Universality conjecture for critical exponents of short-range interacting systems

Establish that for statistical mechanics models with short-range interactions (including lattice spin systems such as the Ising model), the critical behavior—and in particular the critical exponents—depend only on the spatial dimension and on the dimension of the local spin space, independently of microscopic details, as predicted by the universality conjecture.

Background

The paper discusses how many materials exhibit critical phenomena that appear to be largely independent of microscopic details, suggesting a universal behavior near critical points. This observation, supported by experiments from the 1970s, led to the universality conjecture, positing that critical exponents depend only on broad features such as spatial dimension and the dimension of the local spin space.

While exact exponents are rigorously known in special cases (e.g., two-dimensional Ising via Onsager and subsequent work) and mean-field behavior is established in high dimensions for certain models using techniques like the lace expansion, a general, rigorous proof of universality across all short-range interacting systems remains out of reach. The paper situates this conjecture within broader themes including scaling, renormalization, and the mathematical challenges of justifying renormalization group heuristics.

The conjecture aims to unify diverse systems under common critical behavior classes, connecting probabilistic and mathematical-physics methods to experimentally observed power laws. A complete resolution would rigorously explain why different short-range models in the same dimension share identical critical exponents.