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Emergent planarity in two-dimensional Ising models with finite-range interactions. (English) Zbl 1417.82022
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.

##### MSC:
 82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics 82C22 Interacting particle systems in time-dependent statistical mechanics 82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics 58A17 Pfaffian systems
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