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We study trace theoretic concurrent systems. This setting encompasses safe(1-bounded) Petri nets. We introduce a notion of irreducible concurrent systemand we prove the equivalence between irreducibility and a "spectral property".The spectral property states a strict inequality between radii of convergenceof certain growth series associated with the system. The proof that we presentrelies on analytic combinatorics techniques. The spectral property is thecornerstone of our theory, in a framework where the Perron-Frobenius theorydoes not apply directly. This restriction is an inherent difficulty in thestudy of concurrent systems.  We apply the spectral property to the probabilistic theory of concurrentsystems. We prove on the one hand the uniqueness of the uniform measure, aquestion left open in a previous paper. On the other hand, we prove that thisuniform measure can be realized as a Markov chain of states-and-cliques on astate space that can be precisely characterized.

A spectral property for concurrent systems and some probabilistic applications