Short talks

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Chiara Franceschini

Duality for contact-type processes of epidemic spreading In this talk I will discuss two duality results for the one dimensional diffusive contact process with open boundary reservoirs and a generalized version which allows for different death rates. Duality allows writing the correlation functions of the steady state via the dual process, which is simpler. In particular, an explicit expression of the one point correlation function is shown. Joint work with Ellen Saada, Gunter Schütz, and Sonia Velasco.

Carla Crucianelli

Interacting particle systems on sparse \(W\)-random graphs We consider a general interacting particle system with interactions on a random graph and study the large population limit of this system. When the sequence of underlying graphs converges to a graphon, we show convergence of the interacting particle system to a so called graphon stochastic differential equation. This is a system of uncountable many SDEs of McKean-Vlasov type driven by a continuum of Brownian motions. We make sense of this equation in a way that retains joint measurability and essentially pairwise independence of the driving Brownian motions of the system by using the framework of Fubini extensions. The convergence result is general enough to cover nonlinear interaction, as well as various examples of sparse graphs (joint work with L. Tangpi).

Daniel Yukimura

Quantitative results on sampling from quasi-stationary distributions We study the problem of approximating the quasi-stationary distribution (QSD) of Markov processes using simulation methods. I will present both a lower bound and an upper bound for the rate of convergence of these methods to the QSD. We show that the rates of convergence depend on intrinsic properties of the process, like a notion of mixing time, and a uniform notion of how fast the process gets killed. Finally, we apply these to get quantitative estimates for a large class of killed processes in \(\mathbb{Z}^d\) and obtain bounds on a ''natural'' time scale.

Angeliki Koutsimpela

Tagged particles and size-biased dynamics in mean-field interacting particle systems We establish a connection between tagged particles and size-biased empirical processes in interacting particle systems, in analogy to classical results on the propagation of chaos. In a mean-field scaling limit, the evolution of the occupation number on the tagged particle site converges to a time-inhomogeneous Markov process with non-linear master equation given by the law of large numbers of size-biased empirical measures.

María Cecilia De Vita

Synchronization in Random Geometric Graphs The Kuramoto model is a system of ordinary differential equations that describes the behavior of coupled oscillators. In this talk, we consider the case where the coupling is given by a random geometric graph on the unit circle. We ask whether it is possible to guarantee global synchronization (for almost any initial condition, the system converges to a state where all phases coincide) or if there exist other stable equilibria. To address this question, we work with the energy function of the Kuramoto model on these graphs and prove the existence of at least one local minimum for each winding number \(q\) (with high probability). There is a correspondence between these states and the explicit 'twisted states' found in cycle graphs, although in this case without an explicit formula.