Emergent Planarity in two-dimensional Ising Models with finite-range Interactions

TL;DR

This paper demonstrates that boundary correlations in 2D finite-range Ising models exhibit a Pfaffian structure at criticality, revealing universal fermionic features beyond exactly solvable models through topological and stochastic geometric methods.

Contribution

It provides a new topological explanation for the Pfaffian structure of boundary correlations and proves its emergence at criticality in finite-range 2D Ising models, extending universality insights.

Findings

Pfaffian structure of boundary correlations emerges asymptotically at criticality.

New topological perspective explains order-disorder correlation structures.

Results extend fermionic universality beyond solvable Ising models.

Abstract

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.