The subleading order of two dimensional cover times

Abstract

Abstract. The ε-cover time of the two dimensional torus by Brownian motion is the time it takes for the process to come within distance ε > 0 from any point. Its leading order in the small ε-regime has been established by Dembo, Peres, Rosen and Zeitouni [Ann. of Math., 160 (2004)]. In this work, the second order correction is identified. The approach relies on a multi-scale refinement of the second moment method, and draws on ideas from the study of the extremes of branching Brownian motion.

Two-Dimensional Brownian Random Interlacements

“…it was proved in the seminal paper [14] that α = 1 corresponds to the leading-order term of the expected cover time of the torus, see also [4] for the next leading term and [1] in the discrete case. In the following theorem, we prove that, given that the unit ball is unvisited by the Brownian motion, the law of the uncovered set around 0 at time t α is close to that of BRI(α):…”

“…Proof. This is Proposition 8.10 of [4], with small adaptations, since we are working with torus of size n in the continuous setting as well. , we see that the trace of BRI(α) on B(r) can be obtained in the following way: take N ∼ Poisson(2α ln r) particles and place them on ∂B(r) uniformly and independently; then let these particles perform independent W -diffusions.…”

A simple proof of the DPRZ theorem for 2d cover times

“…We believe the route taken here also considerably streamlines the deep DPRZ-results on late and thin/thick points of BM [2], and, what is perhaps more, it will be useful in the study of the finer properties. In fact, our approach carries over, mutatis mutandis, to these issues as well: when backed with [1], the present notes suggest that in order to address lower order corrections, one "simply" needs to increase the number of scales.…”

“…In particular, we implement the multiscale refinement of the second moment method emerged in the recent studies of Derrida's GREM and branching Brownian motion [5]. This tool brings to the fore the true process of covering [1] with the help of minimal infrastructure only; it also efficiently replaces the delicate tracking of points which DPRZ refer to as 'n-successful', and requires the use of finitely many scales only. All these features simplify substantially the proof of the DPRZ-theorem.…”

A Spectral Characterization for Concentration of the Cover Time

“…This is a topic with rich ties to other objects such as the Gaussian free field [22]. There has been many works providing general bounds on the cover time, studying its evolution and its fluctuations in general [40,20,35,6,30], and in particular for the giant component of various random graphs [16], for trees [2,23], for the two dimensional torus [18,21,15,9] and for higher dimensional tori [8]. Feige [24,25] proved tight extremal upper and lower bounds on cover times of graphs (by SRW).…”