The Thermodynamic Limit in Mean Field Spin Glass Models |
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Authors: | Francesco Guerra Fabio Lucio Toninelli |
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Institution: | (1) Dipartimento di Fisica, Università di Roma ``La Sapienza' and INFN, Sezione di Roma, Piazzale A. Moro 2, 00185 Roma, Italy. E-mail: francesco.guerra@roma1.infn.it, IT;(2) Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy, and Istituto Nazionale di Fisica Nucleare, Sezione di Pisa. E-mail: f.toninelli@sns.it, IT |
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Abstract: | 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 N
1
and N
2
sites, respectively, with N
1
+N
2
=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.
Received: 19 April 2002 / Accepted: 22 April 2002 Published online: 6 August 2002 |
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Keywords: | |
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