Asymptotic evolution of acyclic random mappings
Steven Neil Evans, University of California at Berkeley
Tye Lidman, University of California at Berkeley
Abstract
An acyclic mapping from an $n$ element set into itself
is a mapping $varphi$ such that if $varphi^k(x) = x$
for some $k$ and $x$, then $varphi(x) = x$. Equivalently,
$varphi^ell = varphi^{ell+1} = ldots$ for $ell$ sufficiently large.
We investigate the behavior as
$n rightarrow infty$ of a sequence of a Markov chain
on the collection of such mappings. At each step of the chain,
a point in the $n$ element set is chosen uniformly at random
and the current mapping is modified by replacing
the current image of that point by a new one
chosen independently and uniformly at random, conditional
on the resulting mapping being again acyclic.
We can represent an acyclic mapping as a directed graph
(such a graph will be a collection of rooted trees)
and think of these directed graphs as metric
spaces with some extra structure. Informal
calculations indicate that the metric space valued
process associated with the Markov chain should,
after an appropriate time and ``space'' rescaling,
converge as $n rightarrow infty$ to a real tree ($R$-tree)
valued Markov process that is reversible with respect
to a measure induced naturally by the standard reflected
Brownian bridge. Although we don't prove
such a limit theorem, we use
Dirichlet form methods to construct a Markov process
that is Hunt with respect to a suitable Gromov-Hausdorff-like
metric and evolves according to the dynamics
suggested by the heuristic arguments. This process is similar to
one that appears in earlier work by Evans and Winter
as a similarly informal limit of a Markov chain related to the
subtree prune and regraft tree (SPR) rearrangements from
phylogenetics.
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