We solve the Chapman-Kolmogorov equation and study the exact splitting
probabilities of the general stochastic process which describes polymer
translocation through membrane pores within the broad class of Markov chains.
Transition probabilities which satisfy a specific balance constraint provide a
refinement of the Chuang-Kantor-Kardar relaxation picture of translocation,
allowing us to investigate finite size effects in the evaluation of dynamical
scaling exponents. We find that (i) previous Langevin simulation results can be
recovered only if corrections to the polymer mobility exponent are taken into
account and that (ii) the dynamical scaling exponents have a slow approach to
their predicted asymptotic values as the polymer's length increases. We also
address, along with strong support from additional numerical simulations, a
critical discussion which points in a clear way the viability of the Markov
chain approach put forward in this work.Comment: 17 pages, 5 figure