Long crossover times in a finite system

We sutdy a stochastic interacting particle system which displays a nonequilibrium transition in its relaxation dynamics in the infinite volume limit. The transition is destroyed by restriction to a finite volume, but its remnants remain until a crossover time $T_c$(L,ε), where L is the system size and ε is the control parameter measuring the distance from the bulk transition. We find that the crossover time $T_c$(L,ε) diverges where ε→0 in a fixed volume. Thus this finite volume system displays arbitrarily long time scales near the transition.