In this work, we investigate the spectrum of singularities of random stable
trees with parameter γ∈(1,2). We consider for that purpose the scaling
exponents derived from two natural measures on stable trees: the local time
ℓa and the mass measure m, providing as well a purely
geometrical interpretation of the latter exponent. We first characterise the
uniform component of the multifractal spectrum which exists at every level
a>0 of stable trees and corresponds to large masses with scaling index
h∈[γ1+γ,γ−1γ] for the mass measure
(or equivalently h∈[γ1,γ−11] for the local
time). In addition, we investigate the distribution of vertices appearing at
random levels with exceptionally large masses of index
h∈[0,γ1+γ). Finally, we discuss more precisely the
order of the largest mass existing on any subset T(F) of a stable
tree, characterising the former with the packing dimension of the set F.Comment: 50 pages. Major overhaul of the paper, correcting Theorem 4 and
adding the study of the mass measure spectru