  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 $\mu_k$. If $\chi_k$ is $1$ for one-sided
half-lines $k$ and $1/2$ for two-sided half-lines, and $\alpha_k$ is the tail
exponent of the jumps on half-line $k$, we show that the recurrence
classification for the case where all $\alpha_k \chi_k \in (0,1)$ is determined
by the sign of $\sum_k \mu_k \cot ( \chi_k \pi \alpha_k )$. In the case of two
half-lines, the model fits naturally on $\mathbb{R}$ and is a version of the
oscillating random walk of Kemperman. In that case, the cotangent criterion for
recurrence becomes linear in $\alpha_1$ and $\alpha_2$; our general setting
exhibits the essential non-linearity 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 $\mathbb{R}$ with symmetric increments of tail
exponent $\alpha \in (1,2)$.
