Abstract
In this letter we study the thermodynamic stability problem for a
generic geometrical structure by considering the harmonic vibrational
dynamics of a network of masses and springs. We relate the stability
properties of the network to the recurrence properties of random walks
or, equivalently, to the vibrational spectral dimension 
.
This is an extension of the Peierls theorem for the thermodynamic instability
of low-dimensional crystalline structures,
proving that stability is possible if and only if > 2.
We predict the existence of an instability critical length
on structurally disordered materials. Our results are discussed on the specific
case of a Sierpinki-gasket fractal, which is exactly solvable.