We consider a finite-range spin glass model in arbitrary dimension, where the strength of the two-body coupling decays to zero over some distance gamma(-1). We show that, under mild assumptions on the interaction potential, the infinite-volume free energy of the system converges to that of the Sherrington-Kirkpatrick one, in the Kac limit gamma-->0. This could be a first step toward an expansio...

Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 19 February 2020Accepted: 24 August 2020Published online: 07 January 2021KeywordsSherrington--Kirkpatrick, spin glasses, replica symmetry breaking, message passing algorithmsAMS Subject Headings68Q87, 82B44, 60K35Publication DataISSN (print): 0097-5397ISSN (online): 1095-...

We survey the recent mathematical results about aging in certain simple disordered models. We start by the Bouchaud trap model. We then survey the results obtained for simple models of spin-glass dynamics, like the REM (the Random Energy Model, which is well approximated by the Bouchaud model on the complete graph), then the spherical Sherrington-Kirkpatrick model. We will insist on the differe...

The touching graph of balls is a graph that admits a representation by non-intersecting balls in the space (of prescribed dimension), so that its edges correspond to touching pairs of balls. By a classical result of Koebe [?], the disc touching graphs are exactly the planar graphs. This paper deals with a recognition of unit-ball touching graphs. The 2– dimensional case was proved to be NP-hard...

We study the asymptotic behavior of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove, without any condition on time and temperature, an annealed propagation of chaos result. Extending this result to replicated systems, we conclude that the law of a single spin converges to a non markovian probability measure, in law with respect ...

It was proved by Michel Talagrand in [10] that the Parisi formula for the free energy in the Sherrington-Kirkpatrick model is differentiable with respect to inverse temperature parameter. We present a simpler proof of this result by using approximate solutions in the Parisi formula and give one example of application of differentiability to prove non self-averaging of the overlap outside of the...

Talagrand’s positivity principle, Section 6.6 in [8], states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs’ measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifica...

The expression " spin glasses " was originally coined to describe metallic alloys of a non-magnetic metal with few, randomly substituted, magnetic impurities. Experimental evidences where obtained for a low temperature " spin-glass " phase characterized by a non-periodic freezing of the magnetic moments (the spins) with a very slow, and strongly history dependent, response to external perturbat...