Emergent planarity in two-dimensional Ising models with finite-range Interactions

Michael Aizenman, Hugo Duminil-Copin, Vincent Tassion, Simone Warzel

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


The known Pfaffian structure of the boundary spin correlations, and more generally order–disorder correlation functions, is given a new explanation through simple topological considerations within the model’s random current representation. This perspective is then employed in the proof that the Pfaffian structure of boundary correlations emerges asymptotically at criticality in Ising models on Z 2 with finite-range interactions. The analysis is enabled by new results on the stochastic geometry of the corresponding random currents. The proven statement establishes an aspect of universality, seen here in the emergence of fermionic structures in two dimensions beyond the solvable cases.

Original languageEnglish (US)
JournalInventiones Mathematicae
StatePublished - Jan 1 2019

All Science Journal Classification (ASJC) codes

  • General Mathematics


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