Critical behavior of stochastic rumors on the one-dimensional integer lattice

We study rumor-spreading models on the one-dimensional integer lattice, with particular emphasis on the effect of forgetting mechanisms on the long-term behavior of the rumor. We consider the spatial version of the classical Maki--Thompson model together with two variants that incorporate forgetting. Each model is formulated as an interacting particle system and describes the propagation of information through local interactions between neighboring individuals. Our results show that the introduction of forgetting fundamentally changes the behavior of the process. In the classical Maki--Thompson model, the rumor becomes extinct, whereas the two models with forgetting exhibit nontrivial phase transitions between extinction and survival. More precisely, we establish the existence of critical transmission rates that separate regimes in which the rumor dies out almost surely from regimes in which it survives with positive probability. We also investigate the dependence of these critical thresholds on the stifling rate. Combining rigorous arguments with numerical simulations, we prove the existence of the phase transitions, obtain estimates for the critical values, and explore the corresponding phase diagrams. These findings show that forgetting, despite acting as a mechanism that removes spreaders, can qualitatively alter the dynamics of rumor propagation by creating regimes of sustained information spread. Our work provides new insight into the critical behavior of spatial rumor models and raises several questions for future research.