Aspects of large random Markov kernels

Abstract

We briefly review certain asymptotic properties of random Markov kernels on a finite state space. These models can be thought of as finite Markov chains in random environment. Here, the asymptotics are taken with respect to the cardinality of the state space. We study, for instance, the behaviour of the normalized invariant vector, the global behaviour of the spectrum and the extremal eigenvalues. The analysis of these models involves Random Matrix Theory, convex compact polytopes and finite graphs with random weights. We also give some open problems related to these models.

Keywords:

• random matrices
• finite graphs
• convex polytopes
• symmetric group
• Markov chains
• invariant measures

AMS-MSC:

• 15A52
• 60F15
• 62H99

Acknowledgements

The present version benefited from the comments of an anonymous reviewer. The author would like to warmly thank Habib Ouerdiane and his team for invitation and kind hospitality during the workshop held at Hammamet in November 2007.

Notes

^1. Web: http://djalil.chafai.net/

Additional information

Notes on contributors

Djalil Chafaï

1. 1. Web: http://djalil.chafai.net/