  Low-connectivity polymer networks were recently found to behave auxetically
when subjected to small tensions, that is, their Poisson's ratio $\nu$ becomes
negative. In addition, for specific state points, numerical simulations
revealed that diamond-like networks reach the limit of mechanical stability,
exhibiting values of $\nu = -1$, a condition that we define hyper-auxeticity.
This behavior is interesting per se for its consequences in material science,
but also appealing for fundamental physics because the mechanical instability
is accompanied by evidences of criticality. In this work, we deepen our
understanding of this phenomenon by performing a large set of equilibrium and
stress-strain simulations in combination with a phenomenological elasticity
theory. The two approaches are found to be in good agreement, confirming the
above results. We also extend our investigations to disordered polymer networks
and find that the hyper-auxetic behavior also holds in this case, still
manifesting a similar critical-like behavior as in the diamond one. Finally, we
highlight the role of the number density, that is found to be a relevant
control parameter determining the elastic properties of the disordered system.
The validity of the results in disordered conditions paves the way to an
experimental investigation of this phenomenon in real systems, such as
hydrogels.
