  For a directed polymer in a random medium lying on an infinite cylinder, that
is in 1+1 dimensions with finite width and periodic boundary conditions on the
transverse direction, the winding number is simply the algebraic number of
turns the polymer does around the cylinder. This paper presents exact
expressions of the fluctuations of this winding number due to, first, the
thermal noise of the system and, second, the different realizations of the
disorder in the medium.
