Abstract
We investigate the behavior of the rare fluctuations of the free energy in the p-spin spherical model, evaluating the corresponding rate function via the G\"artner-Ellis theorem. This approach requires the knowledge of the analytic continuation of the disorder-averaged replicated partition function to arbitrary real number of replicas. In zero external magnetic field, we show via a one-step replica symmetry breaking (1RSB) calculation that the rate function is infinite for fluctuations of the free energy above its typical value, corresponding to an anomalous, super-extensive suppression of rare fluctuations. We extend this calculation to non-zero magnetic field, showing that in this case this very large deviation disappears and we try to motivate this finding in light of a geometrical interpretation of the scaled cumulant generating function.
Highlights
We investigate the behavior of the rare fluctuations of the free energy in the p-spin spherical model, evaluating the corresponding rate function via the Gärtner-Ellis theorem
Rivoire [4], Parisi and Rizzo [5,6,7,8], and others [9,10,11] followed this line of thought, providing a bridge between spin glasses and the theory of large deviations, which deals with rare events whose probability decays exponentially in the system size
In zero external magnetic field, we show that the 1RSB calculation at finite k produces a scaled cumulant generating function (SCGF) with a linear behavior below a certain value kc; a nice geometrical interpretation of this, dating back to Kondor’s work on the SK model [18], is discussed
Summary
The theory of disordered systems has been mainly developed to describe the typical behavior of physical observables. From the disordered systems perspective, most of the standard results of spin glass theory obtained within the replica method concern only the very special limit k → 0, since ftyp = f = ψ (0), whereas to obtain the full form of I (x) that describes arbitrary rare fluctuations of the free energy, one needs to work out the SCGF for finite replica index k. This problem is clearly equivalent to determine the full analytical continuation of the averaged replicated partition. In the Appendix, we discuss the details of the geometrical interpretation of the 1RSB ansatz
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