Equality of the critical inverse temperatures for the one- and two-sided Dyson models
Noam Berger, Anders Johansson, Anders Öberg
TL;DR
The paper addresses whether the critical inverse temperatures for the one-sided and two-sided Dyson long-range Ising models coincide in the regime 1<α<2. It employs a random-cluster representation, a block-renormalization scheme, and a sprinkling argument to propagate percolation from the two-sided to the one-sided model, leveraging stochastic domination and Erdős–Rényi-type bounds. The main result is the exact equality β_c^N(α)=β_c^Z(α) for 1<α<2, reinforcing boundary-condition independence in this regime and providing groundwork toward the α=2 case. The work integrates transfer-operator perspectives with percolation theory and builds on prior bounds and renormalization ideas to establish the claimed equality through a detailed probabilistic coupling and hierarchical analysis.
Abstract
We prove that the critical inverse temperatures $β_c^{\mathbb N}(α)$ and $β_c^{\mathbb Z}(α)$ for the one- and two-sided Dyson models are the same when the power of the interaction strength $α$ satisfies $1<α<2$. We conjecture that this is true also in the remaining case of $α=2$.