VRRW on complete-like graphs: Almost sure behavior

By a theorem of Volkov (2001) we know that on most graphs with positive probability the linearly vertex-reinforced random walk (VRRW) stays within a finite "trapping" subgraph at all large times. The question of whether this tail behavior occurs with probability one is open in general. In his thesis, Pemantle (1988) proved, via a dynamical system approach, that for a VRRW on any complete graph the asymptotic frequency of visits is uniform over vertices. These techniques do not easily extend even to the setting of complete-like graphs, that is, complete graphs ornamented with finitely many leaves at each vertex. In this work we combine martingale and large deviation techniques to prove that almost surely the VRRW on any such graph spends positive (and equal) proportions of time on each of its nonleaf vertices. This behavior was previously shown to occur only up to event of positive probability (cf. Volkov (2001)). We believe that our approach can be used as a building block in studying related questions on more general graphs. The same set of techniques is used to obtain explicit bounds on the speed of convergence of the empirical occupation measure.