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

Using the symmetric group $S_Q$ symmetry of the $Q$-state Potts model, we classify the (scalar) operator content of its underlying field theory in arbitrary dimension. In addition to the usual identity, energy and magnetization operators, we find fields that generalize the $N$-cluster operators well-known in two dimensions, together with their subleading counterparts. We give the explicit form of all these operators -- up to non-universal constants -- both on the lattice and in the continuum limit for the Landau theory. We compute exactly their two- and three-point correlation functions on an arbitrary graph in terms of simple probabilities, and give the general form of these correlation functions in the continuum limit at the critical point. Specializing to integer values of the parameter $Q$, we argue that the analytic continuation of the $S_Q$ symmetry yields logarithmic correlations at the critical point in arbitrary dimension, thus implying a mixing of some scaling fields by the scale transformation generator. All these logarithmic correlation functions are given a clear geometrical meaning, which can be checked in numerical simulations. Several physical examples are discussed, including bond percolation, spanning trees and forests, resistor networks and the Ising model. We also briefly address the generalization of our approach to the $O(n)$ model.Comment: 35 pages, 6 figure

Resurrecting Dead-water Phenomenon

We revisit experimental studies performed by Ekman on dead-water using modern techniques in order to present new insights on this peculiar phenomenon. We extend its description to more general situations such as a three-layer fluid or a linearly stratified fluid in presence of a pycnocline, showing the robustness of dead-water phenomenon. We observe large amplitude nonlinear internal waves which are coupled to the boat dynamics, and we emphasize that the modeling of the wave-induced drag requires more analysis, taking into account nonlinear effects