Laboratoire de Physique Théorique
Toulouse - UMR 5152

In this seminar, we delve into the dynamics of complex multistate systems, framed within the context of continuous-time Markovian processes. Our primary focus is on the spectral properties of the Markov process generator, which are crucial for understanding the relaxation dynamics of these systems. The complexity of explicitly formulating this generator for real-world systems often renders it a daunting task, leading us to employ random matrices as a practical modeling approach.

We will explore how the sparsity of the generator influences its spectral characteristics, particularly noting the reduction of the large spectral gap commonly associated with dense random generators. We then pivot to examine a paradigmatic model of Markovian dynamics : the asymmetric exclusion process (ASEP). The ASEP’s spectral boundary exhibits unique features, notably the presence of pronounced spikes. I will elucidate the connections between these spectral anomalies and non-Hermitian fermionic models, as well as their relationship to random graphs with specific cycle structures.

This presentation aims to meld concepts from random matrix theory, stochastic processes, and quantum mechanics.