We apply an analytic method to predict peak counts in weak-lensing surveys. It is based on the theory of Gaussian random fields and suitable to quantify the level of detections caused by chance projections of large-scale structures as well as the shape and shot noise contributed by the background galaxies. A simple analytical recipe is given to compute the signal-to-noise distribution of those detections. We compare our method to peak counts obtained from numerical ray-tracing simulations and find good agreement at the expected level. The number of peak detections depends substantially on the shape and size of the filter applied to the gravitational shear field. We confirm that weak-lensing peak counts are dominated by spurious detections up to signal-to-noise ratios of 3-5 and that most filters yield only a few detections per square degree above this level, while a filter optimised for suppressing large-scale structure noise returns up to an order of magnitude more. Galaxy shape noise and noise from large-scale structures cannot be treated as two independent components since the two contributions add in a non-trivial way.Key words: cosmology: theory - large-scale structure of Universe - galaxies: clusters: general - gravitational lensing: weak
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