The stability of long-range order in disordered systems: A generalized Ding-Zhuang argument

TL;DR

This paper introduces a generalized theoretical framework combining Ding-Zhuang and Pirogov-Sinai methods to analyze the stability of long-range order in disordered systems, proving persistence under certain conditions.

Contribution

It develops a systematic scheme extending Peierls methods to disordered systems, enabling rigorous analysis of phase stability in high-dimensional models.

Findings

Proves long-range order persists in systems with weak disorder in dimensions d ≥ 3.

Establishes multiple coexisting Gibbs states under low-temperature and weak-disorder conditions.

Demonstrates the framework's applicability across diverse disordered models.

Abstract

The stability of long-range order against quenched disorder is a central problem in statistical mechanics. This paper develops a generalized framework extending the Ding-Zhuang method and integrated with the Pirogov-Sinai framework, establishing a systematic scheme for studying phase transitions of long-range order in disordered systems. We axiomatize the Ding-Zhuang approach into a theoretical framework consisting of the Peierls condition and a local symmetry condition. For systems in dimensions $d \geq 3$ satisfying these conditions, we prove the persistence of long-range order at low temperatures and under weak disorder, with multiple coexisting distinct Gibbs states. The framework's versatility is demonstrated for diverse models, providing a systematic extension of Peierls methods to disordered systems.