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.

翻译：我们解析研究了一个简单蒙特卡洛算法的弛豫本征模，该算法对应于一个在盒内通过均匀随机跳跃运动的粒子。超出盒边界的移动被拒绝。在长时间尺度下，系统趋近于平衡概率密度，该密度在盒内是均匀的。我们证明，向这个平衡态的弛豫是非同寻常的：当跳跃长度与盒子尺寸相当时，弛豫本征模的数量可能出奇地少，仅为一个或两个。我们提供了这两种状态之间转变的完整解析描述。当仅存在单个弛豫本征模时，对初始条件对称性的适当选择会产生向平衡态的局域衰减。在这种情况下，对平衡态的偏离集中在盒子边缘（拒绝概率最大的区域）。最后，除了主方程的弛豫分析之外，我们还描述了主方程的完整本征谱，包括其次主导本征模。