Equality of the critical inverse temperatures for the one- and two-sided Dyson models

TL;DR

This paper proves that the critical inverse temperatures for one- and two-sided Dyson models are equal for interaction powers between 1 and 2, and conjectures this equality extends to the case when the power equals 2.

Contribution

The paper establishes the equality of critical inverse temperatures for one- and two-sided Dyson models within a specific parameter range and proposes a conjecture for the boundary case.

Findings

Critical inverse temperatures are equal for 1<α<2.

Conjecture that equality holds at α=2.

Provides theoretical proof for the specified range.

Abstract

We prove that the critical inverse temperatures $\beta_c^{\mathbb N}(\alpha)$ and $\beta_c^{\mathbb Z}(\alpha)$ for the one- and two-sided Dyson models are the same when the power of the interaction strength $\alpha$ satisfies $1<\alpha<2$. We conjecture that this is true also in the remaining case of $\alpha=2$.