Electronic Journal of Probability
- Electron. J. Probab.
- Volume 22 (2017), paper no. 88, 19 pp.
No percolation in low temperature spin glass
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.
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
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
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
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