When: Sunday, October 22, 10am
Where: Schreiber 309
Speaker: Asaf Nachmias, Tel Aviv University
Title: The uniform spanning tree of dense graphs
Let G be a connected graph in which almost all vertices have linear
degrees and let T be a uniform spanning tree of G. For any fixed
rooted tree F of height r>0 we compute the asymptotic density of
vertices v for which the r-ball around v in T is isomorphic to F.
As an application, we prove that in such dense graphs, with high
probability, the density of leaves in a uniform spanning tree is
at least e-1-o(1), the density of vertices of degree 2 is
at most e-1+o(1), and the density of vertices of degree k>2 is
at most ((k-2)/e)k-2/(k-1)!+o(1). These bounds are sharp.
Joint work with Jan Hladky and Tuan Tran.