Thick points of 4D critical branching Brownian motion

We study the thick points of branching Brownian motion and branching random walk with a critical branching mechanism, focusing on the critical dimension . We determine the exponent governing the probability to hit a small ball with an exceptionally high number of pioneers, showing that this has a second-order transition between an exponential phase and a stretched exponential phase at an explicit value () of the thickness parameter . We apply the outputs of this analysis to prove that the associated set of thick points T(a) has dimension (4 - a)₊, so that there is a change in behaviour at a = 4 but not at a = 2 in this case. Along the way, we obtain related results for the non-positive solutions of a boundary value problem associated to the semi-linear partial differential equation (PDE) Δv = v² and develop a strong coupling between tree-indexed random walk and tree-indexed Brownian motion that allows us to deduce analogues of some of our results in the discrete case. We also obtain in each dimension d ⩾ 1 an infinite-order asymptotic expansion for the probability that critical branching Brownian motion hits a distant unit ball, finding that this expansion is convergent when d ≠ 4 and divergent when d = 4. This reveals a novel, dimension-dependent critical exponent governing the higher order terms of the expansion, which we compute in every dimension.