This entails a short proof for the asymptotics of the high eigenvalues of sparse directed Erdos-Réniy matrices, which was proved in Arxiv link.
Work in preparation (April 11, 2021); I'm also working on an extension for random directed regular graphs.
In parallel, I'm interested in the rigidites of random point processes, such as number-rigidities, fluctuations reductions, hyperuniformity, and the possible links between these notions. There are different ways in which point processes in can exhibit a stronger order than the totally chaotic Poisson process; hyperuniformity is when the (random) number of points falling in a large domain of radius has a reduced variance, that is, when
In this survey, I try to give a mathematical overview of this rich domain. Topics: the Fourier caracterization of hyperuniformity, the fluctuation scale, the links with number-rigidity and maximal rigidity for stealthy processes, the example of pertubed lattices.
Here is a version of this survey. It's still work in progress.
Hyperuniformity survey (april 2021)
Joint work with Charles Bordenave.
Arxiv link – Published in Journal of Combinatorial Theory (series B).
This is a short note on a generalization of the Erdös-Gallai theorem on graphical sequences.