Current Developments in Mathematics

Volume 2020

A geometric perspective on the scaling limits of critical Ising and $\varphi^4_d$ models

Pages: 1 – 39



Michael Aizenman (Departments of Physics and Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


The lecture delivered at the Current Developments in Mathematics conference (Harvard-MIT, Jan. 2021) focused on the recent proof of the Gaussian structure of the scaling limits of the critical Ising and $\varphi^4$ fields in the marginal case of four dimensions (joint work with Hugo Duminil-Copin). These notes expand on the background of the question addressed by this result, approaching it from two partly overlapping perspectives: one concerning critical phenomena in statistical mechanics and the other functional integrals over Euclidean spaces which could serve as a springboard to quantum field theory. We start by recalling some basic results concerning the models’ critical behavior in different dimensions. The analysis is framed in the models’ stochastic geometric random current representation. It yields intuitive explanations as well as tools for proving a range of dimension dependent results, including: the emergence in $2D$ of Fermionic degrees of freedom, the non-gaussianity of the scaling limits in two dimensions, and conversely the emergence of Gaussian behavior in four and higher dimensions. To cover the marginal case of 4D the tree diagram bound which has sufficed for higher dimensions needed to be supplemented by a singular correction. Its presence was established through multiscale analysis in the recent work with HDC.

I would like to thank Hugo Duminil-Copin for our fruitful collaboration. It was advanced through mutual visits to Princeton and Geneva University, sponsored by a Princeton–UniGe partnership grant. I am also grateful for the earlier successful collaborations on the subject with R. Graham, D. Barsky, R. Fernandez and S. Warzel, and thank J. Shapiro and S. Warzel for useful comments on the initial draft of this article. The author’s work reported here was supported in parts by NSF grants, and the Weston visiting professorship at the Weizmann Institute. — Michael Aizenman

Published 30 September 2022