On the empty balls of a critical or subcritical branching random walk

Abstract

Let {Z_n}_n≥0 be a critical or subcritical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on ℝ^d. Denote by R_n:= sup{u > 0: Z_n({x ∈ ℝ^d: ∣x∣ < u}) = 0} the radius of the largest empty ball centered at the origin of Z_n. In this work, we prove that after suitable renormalization, R_n converges in law to some non-degenerate distribution as n → ∈. Furthermore, our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk. This completes the results of Révész [13] for the critical binary branching Wiener process.