Organizers
吴昊，杨帆，姜建平，顾陈琳，李文博
Speaker
Eric O. Endo
NYU Shanghai
Time
Thur., 16:00 - 17:00
Dec. 11, 2025
Venue
C548, Shuangqing Complex Building
Contour Expansions for Long-Range Ising models
On the $d$-dimensional lattice $\mathbb{Z}^d$ with $d\ge 2$, the phase transition of the nearest-neighbor ferromagnetic Ising model can be proved by using Peierls argument, that requires a notion of contours, geometric curves on the dual of the lattice to study the spontaneous symmetry-breaking.
The problem of defining contours for the long-range Ising model on the $d$-dimensional square lattice with power decay interaction $J_{xy}=\lVert x-y\lVert^{-\alpha}$ with $\alpha>d$ was first studied by Ginibre, Grossman, and Ruelle in 1966, who proved that the same contours by Peierls can be used for $\alpha>d+1$ to show the symmetry-breaking of the system. Since then, the problem of defining contours for $d<\alpha\le d+1$ has been a big challenge in the community because of the difficulty of estimating the Hamiltonian with long-range interaction.
In this talk, we will define a notion of contours for these models for any $\alpha>d$, solving the conjecture and allowing us to analyze the symmetry-breaking of the system with a geometric approach, consequently studying well-known problems with them, such as the phase transition with non-translation invariant, decaying external fields.
Joint work with Lucas Affonso, Rodrigo Bissacot, and Satoshi Handa.
About the Speaker:
Eric Endo is an Assistant Professor of Practice in Mathematics at NYU Shanghai. Endo holds a double PhD degree, one in Applied Mathematics at the University of São Paulo and one in Mathematics at the University of Groningen. His PhD thesis received the Cum Laude distinction at the University of Groningen. He was a Postdoctoral Instructor at NYU Shanghai until 2021.
Endo's research focuses on the ferromagnetic long-range Ising model and its properties. His recent works has centered on the low-temperature expansions for the $d$-dimensional long-range Ising model, metastate of the model with random boundary conditions, and the local central limit theorem for a sequence of Gibbs measures. He has also contributed to research in probability, thermodynamic formalism, and combinatorics.
