Dimension dependence of critical phenomena in long-range percolation

TL;DR

The article surveys universality and dimension dependence in critical phenomena, focusing on long-range percolation and the associated upper critical dimensions. It reviews the mean-field picture above the critical dimension, the lace expansion, and scaling limits (including super-Brownian motion) for percolation, and then extends the discussion to long-range models where a kernel $J(x,y)$ introduces an effective dimension and Sak's prediction $2-\eta=\alpha$ governs the LR regime. Recent rigorous progress confirms much of the LR theory across LD, HD, and CD regimes, deriving precise two-point, three-point, and volume-tail asymptotics, establishing scaling relations, and proving superprocess limits, with a real-space RG approach complementing traditional lace-expansion methods. Open challenges remain in fully resolving the effectively short-range portion of Sak's picture, achieving epsilon-expansion results in the LR LD regime, and establishing uniqueness and conformal invariance for scaling limits.

Abstract

Statistical mechanical systems at and near their points of phase transition are expected to exhibit rich, fractal-like behaviour that is independent of the small-scale details of the system but depends strongly on the dimension in which the model is defined. Moreover, many models are conjectured to have an upper critical dimension with important quantitative and qualitative differences between critical behaviour at, above, and below the upper critical dimension. For models with long-range interactions, one expects additional transitions between effectively long-range and effectively short-range regimes, with further marginal effects on the boundary of these two regimes, leading to (at least) eight qualitatively distinct forms of critical behaviour in total for each given model. We give a broad overview of these conjectures aimed at a general mathematical audience before surveying the significant recent progress that has been made towards understanding them in the context of long-range percolation.

Figures (3)

• Figure 1.1: Three-dimensional critical phenomena. Left: The five largest clusters in critical site percolation on a three-dimensional box of side-length 2000. Right: A three-dimensional self-avoiding walk with 100,000 steps generated by Ben Wallace (github.com/bencwallace/polymers.jl). Numerically, critical 3d percolation clusters have fractal dimension around $2.52\ldots$xu2014simultaneous while 3d SAWs have fractal dimension around $1.70\ldots$clisby2010accurate.
• Figure 3.1: A schematic illustration of the different conjectured regimes of critical behaviour for long-range models including long-range percolation, self-avoiding walk, Ising models, lattice trees, etc. The upper critical dimension $d_c$ is $6$ for percolation, $4$ for self-avoiding walk and the Ising model, and $8$ for lattice trees. The shape of the curve $\alpha=\alpha_c(d)$ pictured here (in which $\alpha_c(d)$ is slightly larger than $2$ for $d$ slightly below $d_c$) is based on numerical analysis of nearest-neighbour percolation together with Sak's prediction $\alpha_c=2-\eta_\mathrm{SR}$; for the Ising model reflection positivity ensures that $\alpha_c(d)\leq 2$ for all $d\geq 1$.
• Figure 3.2: Sak's predicted value of $2-\eta$ (left) for $d=2$ and its consequences for $\delta$ (center) and $d_f$ (right) assuming the validity of the hyperscaling relations which give $\delta=(d+\alpha)/(d-\alpha)$ and $d_f=(d+\alpha)/2$ for $d/3<\alpha<\alpha_c$. The predicted crossover value $\alpha_c(2)=43/24$ arises as $2-\eta_{\mathrm{SR}}$ with $\eta_{\mathrm{SR}}=5/24$, $\delta_\mathrm{SR}=91/5$, and $d_{f,\mathrm{SR}}=91/48$.

Theorems & Definitions (10)

• Theorem 4.1
• Theorem 4.2
• Theorem 4.3: HD and LR CD critical volume tail
• Theorem 4.4: Superprocess limits
• Definition 4.5: CL
• Theorem 4.6: The location of the effectively long-range regime
• Theorem 4.7: LR critical two-point functions
• Theorem 4.8: LR LD critical cluster volumes
• Theorem 4.9: LR three-point function
• Theorem 4.10: LR scaling relations