We discuss and disprove a conjecture of Pemantle concerning
vertex-reinforced random walks.
¶ The setting is a general theory of non-Markovian discrete-time
random 4 processes on a finite space $E = {1, \dots, d}$, for which the
transition probabilities at each step are influenced by the proportion of times
each state has been visited. It is shown that, under mild conditions, the
asymptotic behavior of the empirical occupation measure of the process is
precisely related to the asymptotic behavior of some deterministic dynamical
system induced by a vector field on the $d - 1$ unit simplex. In particular,
any minimal attractor of this vector field has a positive probability to be the
limit set of the sequence of empirical occupation measures. These properties
are used to disprove a conjecture and to extend some results due to Pemantle.
Some applications to edge-reinforced random walks are also considered.