Efthymios Sofos

The set of the primes p for which a variety over the rational numbers has no p-adic point plays a fundamental role in arithmetic geometry. This set is deterministic, however, we prove that if we choose a typical variety from a family then the set has random behaviour. We furthermore prove that this behaviour is modelled by a random walk in Brownian motion. This has several consequences, the main one being the description of the finer properties of the distribution of the primes in this set via the Feynman-Kac formula.