Two rigidity percolation transitions on binary Bethe networks and the intermediate phase in glass

Rigidity Percolation is studied analytically on randomly bonded networks with two types of nodes, respectively with coordination numbers $z_1$ and $z_2$, and with $g_1$ and $g_2$ degrees of freedom each. For certain cases that model chalcogenide glass networks, two transitions, both of first order, are found, with the first transition usually rather weak. The ensuing intermediate pase, although is not isostatic in its entirety, has very low self-stress. Our results suggest a possible mechanism for the appearance of intermediate phases in glass, that does not depend on a self-organization principle.