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Large Deviations in Disordered Spin Systems

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Large Deviations in Physics

Part of the book series: Lecture Notes in Physics ((LNP,volume 885))

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Abstract

This contribution is an introduction to the use of large deviations to study properties of disordered systems. We present some features of the application of the theory of large deviations to models with random bonds or fields. Proceeding by examples, starting from the mean field Ising model we introduce the notation for the rate function and the cumulant generating function for small and large deviations. By means of the replica theory we analyze sample-to-sample free energy and overlap fluctuations. In particular, we address the random Ising chain, random directed polymers, mean-field spin-glasses both with Ising and spherical spins and the random field Ising model. For pedagogical aims, we put more emphasis on low dimensional systems, where product of random matrices can be employed, leaving out more advanced methods, focusing on the basic ideas behind the application of large deviations.

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Notes

  1. 1.

    The computations differ for two reasons. First, the small perturbative parameter τ is defined in two different ways and \(\tau _{\mathrm{exact}} =\tau _{\mathrm{trunc}} -\tau _{\mathrm{trunc}}^{2}/2\). Second, the truncated model represents the exact one near the critical point but already at the fourth order in τ the free energy behavior differs.

  2. 2.

    The function q(x) is the continuous order parameter characterizing the spin-glass phase in presence of a continuous, that is, an infinite number, of breakings of replica symmetry. It is always zero in the paramagnetic phase and it becomes continuously non zero and x-dependent at the transition point to the spin-glass phase. The variable x, defined in the interval [0, 1], is the so called RSB parameter labeling the RSB’s in the continuous limit [23, 36, 46].

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Roma “Sapienza”, Piazzale Aldo Moro 5, Rome, I-00185, Italy

    Andrea Crisanti

  2. Istituto per i Processi Chimico-Fisici, Consiglio Nazionale delle Ricerche Piazzale Aldo Moro 5, Rome, I-00185, Italy

    Luca Leuzzi

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  1. Andrea Crisanti
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Correspondence to Luca Leuzzi .

Editor information

Editors and Affiliations

  1. Dipartimento di Fisica, Università di Roma Sapienza, Roma, Italy

    Angelo Vulpiani

  2. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Roma, Italy

    Fabio Cecconi

  3. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Roma, Italy

    Massimo Cencini

  4. Istituto Sistemi Complessi, Consiglio Nazionale delle Ricerche, Roma, Italy

    Andrea Puglisi

  5. Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Roma, Italy

    Davide Vergni

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Crisanti, A., Leuzzi, L. (2014). Large Deviations in Disordered Spin Systems. In: Vulpiani, A., Cecconi, F., Cencini, M., Puglisi, A., Vergni, D. (eds) Large Deviations in Physics. Lecture Notes in Physics, vol 885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-54251-0_5

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  • DOI: https://doi.org/10.1007/978-3-642-54251-0_5

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