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.
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Kod dostępu: 276466