  We consider random paths on a square lattice which take a left or a right
turn at every vertex. The possible turns are taken with equal probability,
except at a vertex which has been visited before. In such case the vertex is
left via the unused edge. When the initial edge is reached the path is
considered completed. We also consider families of such paths which together
cover every edge of the lattice once and visit every vertex twice. Because
these paths may touch but not intersect each other and themselves, we call them
osculating walks. The ensemble of such families is also known as the dense
O$(n=1)$ model. We consider in particular such paths in a cylindrical geometry,
with the cylindrical axis parallel with one of the lattice directions. We
formulate a conjecture for the probability that a face of the lattice is
surrounded by $m$ distinct osculating paths. For even system sizes we give a
conjecture for the probability that a path winds round the cylinder. For odd
system sizes we conjecture the probability that a point is visited by a path
spanning the infinite length of the cylinder. Finally we conjecture an
expression for the asymptotics of a binomial determinant.
