Dice Question Streamline Icon: https://streamlinehq.com

Limit law for the cover time of wired planar domains

Determine the limiting distribution of the normalized cover time for the simple random walk on the wired domain D_N ∪ {ρ}; specifically, show that there exists an a.s.-finite positive random variable \overline{\mathfrak{Z}}^D such that, for every u ∈ R, P^\varrho( \sqrt{ \tau_{\mathrm{cov}} / \deg(\overline{D}_N) } ≤ (1/\sqrt{\pi})\,\log N − (1/(4\sqrt{\pi}))\,\log\log N + u ) converges to E\big( exp{ − \overline{\mathfrak{Z}}^D e^{ − \sqrt{4\pi}\, u } } \big).

Information Square Streamline Icon: https://streamlinehq.com

Background

The cover time is the time until every vertex is visited. In two dimensions it is of order N2 (log N)2, but finer fluctuation results analogous to the DGFF maximum remain open. The conjecture proposes a precise normalization with centering (log N − (1/4) log log N) and a randomly-shifted Gumbel-type limit.

The normalization involves \deg(\overline{D}_N), the total degree of the wired graph, reflecting the conversion from the boundary-local-time parametrization to natural time. Proving this conjecture would complete the parallel with DGFF extremes and clarify the random multiplicative factor \overline{\mathfrak{Z}}D capturing domain-dependent fluctuations.

References

Conjecture 5.4 For each admissible domain~D, there exists an almost-surely finite and positive random variable~\overline{\eusb Z}D such that \begin{equation} \label{E:5.20} P\varrho\Biggl(\,\sqrt{\frac{\tau_{\text{\rm cov}}{\deg(\overline D_N)}\le \frac1{\sqrt\pi}\log N -\frac1{4\sqrt\pi}\log\log N+u\Biggr) \,\,\underset{N\to\infty}\longrightarrow\,\,E\bigl({-\overline{\eusb Z}D{-\sqrt{4\pi}\, u}\bigr) \end{equation} holds for all~u\in\mathbb R. Here \text{\rm deg}(\overline D_N) is as in (5.15).

— Extremal properties of the random walk local time (2502.09853 - Biskup, 14 Feb 2025) in Conjecture 5.4, Section “Frequent points and cover time scaling” in Appendix “What lies beyond?”