Hausdorff and packing spectra, large deviations, and free energy for branching random walks in $\R^d$

TL;DR

This paper analyzes the fractal dimensions and free energy of level sets in branching random walks in , extending multifractal analysis to include complex cases with phase transitions and providing comprehensive results on their geometric and energetic properties.

Contribution

It introduces a general framework for computing Hausdorff and packing dimensions of level sets in  branching random walks, including cases with phase transitions and non-singleton limit sets.

Findings

Computed Hausdorff and packing dimensions of level sets for all limit points.

Established a 0- law for measures of these level sets.

Derived the free energy for the associated random energy model.

Abstract

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.