Tightness of the cover-time lower bound below the connectivity threshold

Determine for which negative functions f(n) satisfying |f(n)|=o(log n) the lower bound (n log n |f(n)|) for the cover time of the giant component of a d-dimensional random geometric graph with r^d=(log n+f(n))/V_d is asymptotically tight.

Background

The paper examines the cover time near the connectivity threshold, parameterized by rd=(log n+f(n))/V_d with |f(n)|=o(log n). When f(n)<0 and |f(n)|=(1), the authors derive a lower bound of order n log n |f(n)| from the presence of many degree-one vertices in the giant component. They explicitly leave unresolved whether this lower bound gives the correct order for particular choices of f(n).

References

We do not know for which values of $f(n)$ this lower bound is tight: for some values of $f(n)$ a corresponding upper bound could perhaps be obtained by splitting the cubes into certain good cubes where the flow can be nicely dissipated and bad cubes where not, and combining them in an optimal way.

— On the jump of the cover time in random geometric graphs  (2501.02433 - Martinez et al., 5 Jan 2025) in Section 6, Concluding remarks and future work