  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.
