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# General properties of overlap probability distributions in disordered spin systems. Toward Parisi ultrametricity

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### Abstract

For a very general class of probability distributions in disordered Ising spin systems, in the thermodynamical limit, we prove the following property for overlaps among real replicas. Consider the overlaps among s replicas. Add one replica s+1. Then, the overlap q(a,s+1) between one of the first s replicas, let us say a, and the added s+1 is either independent of the former ones, or it is identical to one of the overlaps q(a,b), with b running among the first s replicas, excluding a. Each of these cases has equal probability 1/s.

### Most cited references4

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### Numerical Evidence for Spontaneously Broken Replica Symmetry in 3D Spin Glasses

(1995)
By numerical simulations of the $$3d$$ Ising spin glass we find evidence that spontaneous replica symmetry breaking theory and not the droplet model describes with good accuracy the equilibrium behavior of the system.
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### About the overlap distribution in mean field spin glass models

(2012)
We continue our presentation of mathematically rigorous results about the Sherrington-Kirkpatrick mean field spin glass model. Here we establish some properties of the distribution of overlaps between real replicas. They are in full agreement with the Parisi accepted picture of spontaneous replica symmetry breaking. As a byproduct, we show that the selfaveraging of the Edwards-Anderson fluctuating order parameter, with respect to the external quenched noise, implies that the overlap distribution is given by the Sherrington-Kirkpatrick replica symmetric Ansatz. This extends previous results of Pastur and Shcherbina. Finally, we show how to generalize our results to realistic short range spin glass models.
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### Non-Mean-Field Behavior of Realistic Spin Glasses

(1995)
We provide rigorous proofs which show that the main features of the Parisi solution of the Sherrington-Kirkpatrick spin glass are not valid for more realistic spin glass models in any dimension and at any temperature.
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### Author and article information

###### Journal
24 July 1998
###### Article
10.1088/0305-4470/31/46/006
cond-mat/9807333
LaTeX2e, 11 pages. Submitted to Journal of Physics A: Mathematical and General. Also available at http://rerumnatura.zool.su.se/stefano/ms/ghigu.ps
cond-mat.dis-nn

Theoretical physics