The Roles of Random Boundary Conditions in Spin Systems

  title={The Roles of Random Boundary Conditions in Spin Systems},
  author={Eric O. Endo and Aernout van Enter and Arnaud Le Ny},
  journal={Progress in Probability},
Random boundary conditions are one of the simplest realizations of quenched disorder. They have been used as an illustration of various conceptual issues in the theory of disordered spin systems. Here we review some of these results. 

Gibbs Measures for Long-Range Ising Models

This review-type paper is based on a talk given at the conference Etats de la Recherche en Mecanique statistique, which took place at IHP in Paris (December 10-14, 2018). We revisit old results from



The order parameter in a spin glass

Various possible precise definitions of an Edwards-Anderson type of order parameter for an Ising model spin glass are considered, using boundary conditions for a finite system, states of an infinite

Topics in Disordered Systems

This chapter discusses ground States of Disordered Ferromagnets, which are states of highly Disordered Systems and Metastates found in high temperature systems and low temperature systems.

The phase transition in the one-dimensional Ising Model with 1/r2 interaction energy

We prove the existence of a spontaneous magnetization at low temperature for the one-dimensional Ising Model with 1/r2 interaction energy.

Random-Field Instability of the Ordered State of Continuous Symmetry

Phase transitions are considered in systems where the field conjugate to the order parameter is static and random. It is demonstrated that when the order parameter has a continuous symmetry, the

Incoherent boundary conditions and metastates

In this contribution we discuss the role which incoherent boundary conditions can play in the study of phase transitions. This is a question of particular relevance for the analysis of disordered

Weak versus strong uniqueness of Gibbs measures: a regular short-range example

We provide an example of a nearest-neighbour random model on a regular lattice which, almost certainly, has a unique (disordered) Gibbs state for every boundary condition, although by choosing

Condensation of a one-dimensional lattice gas

We consider a one-dimensional lattice gas in the canonical ensemble with interaction energy 1/rα, 1<α≤2. Using an energy-entropy argument we show that the gas condenses at sufficiently low

On the Ising Model with Random Boundary Condition

The infinite-volume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic size-dependence at low temperatures and we prove that

A mean field spin glass with short-range interactions

We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties.

Phase Transitions and Reflection Positivity . I . General Theory and Long Range Lattice Models

We systematize the study of reflection positivity in statistical mechanical models, and thereby two techniques in the theory of phase transitions: the method of infrared bounds and the chessboard