Ecosystems represent archetypal complex dynamical systems, often modelled by coupled
differential equations of the form dxidt= xi ϕ i (x 1,…, x N), where N represents the number of …

This book is an invitation to the world of free probability theory. Free probability is a quite
young mathematical theory with many avatars. It owes its existence to the visions of one …

Sparse non-Hermitian random matrices arise in the study of disordered physical systems
with asymmetric local interactions, and have applications ranging from neural networks to …

The Ginibre unitary ensemble (GinUE) consists of $ N\times N $ random matrices with
independent complex standard Gaussian entries. This was introduced in 1965 by Ginbre …

It is a classical result of Ginibre that the normalized bulk k-point correlation functions of a
complex n*n Gaussian matrix with independent entries of mean zero and unit variance are …

We prove that the spectral radius of a large random matrix $ X $ with independent,
identically distributed complex entries follows the Gumbel law irrespective of the distribution …

Most recurrent neural networks (RNNs) do not include a fundamental constraint of real
neural circuits: Dale's Law, which implies that neurons must be excitatory (E) or inhibitory (I) …

Let ξ be a real random variable with mean zero and variance one and A= a 1;…; an be a
multi-set in R d. The random sum S_A:= a_1 ξ _1+ ⋯+ a_n ξ _n where ξ i are iid copies of ξ is …

We consider large non-Hermitian real or complex random matrices XX with independent,
identically distributed centred entries. We prove that their local eigenvalue statistics near the …

We study a physical system of N interacting particles in R^d, d\geq1, subject to pair
repulsion and confined by an external field. We establish a large deviations principle for …