Exactly Solvable Model of Random Walks with Stochastic Exchange

We solve exactly the non-equilibrium dynamics of two discrete random walkers moving in channels with transition rates $p \neq q$ that swap positions at a rate $s$. We compute exactly the joint probability distribution $P_{n,m}(t)$ for the walkers, revealing the existence of two dynamical crossovers. The first signals the passage from independent diffusion to a swap-dominated regime where the particles act as identical random walkers swapping positions. The second crossover occurs when both channels become indistinguishable and the walkers move around the same position. Furthermore, we demonstrate the existence of a persistent spatial anisotropy defined by the difference between the second moments of the probability distributions in the two channels. Our results may provide a quantitative framework to understand diverse systems. In biology, it is motivated by motor proteins (kinesin/dynein) exchanging cargo leadership, membrane receptors swapping binding partners, or brain synapses with activity-dependent plasticity. In finance, it models traders with distinct risk profiles swapping positions in limit-order books, or volatility spillover between coupled markets. These diverse systems share a unifying theme: exchange processes mediate macroscopic correlations despite individual heterogeneity.