Electronic Journal of Probability

No percolation in low temperature spin glass

Noam Berger and Ran J. Tessler

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We consider the Edwards-Anderson Ising Spin Glass model for temperatures $T\geq 0.$ We define notions of Boltzmann-Gibbs measure for the Edwards-Anderson spin glass at a given temperature, and of unsatisfied (frustrated) edges, and recall the notion of ground states. We prove that for low positive temperatures, in almost every spin configuration the graph formed by the unsatisfied edges is made of finite connected components. Similarly, for zero temperature, we show that in almost every ground state the graph of unsatisfied edges is a forest all of whose components are finite. In other words, for low enough temperatures the unsatisfied edges do not percolate.

Article information

Electron. J. Probab., Volume 22 (2017), paper no. 88, 19 pp.

Received: 8 June 2017
Accepted: 4 September 2017
First available in Project Euclid: 18 October 2017

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82D40: Magnetic materials

Edwards Anderson spin glass percolation

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Berger, Noam; Tessler, Ran J. No percolation in low temperature spin glass. Electron. J. Probab. 22 (2017), paper no. 88, 19 pp. doi:10.1214/17-EJP103. https://projecteuclid.org/euclid.ejp/1508292259

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