Logarithmic gap on the square torus

Show that endpoint-local monotone allocation on the square torus C_L\square C_L has a gap of order at least log L at diffusive times, and determine whether the tagged-response method can be combined with a Sudakov-type or multiregion anti-concentration principle without assuming Gaussianity or independence.

Background

For the two-dimensional torus C_L\square C_L, the paper's transport-volume argument yields a lower bound of order \sqrt{\log L}. The authors contrast this with the order \log L maximum of the two-dimensional discrete Gaussian free field and with a sharp O(\log L) upper bound known for a different dynamic-averaging process.

The unresolved issue is whether endpoint-local allocation exhibits the larger logarithmic gap scale at diffusive times. The proposed methodological challenge is to obtain joint fluctuation information across many separated regions rather than relying on a single averaged variance budget, without imposing Gaussianity or independence assumptions.