In-law limit for the largest disc covered by the planar Wiener sausage

How quickly does the Wiener sausage of planar Brownian motion fill a disc around its starting point? Meyre and Werner (1994) proved that the radius $R(t)$ of the largest origin-centered disc covered by the radius-1 sausage by time $t$ satisfies, almost surely, $\limsup_{t\to\infty}(\log R(t))^2/(\log t\,\log\log\log t)=c$ for some $c\in[1/8,1]$, and that $\liminf_{t\to\infty}P((\log R(t))^2/\log t>x)\ge e^{-8x}$. Whether $(\log R(t))^2/\log t$ converges in law to a multiple $c'$ of a standard exponential, and the value of $c'$, remains open. Later work resolved the neighboring questions: the a.s. constant is $c=1/4$ (Ben Hough–Peres 2007), and the simple-random-walk analogue converges in law, $(\log R_n)^2/\log n\Rightarrow\mathbf e/4$ (Dembo–Peres–Rosen–Zeitouni 2004, proving a conjecture of Kesten and Révész).

Background

Write $\log_2 t=\log\log t$ and $\log_3 t=\log\log\log t$ for the iterated logarithms, let $\mathbf e$ denote a mean-one exponential random variable, and let $\Rightarrow$ denote convergence in law.

Planar Brownian motion $(B_t)_{t\ge0}$ started at the origin is recurrent: it returns to every neighborhood of the origin infinitely often. Its radius-1 Wiener sausage is the tubular neighborhood $S_t=\bigcup_{0\le s\le t}D(B_s,1)$ of the trajectory up to time $t$, where $D(x,r)$ is the open Euclidean disc of radius $r$ centered at $x$. Recurrence forces $S_t$ to swell and eventually fill the whole plane, so the informative statistics are rates: the area of $S_t$ grows like $2\pi t/\log t$ (Le Gall 1990), and the complementary statistic of the largest disc never met by the trajectory (the 'black holes') was studied by Révész (1992).

This problem concerns the covered-disc radius

$$R(t)=\sup\{r>0:\ D(0,r)\subset S_t\},$$

the radius of the largest disc centered at the origin that the sausage has completely covered by time $t$; $R(t)$ is nondecreasing and tends to infinity almost surely.

The lattice analogue, where $R_n$ is the largest disc about the origin covered after $n$ steps of a planar simple random walk, has a classical history. Erdős and Taylor (1960) studied the fine structure of random-walk paths, including covering radii; Révész (1990) proved almost-sure lower bounds of iterated-logarithm type and the one-sided law bound $\liminf_n P((\log R_n)^2/\log n>z)\ge e^{-120z}$; and around 1990 Révész conjectured that the covering time $T_n$ of the radius-$n$ disc satisfies $\lim_n P(\log T_n\le t(\log n)^2)=e^{-\lambda/t}$ for some unspecified $\lambda>0$, while Kesten (in unpublished work quoted by Lawler (1993)) conjectured the specific value $\lambda=4$, equivalently $(\log R_n)^2/\log n\Rightarrow\mathbf e/4$. Lawler (1993) proved two-sided bounds of this form with the constants $a=2$, $b=4$; Dembo–Peres–Rosen–Zeitouni (2004) proved the full conjecture; and Dembo–Peres–Rosen (2007) further showed that the largest disc covered anywhere (not necessarily centered at the origin) after $n$ steps has radius $n^{1/4+o(1)}$ almost surely.

For the Brownian sausage, Meyre and Werner (1994) proved that almost surely $\limsup_{t\to\infty}(\log R(t))^2/(\log t\log_3 t)=c$ for some $c\in[1/8,1]$, and obtained the one-sided bound $\liminf_{t\to\infty}P((\log R(t))^2/\log t>x)\ge e^{-8x}$. Their analysis rests on excursion asymptotics of Burdzy–Pitman–Yor (1988), where an excursion between two concentric circles is a piece of the path that starts on one circle and runs until it first reaches the other: the number of such excursions completed up to time $t$, normalized by $\log t$, converges in law to $\mathbf e/(2\log 2)$. Ben Hough and Peres (2007) determined the constant $c=1/4$ for both the sausage and the walk, refining Meyre–Werner and answering the walk-side question of Erdős–Taylor (1960).

Problem Statement

Let $(B_t)_{t\ge0}$ be standard planar Brownian motion started at the origin, let $D(x,r)$ denote the open Euclidean disc of radius $r$ centered at $x$, and let $S_t=\bigcup_{0\le s\le t}D(B_s,1)$ be the radius-1 Wiener sausage up to time $t$. Define

$$R(t)=\sup\{r>0:\ D(0,r)\subset S_t\},$$

the radius of the largest disc centered at the origin completely covered by the sausage by time $t$ (an a.s. finite, nondecreasing function with $R(t)\to\infty$ a.s. by recurrence).

(i) Prove or refute the conjecture that there exists a constant $c'\in(0,\infty)$ such that

$$\frac{(\log R(t))^2}{\log t}\ \Longrightarrow\ c'\,\mathbf e\qquad(t\to\infty),$$

where $\mathbf e$ is a mean-one exponential random variable and $\Longrightarrow$ denotes convergence in law.

(ii) If (i) holds, determine $c'$ exactly. Any answer must respect the proven constraints: Meyre–Werner's one-sided bound forces $c'\ge1/8$ if the exponential form holds; the walk analogue has $c'=1/4$ (Dembo–Peres–Rosen–Zeitouni 2004); and the a.s. result $\limsup_{t\to\infty}(\log R(t))^2/(\log t\log_3 t)=1/4$ (Ben Hough–Peres 2007), writing $\log_3 t=\log\log\log t$, does not by itself determine $c'$.

References

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