On empty balls of critical 2-dimensional branching random walks

Let ₙ≥ 0 be a critical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesgue measure on Rᵈ. Denote by Rₙ:=>0:Zₙ(ᵈ:|x|<u\)=0\ the radius of the largest empty ball centered at the origin of Zₙ. In {reves02}, Révész shows that if $d=1$, then Rₙ/n converges in law to an exponential random variable as n→∞. Moreover, Révész (2002) conjectured that limn→∞Rₙ/√ nlaw=non-trival~distri.,~d=2; limn→∞Rₙlaw=non-trival~distri.,~d≥3. Later, Hu (2005) {hu05} confirmed the case of d≥3. This work confirms the case of $d=2$. It turns out that the limit distribution can be precisely characterized through the super-Brownian motion. Moreover, we also give complete results of empty balls of the branching random walk with infinite second moment offspring law. As a by-product, this article also improves the assumption of maximal displacements of branching random walks [Theorem 1]{lalley2015}.