Abstract:We introduce a commutative-algebraic framework for studying occupation patterns in directed support graphs associated with discrete-time Markov chains. Given an initial state, the support graph determines a regular language whose words are the admissible state trajectories. Applying the Parikh map, each trajectory is represented by its occupation vector, recording the number of visits to each state. Equivalently, each trajectory defines a monomial whose exponents are its occupation numbers. For each time $n$, we associate a monomial ideal generated by the Parikh monomials of all admissible trajectories of length $n$. The minimal generators of this ideal encode the distinct occupation patterns realized at that time. This construction embeds occupation patterns into combinatorial commutative algebra and separates three levels of support complexity: reachability growth, trajectory growth, and occupation-pattern growth. The framework provides an algebraic and combinatorial layer attached to the directed support structure of a Markov chain. It connects regular languages, Parikh images, monomial ideals, symbolic dynamics, and support graphs. Examples illustrate how occupation ideals reflect branching, recurrence, transience, and local oscillation in the underlying graph.
From: Luis Pousa [view email]
[v1]
Fri, 19 Jun 2026 10:11:08 UTC (14 KB)
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