Finite-size scaling of the de Almeida-Thouless instability in random sparse networks

We study, in random sparse networks, finite-size scaling of the spin-glass susceptibility χSG, which is a proper measure of the de Almeida-Thouless (AT) instability of spin-glass systems. Using a phenomenological argument regarding the band-edge behavior of the Hessian eigenvalue distribution, we discuss how χSG is evaluated in infinitely large random sparse networks, which are usually identified with Bethe trees, and how it should be corrected in finite systems. In the high-temperature region, data of extensive numerical experiments are generally in good agreement with the theoretical values of χSG determined from the Bethe tree. In the absence of external fields, the data also show a scaling relation χSG = N1/3F (N1/3 | T- Tc |/Tc), which has been conjectured in the literature, where Tc is the critical temperature. In the presence of external fields, on the other hand, the numerical data are not consistent with this scaling relation. A numerical analysis of Hessian eigenvalues implies that strong finite-size corrections of the lower band edge of the eigenvalue distribution, which seem relevant only in the presence of the fields, are a major source of inconsistency. This may be related to the known difficulty in using only numerical methods to detect the AT instability. © 2010 The American Physical Society.