  We construct random walks on free products of the form Z 3 * Z d , with d = 5
or 6 which are divergent and not spectrally positive recurrent. We then derive
a local limit theorem for these random walks, proving that $\mu$ * n (e) $\sim$
CR --n n --5/3 if d = 5 and $\mu$ * n (e) $\sim$ CR --n n --3/2 log(n) --1/2 if
d = 6, where $\mu$ * n is the nth convolution power of $\mu$ and R is the
inverse of the spectral radius of $\mu$. This disproves a result of Candellero
and Gilch [7] and a result of the authors of this paper that was stated in a
rst version of [11]. This also shows that the classication of local limit
theorems on free products of the form Z d 1 * Z d 2 or more generally on
relatively hyperbolic groups with respect to virtually abelian subgroups is
incomplete.
