  There have been some interesting recent advances in understanding the notion
of mechanical disorder in structural glasses and the statistical mechanics of
these systems' low-energy excitations. Here we contribute to these advances by
studying a minimal model for structural glasses' elasticity in which the degree
of mechanical disorder -- as characterized by recently introduced dimensionless
quantifiers -- is readily tunable over a very large range. We comprehensively
investigate a number of scaling laws observed for various macro-, meso- and
microscopic elastic properties, and rationalize them using scaling arguments.
Interestingly, we demonstrate that the model features the universal quartic
glassy vibrational density of states as seen in many atomistic and molecular
models of structural glasses formed by cooling a melt. The emergence of this
universal glassy spectrum highlights the role of self-organization (towards
mechanical equilibrium) in its formation, and elucidates why models featuring
structural frustration alone do not feature the same universal glassy spectrum.
Finally, we discuss relations to existing work in the context of
strain-stiffening of elastic networks and of low-energy excitations in
structural glasses, in addition to future research directions.
