Let \((h_t(x,y))_{t\geq 0}\) be the heat semigroup on the homogeneous tree \(\Gamma\) of degree \(q+1\) with \(q\geq 2\) which describes the transitions of the nearest neighbor random walk on \(\Gamma\) in continuous time. It is shown in this paper that for any \(y\in\Gamma\) and any positive function \(r\) with \(r(t)\cdot t^{-1/2}\to\infty\) for \(t\to\infty\), \[ \lim_{t\to\infty}\sum_{x\in\Gamma,|d(x,y)-t(q-1)/(q+1)|\leq r(t)}h_t(x,y)=1, \] where \(d\) denotes the usual distance on \(\Gamma\). A similar result for \(q=1\) is also derived. The proofs given in the paper are similar to those for the corresponding well-known results for Riemannian spaces and, in particular, the hyperbolic spaces.
It should, however, be noticed that the results of this paper (even with additional estimates for the order of convergence) are easy consequences of Berry-Esseen-type central limit theorems for symmetric random walks on homogeneous trees. In the case \(q=1\) this is an absolutely classical result, and for \(q\geq 2\), Berry-Esseen-type results were derived by the reviewer [J. Multivariate Anal. 34, 290-322 (1990; Zbl 0722.60021)] in a more general context and by M. Lindlbauer [J. Comput. Appl. Math. 99, 287-297 (1998; Zbl 0929.60010)].