Phase transitions in number theory: from the birthday problem to Sidon sets

In this work, we show how number theoretical problems can be fruitfully approached with the tools of statistical physics. We focus on g-Sidon sets, which describe sequences of integers whose pairwise sums are different, and propose a random decision problem which addresses the probability of a random set of k integers to be g-Sidon. First, we provide numerical evidence showing that there is a crossover between satisfiable and unsatisfiable phases which converts to an abrupt phase transition in a properly defined thermodynamic limit. Initially assuming independence, we then develop a mean-field theory for the g-Sidon decision problem. We further improve the mean-field theory, which is only qualitatively correct, by incorporating deviations from independence, yielding results in good quantitative agreement with the numerics for both finite systems and in the thermodynamic limit. Connections between the generalized birthday problem in probability theory, the number theory of Sidon sets and the properties of q-Potts models in condensed matter physics are briefly discussed

Surface-Assisted Luminescence: The PL Yellow Band and the EL of n-GaN Devices

Although everybody should know thatmeasurements are never performed directly onmaterials but on devices, this is not generally true. Devices are physical systems able to exchange energy and thus subject to the laws of physics, which determine the information they provide. Hence, we should not overlook device effects in measurements as we do by assuming naively that photoluminescence (PL) is bulk emission free fromsurface effects. By replacing this unjustified assumption with a propermodel forGaN surface devices, their yellow band PL becomes surface-assisted luminescence that allows for the prediction of the weak electroluminescence recently observed in n-GaN devices when holes are brought to their surfaces.