Chaotic size dependence in the Ising model with random boundary conditions

TL;DR

This paper investigates the behavior of the Ising model with random boundary conditions in dimensions 2 to 4 and higher, showing that the system's limit points are primarily the plus and minus states, confirming a conjecture by Newman and Stein.

Contribution

It proves that in higher dimensions, the Ising model with random boundary conditions almost surely converges to plus or minus states, extending understanding of boundary effects in statistical physics.

Findings

Almost sure convergence to plus and minus states in dimensions ≥4.

Similar convergence results for sparse sequences in dimensions 2 and 3.

Supports the concentration of the Newman-Stein metastate on extremal states.

Abstract

We study the nearest-neighbour Ising model with a class of random boundary conditions, chosen from a symmetric i.i.d. distribution. We show for dimensions 4 and higher that almost surely the only limit points for a sequence of increasing cubes are the plus and the minus state. For d=2 and d=3 we prove a similar result for sparse sequences of increasing cubes. This question was raised by Newman and Stein. Our results imply that the Newman-Stein metastate is concentrated on the plus and the minus state.