Abstract
We study, in random sparse networks, finite-size scaling of the spin-glass susceptibility ${\ensuremath{\chi}}_{\text{SG}}$, 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 ${\ensuremath{\chi}}_{\text{SG}}$ 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 ${\ensuremath{\chi}}_{\text{SG}}$ determined from the Bethe tree. In the absence of external fields, the data also show a scaling relation ${\ensuremath{\chi}}_{\text{SG}}={N}^{1/3}F({N}^{1/3}|T\ensuremath{-}{T}_{c}|/{T}_{c})$, which has been conjectured in the literature, where ${T}_{c}$ 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.
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