Time-Dependent Probability Distribution of Two-Dimensional Moran Walk

In this paper, we introduce and analyze a novel two-dimensional Moran random walk model, where each component evolves by either increasing by one unit or resetting to zero at each time step, with transition probabilities dependent on time. The primary contribution of this work is the derivation of an explicit closed-form expression for the probability distribution of the final maximum altitude, Un=max(Un(1),Un(2)). Additionally, we provide a detailed analysis of the height statistics and cumulative distribution associated with the model. Using probabilistic techniques, we establish the exact distribution of the final altitude and its moments. Furthermore, we conduct numerical simulations to illustrate the behavior of the model for various parameter values. These results offer new insights into the statistical properties of two-dimensional Moran processes and have potential applications in population genetics, nuclear physics, and related fields.