English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

The lecture delivered at the \emph{Current Developments in Mathematics}conference (Harvard-MIT, 2021) focused on the recent proof of the Gaussianstructure of the scaling limits of the critical Ising and $\varphi^4$ fields inthe marginal case of four dimensions(joint work with Hugo Duminil-Copin). Thesenotes expand on the background of the question addressed by this result,approaching it from two partly overlapping perspectives: one concerningcritical phenomena in statistical mechanics and the other functional integralsover Euclidean spaces which could serve as a springboard to quantum fieldtheory. We start by recalling some basic results concerning the models'critical behavior in different dimensions. The analysis is framed in themodels' stochastic geometric random current representation. It yields intuitiveexplanations as well as tools for proving a range of dimension dependentresults, including: the emergence in $2D$ of Fermionic degrees of freedom, thenon-gaussianity of the scaling limits in two dimensions, and conversely theemergence of Gaussian behavior in four and higher dimensions. To cover themarginal case of $4D$ the tree diagram bound which has sufficed for higherdimensions needed to be supplemented by a singular correction. Its presence wasestablished through multi-scale analysis in a recent joint work with HDC.

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