Hausdorff and Packing Spectra, Large Deviations, and Free Energy for Branching Random Walks in ℝd

Abstract

Consider an ℝd-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 {α}, α ∈ ℝd. We also give a 0-∞ law for the Hausdorff and packing measures of the level sets E({α}), 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 α which do not belong to the range of the gradient of the free energy. This covers in particular a situation that was 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. © 2014 Springer-Verlag Berlin Heidelberg.