Why would the curve peak? It would start out flat, then fall off increasingly rapidly as the average inter-event interval was reached, then surpassed. There should be no interval with a rising probability.

The question is whether it falls off even more rapidly than a Poisson process, which might indicate anomalous clustering.

Funny, I had this exact discussion with a paleontologist, who misunderstood the likelihood of finding fossils near an extinction boundary.