Disorder chaos and multiple valleys in spin glasses
Abstract
We prove that the SherringtonKirkpatrick model of spin glasses is chaotic under small perturbations of the couplings at any temperature in the absence of an external field. The result is proved for two kinds of perturbations: (a) distorting the couplings via OrnsteinUhlenbeck flows, and (b) replacing a small fraction of the couplings by independent copies. We further prove that the SK model exhibits multiple valleys in its energy landscape, in the weak sense that there are many states with nearminimal energy that are mutually nearly orthogonal. We show that the variance of the free energy of the SK model is unusually small at any temperature. (By `unusually small' we mean that it is much smaller than the number of sites; in other words, it beats the classical Gaussian concentration inequality, a phenomenon that we call `superconcentration'.) We prove that the bond overlap in the EdwardsAnderson model of spin glasses is not chaotic under perturbations of the couplings, even large perturbations. Lastly, we obtain sharp lower bounds on the variance of the free energy in the EA model on any bounded degree graph, generalizing a result of Wehr and Aizenman and establishing the absence of superconcentration in this class of models. Our techniques apply for the pspin models and the Random Field Ising Model as well, although we do not work out the details in these cases.
 Publication:

arXiv eprints
 Pub Date:
 July 2009
 DOI:
 10.48550/arXiv.0907.3381
 arXiv:
 arXiv:0907.3381
 Bibcode:
 2009arXiv0907.3381C
 Keywords:

 Mathematics  Probability;
 Mathematical Physics;
 60K35;
 60G15;
 82B44;
 60G60;
 60G70
 EPrint:
 38 pages. Theorem 1.7 is new