Universal finite-size scaling in high-dimensional critical phenomena

TL;DR

This work develops a unified finite-size scaling framework for lattice models above their upper critical dimension under periodic boundary conditions by unwrapping finite tori to infinite lattices, yielding a general plateau theorem that ties the scaling window, susceptibility, and two-point plateau to universal exponents governed by the SR upper critical dimension. The authors verify the core hypotheses for long-range self-avoiding walk via the lace expansion and discuss extensions to other models (Ising, $|\varphi|^4$, percolation, branched polymers), establishing model-independent scaling relations and proposing universal profile functions $f(s)$ that interpolate through the critical window. They also address the role of free boundary conditions, conjecturing that the same universal profiles persist around a shifted pseudocritical point, with known proofs for SR Ising and percolation supporting this view. The work connects rigorous high-dimensional FSS with universal profiles, bridging SR and LR physics and providing a rigorous foundation for finite-size effects in a broad class of critical phenomena.

Abstract

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.

Figures (3)

• Figure 1: The unwrapping of a lattice tree from the 2-dimensional discrete torus of period 5 to the infinite lattice.
• Figure 2: $\mathbb{Z}^d$ configurations contributing to $\Gamma_{R,\beta}(x)-G_{R,\beta}(x)$. First line, left to right: SAW, Ising, percolation. Second line: two topologies for BP. Thin lines represent $G_\beta$; the bold line for Ising represents $\Gamma_{R,\beta}$. Hollow vertices are $0$ and $x' = x + Ru$, filled vertices are summed over $\mathbb Z^d$, and box vertices are summed over torus equivalent, distinct, unordered pairs in $\mathbb{Z}^d$.
• Figure 3: Torus configurations contributing to $\Gamma_{R,\beta}(x)-G_{R,\beta}(x)$. First line, left to right: SAW, Ising, percolation. Second line: three diagrams for BP. Solid lines represent $\Gamma_{R,\beta}$, dashed lines represent $\chi/V$, two dashed lines connected by $\times$ represent $\chi^2/V$. Hollow vertices are $0,x$ and filled vertices are summed over the torus.

Theorems & Definitions (14)

• Theorem 1
• Lemma 2
• Proposition 3
• proof
• proof : Proof of Theorem \ref{['thm:main']}
• proof : Verification of Hypothesis \ref{['ass:G']}.
• proof : Verification of Hypothesis \ref{['ass:Gbar']}.
• Conjecture 4
• Conjecture 5
• Conjecture 6