Existence of a tricritical point for the
Blume–Capel model on ℤd

Trishen S. Gunaratnam; Dmitrii Krachun; Christoforos Panagiotis

doi:10.48550/arXiv.2210.13394

Existence of a tricritical point for the Blume–Capel model on ℤd

Trishen S. Gunaratnam, Dmitrii Krachun, Christoforos Panagiotis

Department of Mathematical Sciences

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Abstract

We prove the existence of a tricritical point for the Blume–Capel model on Z ^d for every d ≥ 2. The proof for d ≥ 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. 334:2 (2015), 719–742), and the celebrated infrared bound. For d = 2, the proof relies on a quantitative analysis of crossing probabilities of the dilute random cluster representation of the Blume–Capel model. In particular, we develop a quadrichotomy result in the spirit of Duminil-Copin and Tassion (Moscow Math. J. 20:4 (2020), 711–740), which allows us to obtain a fine picture of the phase diagram for 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 ≥ 2.

Original language	English
Pages (from-to)	785-845
Number of pages	61
Journal	Probability and Mathematical Physics
Volume	5
Issue number	3
Early online date	30 Jun 2024
DOIs	https://doi.org/10.48550/arXiv.2210.13394 https://doi.org/10.2140/pmp.2024.5.785
Publication status	Published - 30 Jun 2024

Funding

Above all we thank Hugo Duminil-Copin for encouraging this collaboration. We thank Hendrik Weber for introducing us to this model. We thank Amol Aggarwal, Roman Koteck\u00FD, Vieri Mastropietro, Romain Panis, S\u00E9bastien Ott, Tom Spencer and Yvan Velenik for exciting discussions at various stages of this project. This project was completed when all three authors were at the University of Geneva. Gunaratnam was supported by the Simons Foundation, Grant 898948, HDC. Krachun and Panagiotis were supported by the Swiss National Science Foundation and the NCCR SwissMAP.

Funders	Funder number
NCCR Catalysis
Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung
Horticultural Development Company
Simons Foundation	898948

Keywords

Blume–Capel model
Ising model
critical phenomena
percolation
tricritical point

ASJC Scopus subject areas

Statistics and Probability
Atomic and Molecular Physics, and Optics
Statistical and Nonlinear Physics

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Cite this

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