Week 09.12.2024 – 15.12.2024

We consider ‘λ-shaped random matrices’, whose entries are i.i.d. in the boxes of a given Young diagram λ and zero elsewhere. In particular, we study their limiting spectral distribution when the shape λ is dilated by a growing factor N. The moments of such a distribution are a generalisation of Catalan numbers, and enumerate combinatorial objects which we call λ-plane trees: these are trees whose vertices are labelled in a way that is ‘compatible’ with λ. Based on joint works with Elia Bisi and Marilena Ligabò. 

The assessment and quantification of spatial correlations in complex
systems is of central relevance for a variety of processess, ranging
from chemical reactions to the emergence of social deprivation. The
traditional approaches to the quantification of spatial correlation and
heterogeneity are either based on the comparison with a "well-mixed"
case, which is often pretty artificial and almost irrelevant, or depend
on the choice of a scale parameter, or are affected by the actual size
and peculiarity of the system at hand. In this talk we will explore a
family of methods, inspired by dynamical systems and statistical
physics, to quantify spatial correlations from first principles. These
methods are intrinsically non-parametric, and are able to encompass
information about segregation at all the relevant scales of a system. We
show how these methods allow to uncover spatial patterns in different
real-world systems, and how they correlate with exogenous measures of
segregation and social deprivation.