Strong disorder for a certain class of directed polymers in a random environment

We study a model of directed polymers in a random environment with a positive recurrent Markov chain, taking values in a countable space epsilon. The random environment is a family (g(i,x), i >= 1, x is an element of Sigma ) of independent and identically distributed real-valued variables. The asymptotic behaviour of the normalized partition function is characterized: when the common law of the g(.,.) is infinitely divisible and the Markov chain is exponentially recurrent we prove that the normalized partition function converges exponentially fast towards zero at all temperatures.