Probability Seminar
Topic #1 (Aru): A few recent stories about the Gaussian free field; Topic #2 (Ding): Long range order for three-dimensional random field Ising model
Aru Abstract: In the first story we wonder about the ubiquity of the free field and look at a few characterisation theorems. In the second story we discuss the mutually benefiting relationship between the discrete free field and the O(N) spin model. Finally, in a third story we see how twisted/orbifold free fields, CLE_4 and Brownian loop soup come together to help describe correlation functions of the Ashkin-Teller model.
Ding Abstract: Consider the Ising model in three and higher dimensions with i.i.d. mean-zero Gaussian external field. A classic result of Imbrie (1985) and Bricmont-Kupiainen (1988) proved that at very low temperature long range order exists as long as the variance of the external field is small enough. In this talk, I will present a couple of recent results: (1) A new and simple proof (with Zijie Zhuang) for the result of Imbrie and Bricmont-Kupiainen. The proof combines an extension of the Peierls argument with an ingredient from Chalker (1983) and Fisher-Fröhlich-Spencer (1984). (2) A result (with Yu Liu and Aoteng Xia) that takes (1) as a starting point and extends it to the entire low temperature regime. In addition, I will mention a new correlation inequality (with Jian Song and Rongfeng Sun) for the Ising model which implies exponential decay in the entire high temperature regime.