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

TL;DR

This paper derives an explicit formula for the Green function of wave equations on the 2-regular Bethe lattice, revealing an unusual wave propagation speed lower than expected, with waves propagating at any speed below a critical threshold.

Contribution

It provides the first explicit analytic expression for the Green function on the 2-regular Bethe lattice and uncovers a novel phenomenon of abnormal wave propagation speed.

Findings

Effective wave speed c_* = 2√2/3 < 1

Waves propagate at any speed c < c_*

Explicit Green function expression derived

Abstract

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 $c_*=2\sqrt 2/3 <1$, and there exists a true propagation at any speed $c<c_*$.