Abstract
We prove existence of the large deviation principle, with a proper convex rate function, for the distribution of the renormalized distance from the origin of a random walk on a free product of finitely generated groups. As a consequence, we derive the same principle for nearest-neighbour random walks on regular trees.
Highlights
Introduction and main resultThe study of random walks on algebraic and geometric structures, most notably graphs and groups, has attracted considerable attention over the last four decades
Our main result establishes the existence of the large deviation principle, with a proper convex rate function, for the collection of non-trivial free products of finitely generated groups, under a non-degeneracy assumption on the semigroup Γ generated by the support of the driving measure μ
By an iterative application of the strong Markov property ([20, Chap. 17]) to the process (Yn)n∈N, it follows that H-valued process (Yτn )n∈N is a right random walk on H driven by a measure μ having finite momentgenerating function with respect to the word length determined by S; if H is a non-trivial free product of finitely generated groups, all conclusions of Theorem 1.4 hold
Summary
The study of random walks on algebraic and geometric structures, most notably graphs and groups, has attracted considerable attention over the last four decades. Our main result establishes the existence of the large deviation principle, with a proper convex rate function, for the collection of non-trivial free products of finitely generated groups, under a non-degeneracy assumption on the semigroup Γ generated by the support of the driving measure μ. Suppose that μ is a probability measure on G whose support generates a pattern-avoiding semigroup, and let (Yn)n≥0 be a right random walk on G with increments distributed according to μ. 17]) to the process (Yn)n∈N, it follows that H-valued process (Yτn )n∈N (where we agree that Yτ0 = e) is a right random walk on H driven by a measure μ having finite momentgenerating function with respect to the word length determined by S; if H is a non-trivial free product of finitely generated groups, all conclusions of Theorem 1.4 hold.
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