I present some recent results on the non-equilibrium dynamics of a Brownian particle linearly coupled to a fluctuating field at or near criticality, modelling colloidal diffusion in complex liquids. This microrheological model can be applied to study the rich phenomenology of transport in disordered media, such as biological cells or tissues, or spin glasses. In my talk, I will focus on how the dynamical properties of the colloid can be used to infer the critical properties of the surrounding field. In so doing, this provides an experimental protocol to study the critical properties of fields via colloids, which is useful in situations where the field itself is difficult to observe experimentally.

We approach this problem by constructing a non-equilibrium field theory describing the joint stochastic evolution of the colloid and the field as a perturbation theory valid for weak coupling strengths. This allows us to go beyond previously found results for Gaussian fields at equilibrium, and to study self-interacting fields out of equilibrium through systematic perturbative expansion.