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

TL;DR

This paper proves the existence of a tricritical point in the Blume-Capel model on integer lattices of dimension two or higher, using novel combinatorial and probabilistic techniques to analyze phase transitions.

Contribution

It introduces new methods for establishing the tricritical point and phase diagram structure for the Blume-Capel model across all dimensions $d\,\geq\,2$, including a detailed analysis in two dimensions.

Findings

Existence of a tricritical point for all $d\geq 2$

Development of a quadrichotomy for crossing probabilities in 2D

Extension of sharpness techniques to dilute random cluster models

Abstract

We prove the existence of a tricritical point for the Blume-Capel model on $\mathbb{Z}^d$ for every $d\geq 2$. The proof in $d\geq 3$ 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 $d\geq 2$.