Time: 14:00-15:00, Tuesday, December 10 2024
Venue: E4-233, Yungu Campus
Host: Zhennan Zhou, ITS
Speaker: Eric Endo, NYU Shanghai
Biography：Eric Endo is an Assistant Professor of Practice in Mathematics at NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, specializing in Statistical Mechanics, with emphasis in ferromagnetic Ising models. His research interests lie in long-range interactions, cluster expansions, and random boundary conditions. Endo received his double-degree Ph.D. in Applied Mathematics from the University of São Paulo and Mathematics from the University of Groningen in 2018. After completing his doctorate, he was a post-doctoral fellow at NYU Shanghai until 2021.
Title: Contour Expansions for Long-Range Ising models
Abstract: On the d-dimensional lattice Zd with d ≥ 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.
It is known that the one-dimensional nearest-neighbor ferromagnetic Ising model does not undergo a phase transition at any temperature. On the other hand, if we add a polynomially decaying long-range interaction given by Jxy = |x − y|-α for x, y ∈ Z, the works by Dyson and Fröhlich-Spencer show the phase transition at low temperatures for 1 < α ≤ 2. Moreover, Fröhlich and Spencer defined a notion of contours for α = 2. There are many other works that study and extend the definition of contours for other regions of α.
In this talk, we define contours for d-dimensional long-range Ising model with d ≥ 2, where the interactions are given by Jxy = |x − y|-α where α > d and x, y ∈ Zd. As an application, we add non-homogeneous external fields hx = |x|-γ in the Hamiltonian and give conditions for γ and α so that the model undergoes a phase transition at low temperatures.
Joint work with Lucas Affonso, Rodrigo Bissacot, and Satoshi Handa.
