Long time heat diffusion on homogeneous trees

Let ${\mathfrak{X}}$ be a homogeneous tree of degree $q+1$, $q\geq 2$, ${\mathcal{L}}$ the Laplace operator of ${\mathfrak{X}}$ and $h_{t}(x)$ the fundamental solution of the heat equation $(\partial _{t} +{\mathcal{L}}) u=0$ on $\mathfrak{X}$. We show that the heat kernel $h_{t}(x)$ is asymptotically concentrated in an annulus moving to infinity with finite speed $R_{1}=(q-1)/(q+1)$. Asymptotic concentration of heat in the $L^{p}$ norm is also investigated.