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.
The main part of Theorem is the continuity of @math around @math , which is the zero temperature asymptotic result for the free energy. This type of problems does not seem to attract much interest in the discrete time setting since in some cases the answers are simple. For instance, consider the (nearest-neighbor) simple random walk model with an i.i.d. random environment with @math . Then it is easy to see that as @math , the free energy is asymptotic to @math times the time constant of the directed last passage percolation. However, if @math is Bernoulli distributed and we send @math , the situation is not so simple. As we mentioned at the beginning, the existence of @math proved in @cite_19 is already highly nontrivial and the continuity as @math remains an open question at the moment.
@cite_19
2039765418
We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random @math -regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for @math the cover time is asymptotic to @math , where @math . In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph @math . For @math independent walks on @math , the cover time @math is asymptotic to @math , where @math is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is @math . If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for @math walks to broadcast a single piece of information to each other is asymptotic to @math as @math . We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of @math explosive particles ( @math even) tends to @math as @math . The case of @math coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to @math . Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with @math predators and @math prey the expected detection time is @math , where @math is the @math th harmonic number.