  In this paper, we study the following detection problem. There are $n$
detectors randomly placed in the unit square $S =
\left[-\frac{1}{2},\frac{1}{2}\right]^2$ assigned to detect the presence of a
source located at the origin. Time is divided into slots of unit length and
$D_i(t) \in \{0,1\}$ represents the (random) decision of the $i^{\rm th}$
detector in time slot $t$. The location of the source is unknown to the
detectors and the goal is to design schemes that use the decisions
$\{D_i(t)\}_{i,t}$ and detect the presence of the source in as short time as
possible.
  We first determine the minimum achievable detection time $T_{cap}$ and show
the existence of \emph{randomized} detection schemes that have detection times
arbitrarily close to $T_{cap}$ for almost all configuration of detectors,
provided the number of detectors $n$ is sufficiently large. We call such
schemes as \emph{capacity achieving} and completely characterize all capacity
achieving detection schemes.
