Matthias Keller, Daniel Lenz, Florentin Münch, Marcel Schmidt, András Telcs; Note on short-time behavior of semigroups associated to self-adjoint operators. Bull London Math Soc 2016; 48 (6): 935-944. doi: 10.1112/blms/bdw054
We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $$t$$ roughly like $$t^d$$, where $$d$$ is the combinatorial distance. This is very different from the classical Varadhan-type behavior on manifolds. Moreover, this also gives that short-time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.