text: We study asymptotic behaviors of the free energy for the directed polymer in random environment. The polymer is allowed to make unbounded jumps and the environment is given by the Bernoulli variables. We first establish the existence and continuity including the negative infinity value of the coupling constant β. Our proof of existence at β=−∞ differs from existing ones in that it avoids the direct use of subadditivity. Secondly, we identify the asymptotics of the free energy at β=−∞ in the limit of the success probability of the Bernoulli variables tending to one. It is described by using the so-called time constant of a certain directed first passage percolation. Our proof relies on a certain continuity property of the time constant, which is of independent interest.
  @cite_25: We study a model of directed polymers in random environment in dimension 1 + d, given by a Brownian motion in a Poissonian potential. We study the effect of the density and the strength of inhomogeneities, respectively the intensity parameter ν of the Poisson field and the temperature inverse β. Our results are: (i) fine information on the phase diagram, with quantitative estimates on the critical curve; (ii) pathwise localization at low temperature and or large density; (iii) complete localization in a favourite corridor for large νβ 2 and bounded β.
  @cite_20: We study the thermodynamics of a continuous model of directed polymers in random environment. The environment is given by a space-time Poisson point process, whereas the polymer is defined in terms of the Brownian motion. We mainly discuss: (i) The normalized partition function, its positivity in the limit which characterizes the phase diagram of the model. (ii) The existence of quenched Lyapunov exponent, its positivity, and its agreement with the annealed Lyapunov exponent; (iii) The longitudinal fluctuation of the free energy, some of its relations with the overlap between replicas and with the transversal fluctuation of the path.
summary: Another continuous time polymer model is Brownian directed polymer in Poissonian environment introduced by Comets--Yoshida @cite_20 . The @math limit was studied in the same paper, as well as @math for @math with a specific choice of the other parameters. It is possible to show by a block argument that the finite volume free energy stays bounded as @math in general but, to the best of our knowledge, the existence of the limit at @math is not known. Later in @cite_25 , the asymptotics as the density of the Poisson point process tends to @math was also studied but only for bounded @math , in contrast to Theorem here.