Random Interpolating Sequences in Dirichlet Spaces

Abstract We discuss random interpolating sequences in weighted Dirichlet spaces ${{\mathcal{D}}}_\alpha $, $0\leq \alpha \leq 1$, when the radii of sequence points are fixed a priori and arguments uniformly distributed. Although conditions for deterministic interpolation these depend on capacities, which very hard to estimate general, we show that is driven by surprisingly simple distribution conditions. As consequence, obtain breakpoint at $\alpha =1/2$ behavior showing more precisely almost sure $ exactly separated $0\le <1/2$ (which includes Hardy space $H^2={{\mathcal{D}}}_0$), they zero $1/2 \le 1$ classical ${{\mathcal{D}}}={{\mathcal{D}}}_1$).