Fluctuation limits for long-range percolation graph distance and effective resistance

Determine the fluctuation limits of graph distance and effective resistance in long-range percolation with nearest-neighbor edges always present, distinguishing the regimes d=1 with s>2 and d≥2 with s>2d: establish the expected analogous behavior for d=1, s>2, determine whether fluctuations are Gaussian when s is sufficiently large for d≥2, and resolve whether the limiting process is stable when s is close to 2d.

Background

The paper develops a long-range-percolation coupling to analyze stable fluctuations of graph distance and effective resistance for random-walk traces in dimensions four and five. It proposes extending this framework to long-range percolation itself, including one-dimensional and higher-dimensional parameter regimes.

For one-dimensional long-range percolation with s>2, the authors expect an analogous result. For dimensions d≥2 and s>2d, they predict a different phase diagram: Gaussian fluctuations for sufficiently large s, while near the threshold s=2d they conjecture that the limiting process need not be stable. These predictions are not established in the paper.

References

A similar problem can be considered for the graph distance/effective resistance in long range percolation with nearest-neighbor edges always present for $d=1$ and $s>2$, and we expect an analogous result to hold. On the other hand, for long range percolation with $d\ge 2$ and $s>2d$, our prediction is different. When $s$ is sufficiently large, we expect the fluctuation to converge to a Gaussian. However, when $s$ is close to $2d$, we conjecture that the limiting process is not necessarily stable.

— Graph distance and effective resistance of the random walk trace in four and five dimensions  (2608.23135 - Adhikari et al., 24 Aug 2026) in Section 1, subsection “Open Problems”