Randomized detection and detection capacity of multidetector networks

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 $i^{\mathrm{t}\mathrm{h}}$ 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.