Universal finite-size scaling in high-dimensional critical phenomena

We present a new unified theory of critical finite-size scaling for lattice statistical mechanical models with periodic boundary conditions above the upper critical dimension. Our theory is based on recent mathematically rigorous results for linear and branched polymers, multi-component spin systems, and percolation. Both short-range and long-range interactions are included. The universal finite-size scaling is inherited from the scaling of the system unwrapped to the infinite lattice. We also present conjectures for universal scaling profiles for the susceptibility and two-point function plateau in a critical window. For free boundary conditions, the universal scaling has been proven to apply at a pseudocritical point for hierarchical spins, and we conjecture that this holds generally.

💡 Research Summary

The paper develops a rigorous, unified theory of finite‑size scaling (FSS) for lattice statistical‑mechanical models above their upper critical dimension when periodic boundary conditions (PBC) are imposed. The central idea is to “unwrap” the finite torus T_R^d (volume V=R^d) into the infinite lattice Z^d by replicating each torus site at positions x+Ru for all integer vectors u. The unwrapped two‑point function Γ_{R,β}(x)=∑{u∈Z^d}G_β(x+Ru) is then compared with the original torus two‑point function G{R,β}(x).

Two hypotheses are required. Hypothesis 1 assumes that in the infinite volume the two‑point function decays as |x|^{-(d‑α)} up to the correlation length ξ(β) and that the susceptibility diverges as χ(β)∼t^{‑γ} with t=(β_c‑β)/β_c. The decay function g(u) controls the crossover beyond ξ(β); for short‑range models g(u)=e^{‑cu}, for long‑range models g(u)=(1+cu)^{‑2α}. Hypothesis 2 provides a comparison inequality: χ(β)^{d_c/2}V^{‑1}Γ_{R,β}(x) ≤ G_{R,β}(x) ≤ Γ_{R,β}(x) uniformly when ξ(β)≥s_2R. The exponent d_c/2 (not d_{c,α}) originates from the topology of certain Feynman diagrams and is common to both short‑ and long‑range systems.

The main result, Theorem 1, states that if d>d_c (or d>d_{c,α}) and the two hypotheses hold, then there exists a constant c_1 such that at a temperature shift t_* = c_1 V^{‑2γ/d_c} (the “critical window”) the torus susceptibility satisfies χ_R(β_*)≈V^{2/d_c} and the torus two‑point function exhibits a plateau: \