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Dynamical processes on time-varying complex networks are key to understanding and modeling a broad variety of processes in socio-technical systems. Here we focus on empirical temporal networks of human proximity and we aim at understanding…
We study a class of dynamically constructed point processes in which at every step a new point (particle) is added to the current configuration with a distribution depending on the local structure around a uniformly chosen particle. This…
We study Fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points. The kernels are natural higher order analogues of the Airy kernel and are…
We study the global existence of solutions reaction-diffusion systems with control of mass on multiple domains. Some of these domains overlap, and as a result, an unknown defined on one subdomain can impact another unknown defined on a…
Consider $(X_{i}(t))$ solving a system of $N$ stochastic differential equations interacting through a random matrix $\mathbf J = (J_{ij})$ with independent (not necessarily identically distributed) random coefficients. We show that the…
Airy and Pearcey-like kernels and generalizations arising in random matrix theory are expressed as double integrals of ratios of exponentials, possibly multiplied with a rational function. In this work it is shown that such kernels are…
A new non-conservative stochastic reaction-diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport…
The theory of patterns formation for a reaction-diffusion system defined on a multiplex is developed by means of a perturbative approach. The intra-layer diffusion constants act as small parameter in the expansion and the unperturbed state…
Empirical complex systems must differentially respond to external perturbations and, at the same time, internally distribute information to coordinate their components. While networked backbones help with the latter, they limit the…
Interacting particle systems are continuous time Markov processes which are used to construct models in many disciplines. Monotonicity is a property that some interacting particle systems possess. A monotone interacting particle system is…
We consider heterogeneously interacting diffusive particle systems and their large population limit. The interaction is of mean field type with weights characterized by an underlying graphon. A law of large numbers result is established as…
We consider the symmetric inclusion process on a general finite graph. Our main result establishes universal upper and lower bounds for the spectral gap of this interacting particle system in terms of the spectral gap of the random walk on…
Systems of dynamical interactions between competing species can be used to model many complex systems, and can be mathematically described by {\em random} networks. Understanding how patterns of activity arise in such systems is important…
Reaction-diffusion systems may lead to the formation of steady state heterogeneous spatial patterns, known as Turing patterns. Their mathematical formulation is important for the study of pattern formation in general and play central roles…
We study a new class of matrix models, formulated on a lattice. On each site are $N$ states with random energies governed by a Gaussian random matrix Hamiltonian. The states on different sites are coupled randomly. We calculate the density…
The propagation of an initially localized perturbation via an interacting many-particle Hamiltonian dynamics is investigated. We argue that the propagation of the perturbation can be captured by the use of a continuous-time random walk…
Two processes can influence the evolution of protein interaction networks: addition and elimination of interactions between proteins, and gene duplications increasing the number of proteins and interactions. The rates of these processes can…
Statistical mechanics describes interaction between particles of a physical system. Particle properties of the system can be modelled with a random field on a lattice and studied at different distance scales using renormalization group…
We study the effect of randomness and anisotropy on Turing patterns in reaction-diffusion systems. For this purpose, the Gierer-Meinhardt model of pattern formation is considered. The cases we study are: (i)randomness in the underlying…
In this work, we investigate the dynamics of interacting particle systems subjected to repulsive forces, such as lattices of magnetized particles. To this end, we first develop a general model capable of capturing the complete dynamical…