The Bethe lattice is defined as a graph of infinite points each connected to z neighbors (the coordination number) such that no closed loops exist in the geometry

They are related to Cayley trees.

Several results exist for Percolation on these lattices. For instance, their Percolation threshold is $p_c = 1/(z-1)$, for any $z \geq 3$.

See also the chapter on this on the phase transitions book by Sole.