Communications in Mathematical Physics

, Volume 230, Issue 1, pp 71–79

The Thermodynamic Limit in Mean Field Spin Glass Models


  • Francesco Guerra
    • Dipartimento di Fisica, Università di Roma ``La Sapienza'' and INFN, Sezione di Roma, Piazzale A. Moro 2, 00185 Roma, Italy. E-mail:
  • Fabio Lucio Toninelli
    • Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa. E-mail:

DOI: 10.1007/s00220-002-0699-y

Cite this article as:
Guerra, F. & Toninelli, F. Commun. Math. Phys. (2002) 230: 71. doi:10.1007/s00220-002-0699-y


 We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the Sherrington-Kirkpatrick model, and the Derrida p-spin model. The main argument is based on a smooth interpolation between a large system, made of N spin sites, and two similar but independent subsystems, made of N1 and N2 sites, respectively, with N1+N2=N. The quenched average of the free energy turns out to be subadditive with respect to the size of the system. This gives immediately convergence of the free energy per site, in the infinite volume limit. Moreover, a simple argument, based on concentration of measure, gives the almost sure convergence, with respect to the external noise. Similar results hold also for the ground state energy per site.

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© Springer-Verlag Berlin Heidelberg 2002