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THE PROBABILITY THEORY OF 3-MANIFOLDS. (1 of 4)
We investigate the statistical topology of 3-manifolds by posing the following
question:
To what extent is the
topology of a manifold remembered by purely statistical properties
of certain stochastic
processes executed on the manifold? We focus on compact three-
dimensional manifolds
without boundary, and the stochastic process we consider is a
combinatorial,
discrete-time random walk on the manifold. Our model for the random walk is
based on the well-known
fact due to Moise that every compact 3-manifold M can be
triangulated by a
simplicial complex T with finitely many tetrahedra. Choose and fix such
a T. From a
tetrahedron, the walk proceeds across one of the tetrahedron's 4 two-faces to
the adjacent tetrahedron.
The face is chosen uniformly at random.
We focused on the mean commute
time associated to such random walks, i.e. the expected
number of steps that it takes
to go from a tetrahedron a to a tetrahedron b and back to a.
An important point to be
emphasized is that the walking entity itself can, by maintaining a
"flight log'', calculate
approximations to the mean commute times. The results discussed
below can then be used by the
walking entity to obtain information about which manifold it
might be walking in.
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