks and the Ising model. We also briefly address the generalization of ourapproach to the $O(n)$ model.

Operator content of the critical Potts model in d dimensions and logarithmic correlations