We discuss the asymptotics of the fluctuations of the endpoint of a directed $1+1$ dimensional polymer in a random environment, as the polymer length grows to infinity. We contrast the situation of a spatially periodic environment, with the free space case. We prove a central limit theorem in the first case, while in the second we recover the result of Balazs-Quastel -Seppalainen on the super-diffusive behavior of the endpoint, i.e. that the variance of the position of the endpoint is of order $t^{4/3}$, where $t$ is the length of the polymer. Our proof of the latter result is based entirely on stochastic analysis and simplifies the original argument of Balazs-Quastel-Seppalainen that used a discrete approximation through an exclusion process and the second class particle.
Identyfikator spotkania: 251 152 4038 Kod dostępu: 276466