Discrete Random Walks, DRW'03
Cyril Banderier and Christian Krattenthaler (eds.)
DMTCS Conference Volume AC (2003), pp. 259264
author:  Saibal Mitra and Bernard Nienhuis 

title:  Osculating Random Walks on Cylinders 
keywords: 
Random walks,
O(n)
loop model

abstract: 
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

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reference:  Saibal Mitra and Bernard Nienhuis (2003), Osculating Random Walks on Cylinders, in Discrete Random Walks, DRW'03, Cyril Banderier and Christian Krattenthaler (eds.), Discrete Mathematics and Theoretical Computer Science Proceedings AC, pp. 259264 
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