243 research outputs found
Percolation and coarse conformal uniformization
We formulate conjectures regarding percolation on planar triangulations
suggested by assuming (quasi) invariance under coarse conformal uniformization
On the trace of branching random walks
We study branching random walks on Cayley graphs. A first result is that the
trace of a transient branching random walk on a Cayley graph is a.s. transient
for the simple random walk. In addition, it has a.s. critical percolation
probability less than one and exponential volume growth. The proofs rely on the
fact that the trace induces an invariant percolation on the family tree of the
branching random walk. Furthermore, we prove that the trace is a.s. strongly
recurrent for any (non-trivial) branching random walk. This follows from the
observation that the trace, after appropriate biasing of the root, defines a
unimodular measure. All results are stated in the more general context of
branching random walks on unimodular random graphs.Comment: revised versio
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