UBC Probability Seminar: Romain Panis
Topic
The scaling limit of the high-dimensional Ising model is the Gaussian free field
Speakers
Details
In two independent papers, Aizenman and Fröhlich argued that every scaling limit of the critical Ising model in dimensions is trivial (or Gaussian). This qualitative result can be reformulated as follows: the Schwinger functions of any (reasonable) scaling limit satisfy Wick’s law. This means that the scaling limit of the model’s two-point function fully characterizes the limit.
The lace expansion approach was successfully applied by Sakai to derive exact asymptotic of the model’s two-point function in large enough dimensions. This leaves open a gap to fill to identify the scaling limit in the entire mean-field regime of the model.
We use the so-called random current representation of the model, random walk arguments, and the Messager--Miracle-Solé inequalities to derive exact asymptotic of the critical two-point function in dimensions , thus characterizing the scaling limit to be the Gaussian free field in this setting.
Based on joint works with Hugo Duminil-Copin.