Mechanism of anomalous-dimension renormalization across the long-range regimes

Establish how the correction \(\eta-(2-\sigma)\) emerges in the LR-WF-A regime of two-dimensional long-range bond percolation with connection probability \(p(r)\propto 1/r^{2+\sigma}\), and how this correction vanishes toward the LR-WF-B regime.

Background

The paper studies two-dimensional bond percolation with algebraically decaying connection probabilities p(r)∝1/r2+σp(r)\propto 1/r^{2+\sigma}. It identifies an LR-WF-A regime for 1<σ≤21<\sigma\le2 and an LR-WF-B regime for 2/3<σ≤12/3<\sigma\le1. In the LR-WF-B regime, mathematical and numerical evidence supports the Gaussian-form relation η=2−σ\eta=2-\sigma, whereas in the upper portion of LR-WF-A the measured anomalous dimension shows a positive deviation δη=η−(2−σ)\delta\eta=\eta-(2-\sigma).

The unresolved issue is to provide a theoretical explanation for the onset of this anomalous-dimension correction in LR-WF-A and for its apparent disappearance as σ\sigma approaches the LR-WF-B side, including the behavior near the proposed boundary at σ=1\sigma=1.

References

A theoretical understanding of how the correction \eta-(2-\sigma) emerges from the LR-WF-A regime, and how it vanishes toward the LR-WF-B side, remains an important open problem.

— Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime  (2608.20750 - Liu et al., 21 Aug 2026) in Discussion, Section 4

As shown in Fig.~\ref{fig:delta_eta}, the data near \sigma=1 can be described by this form, with p\simeq1, although we are unable to obtain an accurate estimate of p.

— Two-dimensional percolation with algebraically decaying interactions II: Critical exponents in the long-range regime  (2608.20750 - Liu et al., 21 Aug 2026) in Results, subsection “Anomalous dimension \(\eta\) in the LR-WF-A and LR-WF-B regimes”