Heavy-Tailed Random Walks on Complexes of Half-Lines

We study a random walk on a complex of finitely many half-lines joined at a common origin; jumps are heavy-tailed and of two types, either one-sided (towards the origin) or two-sided (symmetric). Transmission between half-lines via the origin is governed by an irreducible Markov transition matrix, with associated stationary distribution $$\mu _k$$ μ k . If $$\chi _k$$ χ k is 1 for one-sided half-lines k and 1 / 2 for two-sided half-lines, and $$\alpha _k$$ α k is the tail exponent of the jumps on half-line k, we show that the recurrence classification for the case where all $$\alpha _k \chi _k \in (0,1)$$ α k χ k ∈ ( 0 , 1 ) is determined by the sign of $$\sum _k \mu _k \cot ( \chi _k \pi \alpha _k )$$ ∑ k μ k cot ( χ k π α k ) . In the case of two half-lines, the model fits naturally on $${{\mathbb {R}}}$$ R and is a version of the oscillating random walk of Kemperman. In that case, the cotangent criterion for recurrence becomes linear in $$\alpha _1$$ α 1 and $$\alpha _2$$ α 2 ; our general setting exhibits the essential nonlinearity in the cotangent criterion. For the general model, we also show existence and non-existence of polynomial moments of return times. Our moments results are sharp (and new) for several cases of the oscillating random walk; they are apparently even new for the case of a homogeneous random walk on $${{\mathbb {R}}}$$ R with symmetric increments of tail exponent $$\alpha \in (1,2)$$ α ∈ ( 1 , 2 ) .