About the spectral analysis of large random Markov kernels

This tiny note concerns the spectral analysis of models of random Markov kernels, centered around joint works in collaboration with Charles Bordenave and Pietro Caputo, with a “large random matrices” point of view. Some open problems are given. A (rather old now) survey about the spectral analysis of large random Markov kernels can be found in [9]. An up to date review can be found in the Habilitation thesis of Charles Bordenave under preparation (gives a complete bibliography). These models can be seen as random walks or Markov chains in random environment. From the random matrix theory point of view, these random matrices have dependent entries, and what is know concerns mostly first order asymptotics for empirical spectral distributions, spectral edge, and principal eigenvector. In particular, the study of the asymptotic fluctuations is completely open.