Singular relaxation of a random walk in a box with a Metropolis Monte Carlo dynamics

We study analytically the relaxation eigenmodes of a simple Monte Carlo algorithm, corresponding to a particle in a box which moves by uniform random jumps. Moves outside of the box are rejected. At long times, the system approaches the equilibrium probability density, which is uniform inside the box. We show that the relaxation towards this equilibrium is unusual: for a jump length comparable to the size of the box, the number of relaxation eigenmodes can be surprisingly small, one or two. We provide a complete analytic description of the transition between these two regimes. When only a single relaxation eigenmode is present, a suitable choice of the symmetry of the initial conditions gives a localizing decay to equilibrium. In this case, the deviation from equilibrium concentrates at the edges of the box where the rejection probability is maximal. Finally, in addition to the relaxation analysis of the master equation, we also describe the full eigen-spectrum of the master equation including its sub-leading eigen-modes.