Abstract.
Some questions concerning the calculation of the number of “physical” (metastable) states or complexity of the spherical p-spin spin glass model are reviewed and examined further. Particular attention is focused on the general calculation procedure which is discussed step-by-step.
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Not to be confused with the angular part of m i used in Section 3. We leave it like this because it is the standard notation in literature
Usually this function is written as \(G_{x_{\rm P}} = -B (1-q) + B^2/\lambda + \ln[1/(1-q)+B]\). The two expressions correspond to B = 0 and \(B\not=0\), respectively
All integrals are well defined and no extra conditions must be added
The coefficient in (30) contains the term \(|\partial_q f(q)|_{q=q^*}|\), where \(f(q)\) is the value of the TAP functional at its stationary point and not the TAP functional itself. A simple calculation shows that \(\partial_q f(q)= -(q- q_{\rm a})\,x_{\rm P}(q)/2\beta(1-q)^2\) which is vanishes only for \(q=q_{\rm a} {\mathrm or} x_{\rm P}=0\)
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By setting \(ix=-B\) one recovers the standard expression in terms of the variable B. We prefer to use x since in this way the integral is over the real axis
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Received: 24 July 2003, Published online: 19 November 2003
PACS:
75.10.Nr Spin-glass and other random models - 02.30.Mv Approximations and expansions
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Crisanti, A., Leuzzi, L. & Rizzo, T. The complexity of the spherical \(\mathsf{p}\)-spin spin glass model, revisited. Eur. Phys. J. B 36, 129–136 (2003). https://doi.org/10.1140/epjb/e2003-00325-x
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DOI: https://doi.org/10.1140/epjb/e2003-00325-x