Macroscopically, the population density function of random walk models can satisfy self-contained dynamics and exhibit unexpected order. Here, we show how a two-species random walk model that includes diffusion, probabilistic absorption of walkers mediated by a potential function, cross-species annihilation, and normalization of the population density can be implemented semianalytically (by propagating the population density directly instead of simulating individual walkers) to yield steady states that precisely match eigenstates of the Hamiltonian with the corresponding potential function. Key parameters of the random walk model, namely, the absorption coefficient and the diffusion displacement variance, are constrained by establishing an explicit correspondence between the model and imaginary-time propagation of the Schrödinger equation for small time steps. To enforce convergence toward a targeted excited state—rather than merely the ground state—we implement a sparse and periodic orthogonalization where components of the population density proportional to lower-energy eigenstates are removed. This adjustment is sufficient to ensure that the population density propagates within the subspace orthogonal to the span of the lower-energy states, while still converging toward the target state. The model is tested in one- and two-dimensional cases, for different potential functions and different energy levels. The model converges to steady states whose energy uncertainty extrapolates to zero in the limit of vanishing time step, and comparison with theoretical eigenfunctions yields error norms ranging from ${10}^{-3}$ to ${10}^{-11}$. Beyond the application to Hamiltonian eigenvalue problems, this work gives insight into the emergence of nontrivial macroscopic observables based on microscopic, stochastic dynamics.

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• Brownian motion
• Random walks
• Numerical approximation & analysis
• Quantum Monte Carlo