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This paper studies the Shannon regime for the random displacement ofstationary point processes. Let each point of some initial stationary pointprocess in $\R^n$ give rise to one daughter point, the location of which isobtained by adding a random vector to the coordinates of the mother point, withall displacement vectors independently and identically distributed for allpoints. The decoding problem is then the following one: the whole mother pointprocess is known as well as the coordinates of some daughter point; thedisplacements are only known through their law; can one find the mother of thisdaughter point? The Shannon regime is that where the dimension $n$ tends toinfinity and where the logarithm of the intensity of the point process isproportional to $n$. We show that this problem exhibits a sharp threshold: ifthe sum of the proportionality factor and of the differential entropy rate ofthe noise is positive, then the probability of finding the right mother pointtends to 0 with $n$ for all point processes and decoding strategies. If thissum is negative, there exist mother point processes, for instance Poisson, anddecoding strategies, for instance maximum likelihood, for which the probabilityof finding the right mother tends to 1 with $n$. We then use large deviationstheory to show that in the latter case, if the entropy spectrum of the noisesatisfies a large deviation principle, then the error probability goesexponentially fast to 0 with an exponent that is given in closed form in termsof the rate function of the noise entropy spectrum. This is done for twoclasses of mother point processes: Poisson and Mat\'ern. The practical interestto information theory comes from the explicit connection that we also establishbetween this problem and the estimation of error exponents in Shannon'sadditive noise channel with power constraints on the codewords.

Information-Theoretic Capacity and Error Exponents of Stationary Point Processes under Random Additive Displacements