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

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.

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.

• Wouldn't it be more accurate to say that the time it takes is $2/ \pi \cdot |\log \epsilon|^2$? Not sure if I am misinterpreting the abstract of their paper. Apr 24 '20 at 23:55
• @HarryRichman: What I see is $$\lim_{\epsilon\to0} \frac{C_\epsilon}{({\ln \epsilon})^2} = \frac{2}{\pi}$$ where $C_\epsilon$ is the time to come w/in $\epsilon$. Apr 24 '20 at 23:58

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.

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.