Conjectured effect of percolation-cluster tubes on the conditioned Gaussian free field

Investigate the conjectured effect of tube-like regions and associated large-variance maxima in the Gaussian free field on a supercritical Bernoulli percolation cluster when the field is conditioned to remain non-negative over a macroscopic set.

Background

The paper explains that the Gaussian free field on a typical infinite supercritical Bernoulli cluster has substantially larger maxima than the Gaussian free field on the full lattice. This difference is attributed in part to tube-like regions of logarithmic length, where the diagonal Green function and hence the field variance can be large. Such high-variance regions may produce unusually high field values while simultaneously forming narrow pathways that do not improve connectivity of level sets.

The authors point to a related conjectured phenomenon for the Gaussian free field conditioned to stay non-negative over a macroscopic set. The cited conjecture concerns how the same geometric irregularities and competing effects between high maxima and poor connectivity influence the conditioned field; it is not resolved in the present paper.