Extinction, Survival and Fluctuations for the Spatial Maki--Thompson Model on Infinite Graphs

Abstract

We study the spatial Maki--Thompson rumor model on infinite, connected graphs of bounded degree. Spreaders transmit the rumor to ignorant neighbors but become stiflers upon contacting non-ignorant neighbors. We prove extinction on Cayley graphs of linear growth, for every \(λ,α>0\) and every initial configuration with finitely many non-ignorant vertices, and establish an explicit extinction criterion on arbitrary bounded-degree graphs for processes started from finitely many spreaders. On Cayley graphs of superlinear growth, we prove survival from a single spreader whenever the ratio of the stifling to the transmission rate lies below an explicit threshold depending only on the maximum degree. On Cayley graphs of polynomial growth of degree \(D\ge2\), we further show that the range has positive lower density with positive probability. Under a stronger condition, macroscopic annuli contain a surface-order number of simultaneously active spreaders for a total duration bounded uniformly away from zero. When \(α>0\), every finite region eventually contains no spreaders, so global survival forces the rumor to move continually into new regions. Under either subcriticality or a sufficiently small stifling rate, we prove central limit theorems for the final stifler density and the total spreader occupation time, together with a functional central limit theorem for the empirical survival function. These results follow from a central limit theorem for stationary stabilizing functionals of i.i.d.fields on polynomial-growth Cayley graphs; the functionals may depend on the field outside the observation set.