How long for Brownian motion to "fill-out" a torus in d-dimensions?

Question

I've been taken by the concise result¹ that (roughly!), on a $2$-dimensional torus $\mathbb{T}^2$, the time it takes to visit nearly every point (within $\epsilon$, as $\epsilon \to 0$) is: $\frac{2}{\pi}$.

My question is:

Q. Is the same situation known for the $d$-dimensional torus, $\mathbb{T}^d$? What is the time it takes for a Brownian-motion particle to visit within $\epsilon \to 0$ of every point of $\mathbb{T}^d$?

This is probably known—or known to be unknown—so this is just a reference request.

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¹ Dembo, Amir, Yuval Peres, Jay Rosen, and Ofer Zeitouni. "Cover times for Brownian motion and random walks in two dimensions." Annals of Mathematics (2004): 433-464. Annals link.

Answer 1

A very general answer, in dimension $d\geq 3$, is in the following paper of Dembo, Peres and Rosen https://projecteuclid.org/euclid.ejp/1464037588: for compact $d$-dimensional manifolds, $$C_\epsilon(M) /\epsilon^{2-d}\to V(M)$$ where $V(M)$ is the volume.

The $d$ dimensional case for $d\geq 3$ is much simpler than $d=2$ because in the latter, you have long range (=logarithmic) correlations arising from the recurrence of BM.

In passing, let me mention that Belius's paper mentioned in Josiah Park's answer gives a more refined answer, namely a limit law for a centered version of $\sqrt{C_\epsilon}$ (with no scaling), for the random walk case. Similar results can be written for the manifold case in $d\geq 3$. For the critical case of $d=2$, higher precision than the law of large numbers is available but as of this writing, the limit law for $\sqrt{C_\epsilon}-E\sqrt{C_\epsilon}$ has not been derived.

Answer 2

It looks like the $d$-dimensional case is easier generally than the $d=2$ case according to the excerpt below from this paper (see page three).

...the two-dimensional model is also more difficult than its higher-dimensional counterparts. This is because dimension two is critical for the walk, resulting in strong correlations. To illustrate the dimension-based comparison, observe that very fine results are available for d ≥ 3, see e.g. 2 and references therein... In contrast, in two dimensions the first-order asymptotics of the cover time was completed only recently, after a series of intermediate steps over a decade of efforts.

Looking at Aldous and Fill's book (Corollary 7.24) there is an expression for the cover time for $(\mathbb{Z}/n\mathbb{Z})^d$ for $d\geq 3$, $T\sim R_d n \log n$, where

$$R_d=\int_{0}^1\dots\int_{0}^1\frac{1}{\frac{1}{d}\sum_{u=1}^d(1-\cos{2\pi x_u})}dx_1\dots dx_d.$$

In the limit as $n\rightarrow \infty$ with appropriate normalization, it appears this constant $R_d$ is the cover time.

2: D. Belius (2013) Gumbel fluctuations for cover times in the discrete torus. To appear in: Probab. Theory Relat. Fields. Prelininary arXiv abs.