Dynamics of a tracer trapped in a correlated medium in the presence of a wall

We describe the random motion of a particle immersed in a thermally fluctuating medium and harmonically trapped at a certain distance from a wall. The medium, modeled by a Gaussian field with a tunable correlation length (Formula presented) (Formula presented), is linearly coupled to the particle and evolves according to dissipative relaxational dynamics. Dirichlet boundary conditions imposed on the field at the wall give rise to a repulsive fluctuation-induced force acting on the particle, causing a shift in its average position and a renormalization of the strength of the harmonic trap. We describe the effective overdamped dynamics of the particle, which features a nonlinear memory term depending on the wall-particle separation. We show that the two-time correlation function of the particle position features a memory-induced term that depends on the distance of the particle from the wall. At the critical point, this term decays algebraically upon increasing time and it displays a crossover from the behavior observed in the bulk to that corresponding to having the particle at the wall. Our results establish a concrete connection between boundary-induced modifications of critical fluctuations and the non-equilibrium, memory-dependent dynamics of a trapped particle.