The Green function for waves on the $2$-regular Bethe lattice

In this paper, we compute an explicit analytic expression for the Green function of the wave operator on the $2$-regular lattice called the “Bethe lattice” equipped with its standard metric. In particular, we exhibit a phenomena of abnormal speed of propagation for waves: the effective speed of propagation of energy for large time is $c_{\ast }=2\sqrt{2}\mathrm{/}3<1$, and there exists a true propagation at any speed $c<c_{\ast }$.