Vibrational thermodynamic instability of recursive networks

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 (d) over bar. 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 (D) over bar > 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.