Quasi-isometrically rigid subgroups in right-angled Coxeter groups

In the spirit of peripheral subgroups in relatively hyperbolic groups, we exhibit a simple class of quasi-isometrically rigid subgroups in graph products of finite groups, which we call eccentric subgroups. As an application, we prove that if two right-angled Coxeter groups C(Gamma(1)) and C(Gamma(2)) are quasi-isometric, then for any minsquare subgraph Lambda(1) <= Gamma(1), there exists a minsquare subgraph Lambda(2) <= Gamma(2) such that the right-angled Coxeter groups C(Lambda(1)) and C(Lambda(2)) are quasi-isometric as well. Various examples of non-quasi-isometric groups are deduced. Our arguments are based on a study of nonhyperbolic Morse subgroups in graph products of finite groups. As a by-product, we are able to determine precisely when a right-angled Coxeter group has all its infinite-index Morse subgroups hyperbolic, answering a question of Russell, Spriano and Tran.