This entails a short proof for the asymptotics of the high eigenvalues of sparse directed Erdos-Réniy matrices.
Work in preparation (April 11, 2021).
I'm working on the same thing 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.