Special Probability Seminar
Monday, September 23, 2019 at 3:00pm to 4:00pm
Room 2 - 139
Remco Van der Hofstad (Eindhoiven University of Technology)
Title : "Ising models on random graphs"
Abstract:
The Ising model is one of the simplest statistical mechanics models that displays a phase transition. While invented by Ising and Lenz to model magnetism, for which the Ising model lives on regular lattices, it is now widely used for other real-world applications as a model for cooperative behavior and consensus between people. As such, it is natural to consider the Ising model on complex networks. Since complex networks are modelled using random graphs, this leads us to study the Ising model on random graphs. In this talk, we discuss some recent results on the stationary distibution of the Ising model on locally tree-like random graphs as well as on its approach to stationarity.
Due to the randomness of the graphs on which the Ising model lives, there are different settings for the Ising model on it. The quenched setting describes the Ising model on the random graph as it is, while the averaged quenched setting takes the expectation w.r.t. the randomness of the graph. As such, it takes the expectation of the Boltzman distribution, which is a ratio of an exponential involving the Hamiltonian, and the partition function. In the annealed setting, the expectation is taken on both sides of the ratio. These different settings each describe different physical realities.
We discuss thermodynamic limit of the Ising model, which can be used to define the phase transition in the Ising model on locally tree-like random graphs, by describing when spontaneous magnetization exists and when not, extending work by Dembo and Montanari. Wegive an explicit expression for the critical value and the critical exponents for the magnetization close to it. We close by discussing recent results about the fast or slow mixing and metastability for annealed and quenched Ising models on random graphs. This talk is based on several joint works with Sander Dommers, Cristian Giardina, Claudio Giberti, Maria Luisa Prioriello, Takashi Kumagai and Hao Can.
- Event Type
- Events By Interest
- Events By Audience
- Department
- Department of Mathematics
- Add to my calendar