Existence of a tricritical point for the Blume-Capel model on ${\mathbb{Z}}^d$

Abstract

We prove the existence of a tricritical point for the Blume-Capel model on ${\mathbb{Z}}^d$ for every $d\ge 2$. The proof in $d\ge 3$ relies on a novel combinatorial mapping to an Ising model on a larger graph, the techniques of Aizenman, Duminil-Copin, and Sidoravicious (Comm. Math. Phys, 2015), and the celebrated infrared bound. In $d=2$, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume-Capel. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J., 2020), which allows us to obtain a fine picture of the phase diagram in $d=2$, including asymptotic behaviour of correlations in all regions. Finally, we show that the techniques used to establish subcritical sharpness for the dilute random cluster model extend to any $d\ge 2$.

56 pages. 4 figures. Accepted version

References in corpus (2)

• Analysis of phase transitions in the mean-field Blume-Emery-Griffiths model
• Critical and tricritical singularities from small-scale Monte Carlo simulations: The Blume-Capel model in two dimensions

Cited by in corpus (1)

• Information-Geometric Signatures from Nonextensivity in the $1$-D Blume-Capel Model