Mean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions

TL;DR

This paper proves that in high-dimensional ferromagnetic Ising models, the 1-arm critical exponent is bounded above by 1, confirming mean-field behavior under certain assumptions and improving previous bounds.

Contribution

The paper establishes a mean-field bound on the 1-arm exponent for high-dimensional Ising models using the random-current representation and the assumption ta=0, extending results to all dimensions greater than 4.

Findings

The 1-arm exponent ta is bounded above by 1 in dimensions > 4.

The result confirms mean-field behavior for the 1-arm exponent in high dimensions.

The bound improves upon previous hyperscaling inequality bounds.

Abstract

The 1-arm exponent $\rho$ for the ferromagnetic Ising model on $\mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Suppose that the spin-spin coupling $J$ is translation-invariant, $\mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $\eta=0$, we show that the optimal mean-field bound $\rho\le1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.