Limit law for the maximal local time (frequent points) of planar random walk

Establish the limiting distribution of the centered maximum of the local time of a simple random walk on the wired domain D_N ∪ {ρ}, started at the origin and stopped at the hitting time τ_ρ of ρ; specifically, determine constants \bar{\alpha} > 0 and an a.s.-finite positive random variable \mathfrak{Z}^D such that, for every u ∈ R, P^0( max_{x ∈ D_N} \ell_{\tau_\varrho}(x) ≤ (2/\sqrt{\pi})\,\log N − (1/\sqrt{\pi})\,\log\log N + u ) converges to E\big( exp{− \mathfrak{Z}^D e^{− \bar{\alpha} u}} \big).

Background

The frequent points problem asks for the extreme value statistics of the local time field attained by a two-dimensional random walk before exiting a large planar domain. The leading-order growth of the maximum is known: the maximum local time scales like (4/π)(log N)^2. The conjecture refines this by proposing precise centering and a randomly-shifted Gumbel limit, analogously to the DGFF maximum.

Such a result would parallel the now-established extremal theory for the two-dimensional DGFF and is connected to conjectural descriptions via Brownian multiplicative chaos. It would also clarify the role of geometry (through the admissible domain D) in the law of the random shift \mathfrak{Z}^D.