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Mathematics Dept., Princeton University
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Ising model on trees and factor of IID
Lingfu Zhang
Ising model on trees and factor of IID
Abstract: I will talk about the Ising model on the d-regular tree (with free boundary conditions). In different disciplines (information theory, theoretical computer science), this is also known as the ‘binary symmetric channel’, or ‘broadcasting problem’ on trees. We consider the problem of whether this model is a factor of IID, which is (1) a classical and strong ergodic property and (2) can be interpreted as local sampling algorithms. Such property turns out to rely on the inverse temperature $\beta$. There are two thresholds $0 < \beta_c < \beta_r$, which are about the uniqueness of the Gibbs measure, and the solvability of the reconstruction problem, respectively. I will explain why the model is a factor of IID when $\beta\le\beta_c$, and why it is not when $\beta>\beta_r$. Then I will talk about my recent work with Danny Nam and Allan Sly, where we show it is a factor of IID in (part of) the intermediate regime, using a construction via a system of stochastic differential equations.
Speaker: Lingfu Zhang, Mathematics