an:07066467
Zbl 1417.82022
Aizenman, Michael; Duminil-Copin, Hugo; Tassion, Vincent; Warzel, Simone
Emergent planarity in two-dimensional Ising models with finite-range interactions
EN
Invent. Math. 216, No. 3, 661-743 (2019).
00434017
2019
j
82C20 82C22 82C44 58A17
Ising models; Pfaffian structure; order-disorder correlation
Summary: 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 \({\mathbb {Z}}^2\) 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.