  We discuss walking behavior in gauge theories and weak first-order phase
transitions in statistical physics. Despite appearing in very different systems
(QCD below the conformal window, the Potts model, deconfined criticality) these
two phenomena both imply approximate scale invariance in a range of energies
and have the same RG interpretation: a flow passing between pairs of fixed
point at complex coupling. We discuss what distinguishes a real theory from a
complex theory and call these fixed points complex CFTs. By using conformal
perturbation theory we show how observables of the walking theory are
computable by perturbing the complex CFTs. This paper discusses the general
mechanism while a companion paper [1] will treat a specific and computable
example: the two-dimensional Q-state Potts model with Q > 4. Concerning walking
in 4d gauge theories, we also comment on the (un)likelihood of the light
pseudo-dilaton, and on non-minimal scenarios of the conformal window
termination.
