On the volume of the intersection of two Wiener sausages


For $a>0$, let $W_1^a(t)$ and $W_2^a(t)$ be the $a$-neighbourhoods of two independent standard Brownian motions in $\mathrm{R}^d$ starting at 0 and observed until time $t$. We prove that, for $d \geq 3$ and $c>0$, \[ \lim_{t \to \infty} \frac{1}{t^{(d-2)/d}} \log P\Big(|W_1^a(ct) \cap W_2^a(ct)| \geq t\Big) = – I_d^{\kappa_a}(c) \] and derive a variational representation for the rate constant $I_d^{\kappa_a}(c)$. Here, $\kappa_a$ is the Newtonian capacity of the ball with radius $a$. We show that the optimal strategy to realise the above large deviation is for $W_1^a(ct)$ and $W_2^a(ct)$ to “form a Swiss cheese”: the two Wiener sausages cover part of the space, leaving random holes whose sizes are of order 1 and whose density varies on scale $t^{1/d}$ according to a certain optimal profile.

We study in detail the function $c\mapsto I_d^{\kappa_a}(c)$. It turns out that $I_d^{\kappa_a}(c)=\Theta_d(\kappa_a c)/\kappa_a$, where $\Theta_d$ has the following properties: (1) For $d\geq 3$: $\Theta_d(u)<\infty$ if and only if $u\in(u_\diamond,\infty)$, with $u_\diamond$ a universal constant; (2) For $d=3$: $\Theta_d$ is strictly decreasing on $(u_\diamond,\infty)$ with a zero limit; (3) For $d=4$: $\Theta_d$ is strictly decreasing on $(u_\diamond,\infty)$ with a nonzero limit; (4) For $d\geq 5$: $\Theta_d$ is strictly decreasing on $(u_\diamond,u_d)$ and a nonzero constant on $[u_d,\infty)$, with $u_d$ a constant depending on $d$ that comes from a variational problem exhibiting "leakage". This leakage is interpreted as saying that the two Wiener sausages form their intersection until time $c^*t$, with $c^*=u_d/\kappa_a$, and then wander off to infinity in different directions. Thus, $c^*$ plays the role of a critical time horizon in $d\geq 5$. We also derive the analogous result for $d=2$, namely, \[ \lim_{t \to \infty} \frac{1}{\log t} \log P\Big(|W_1^a(ct) \cap W_2^a(ct)| \geq t/\log t\Big) = - I_2^{2\pi}(c), \] where the rate constant has the same variational representation as in $d\geq 3$ after $\kappa_a$ is replaced by $2\pi$. In this case $I_2^{2\pi}(c)=\Theta_2(2\pi c)/2\pi$ with $\Theta_2(u)<\infty$ if and only if $u\in(u_\diamond,\infty)$ and $\Theta_2$ is strictly decreasing on $(u_\diamond,\infty)$ with a zero limit.


Michiel van den Berg

Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Erwin Bolthausen

Institut für Mathematik, Universität Zürich, 8057 Zürich, Switzerland

Frank den Hollander

EURANDOM, P. O. Box 513, 5600 MB Eindhoven, The Netherlands