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.
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.
@cite_25 @cite_20
2004518210 2087295179
We study the directed polymer model in dimension @math when the environment is heavy-tailed, with a decay exponent @math . We give all possible scaling limits of the model in the weak-coupling regime, i.e., when the inverse temperature temperature @math vanishes as the size of the system n goes to infinity. When @math , we show that all possible transversal fluctuations @math can be achieved by tuning properly @math , allowing to interpolate between all super-diffusive scales. Moreover, we determine the scaling limit of the model, answering a conjecture by Dey and Zygouras [cf:DZ] - we actually identify five different regimes. On the other hand, when @math , we show that there are only two regimes: the transversal fluctuations are either @math or @math . As a key ingredient, we use the Entropy-controlled Last Passage Percolation (E-LPP), introduced in a companion paper [cf:BT_ELPP]. Let @math be an i.i.d. sequence of random variables in @math , @math . We show that, for any function @math , under regularity conditions, [n^ 1 2 (n^ -1 i=1 ^n (X_i) f ^(X_i) - (x) ,dx ) P 0, ] where @math is the classical kernel estimator of the density of @math . This result is striking because it speeds up traditional rates, in root @math , derived from the central limit theorem when @math . Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions @math and @math , the dimension @math and the bandwidth of the kernel estimator @math . Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of @math . As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.