On empty balls of critical 2-dimensional branching random walks

Let \Zₙ\₍ 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: =\u>0: Zₙ (\{xᵈ: |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 ₍Rₙ nlaw=non-trival~distri. , ~d=2; ₍Rₙlaw=non-trival~distri. , ~d3. Later, Hu (2005) hu05 confirmed the case of d3. 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 1lalley2015.