  Consider an $\R^d$-valued branching random walk (BRW) on a supercritical
Galton Watson tree. Without any assumption on the distribution of this BRW we
compute, almost surely and simultaneously, the Hausdorff and packing dimensions
of the level sets $E(K)$ of infinite branches in the boundary of the tree
(endowed with its standard metric) along which the averages of the BRW have a
given closed connected set of limit points $K$. This goes beyond multifractal
analysis, which only considers those level sets when $K$ ranges in the set of
singletons $\{\alpha\}$, $\alpha\in\R^d$. We also give a $0$-$\infty$ law for
the Hausdorff and packing measures of the level sets $E(\{\alpha\})$, and
compute the free energy of the associated logarithmically correlated random
energy model in full generality. Moreover, our results complete the previous
works on multifractal analysis by including the levels $\alpha$ which do not
belong to the range of the gradient of the free energy. This covers in
particular a situation until now badly understood, namely the case where a
first order phase transition occurs. As a consequence of our study, we can also
describe the whole singularity spectrum of Mandelbrot measures, as well as the
associated free energy function (or $L^q$-spectrum), when a first order phase
transition occurs.
