Detecting the Energy Spectrum of Bound States with Random Walks*

In the study of complex systems, unexpected macroscopic behaviors often arise from microscopic mechanics. We introduce a simple model consisting of two types of corpuscular particles, 'walkers' and 'antiwalkers', undergoing an unbiased, homogeneous random walk. Walker-antiwalker pairs annihilate when locally brought into contact, and are subject to a survival test, the survival probability of which is determined by the potential energy at their location, following a negative exponential relationship. Analyzing the spatial distribution of survivors (walkers and antiwalkers) and renormalizing it, our system converges to a stationary state resembling the fundamental eigenfunction of the related Hamiltonian. Moreover, when subtracting the common component from a random initial condition with the first (N-1) eigenfunctions of the Hamiltonian, the system reaches a stable state corresponding to the N-th eigenfunction. This discovery enables the development of an eigenfunction detection algorithm that can be linked to Imaginary Time Propagation (ITP). We present analytical and numerical results, including the relationship between random walk variance and result precision. Finally, our model also offers a corpuscular interpretation of ITP.