Phase Transitions in Long-Range Ising Models and Multiscale Methods
Phase Transitions in Long-Range Ising Models and Multiscale Methods
Classes: Monday, Tuesday, Wednesday, and Thursday 16:00 - 18:00.
Room: B-139
Dates: From January 08 to February 13, 2026.
Objectives:
Introduce the students to multiscale techniques applied to statistiscal mechanics with focus on the long-range Ising models in the lattice.
Summary and goals:
Multiscale methods were introduced in the seminal work of Fröhlich and Spencer during the 1980s in the study of phase transitions in statistical mechanics, solving Kac and Thompson conjecture on phase transitions for unidimensional models, used in the rigorous proof of the BKT transition and also important results on the study of random Schrodinger operators (later simplified by Klein and Dreifus). Due to its technical difficulty, the use of this method to long-range systems were restricted to unidimensional models, after being revisited in many works by Cassandro, Picco, Merola, Pressuti, Rozikov, Ferrari and many others. This course aims to introduce the students to the recent progress made on the use of the multiscale technique to study phase transitions in multidimensional long-range models. We will also discuss applications in the problem of phase transitions for the random field, nonuniform fields and cluster expansions.
Content:
Gibbs measures in finite volume. The Ising model. Peierls argument. Imry-Ma argument. Nonuniform magnetic field. The unidimensional long-range Ising model. The multidimensional long-range Ising model. The Ding-Zhuang strategy on the random field case. Cluster expansions.
Type of Assessment:
The evaluation will be carried out through assignments, seminars, and exams. In the first two weeks of classes, the course instructor will set the number of assignments, seminars, and exams, the dates for the exams, the criteria to be used in the computation of the numeric grade, as well as the criteria for attributing the final letter grade.
Bibliography:
1. Affonso, L.; Multidimensional Contours à la Fröhlich-Spencer and Boundary Conditions for Quantum Spin Systems, Ph.D. Thesis - University of São Paulo, arXiv:2310.07946, (2023).
2. Affonso, L., Bissacot, R., Endo, E. O., and Handa, S.; Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields, Journal of the European Mathematical Society (JEMS), v. 27, no. 4, pp. 1679–1714, (2025).
3. Affonso, L., Bissacot, R., Faria, G. and Welsch, K. ; Phase Transition in Long-Range q-state Models via Contours. Clock and Potts models with Fields. arXiv:2410.01234, (2025).
4. Affonso, L., Bissacot, R., Corsini, H. , Welsch, K.; Phase Transitions on 1d Long-Range Ising Models with Decaying Fields: A Direct Proof via Contours. arXiv:2412.07098, (2025).
5. Affonso, L., Bissacot, R., Maia, J., Rodrigues, J.F., Welsch, K.; Cluster Expansion and Decay of Correlations for Multidimensional Long-Range Ising Models, arXiv:2508.15666 (2025).
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