# Random interpolating sequences in Dirichlet spaces

## TL;DR

This paper investigates random interpolation in weighted Dirichlet spaces, revealing a phase transition at alpha=1/2 where the nature of interpolating sequences shifts from separated to zero sequences, based on simple distribution conditions.

## Contribution

It demonstrates that random interpolation in Dirichlet spaces depends on simple distribution conditions, uncovering a phase transition at alpha=1/2 in the behavior of interpolating sequences.

## Key findings

- For alpha<1/2, almost sure interpolating sequences are separated sequences.
- For alpha≥1/2, almost sure interpolating sequences are zero sequences.
- A phase transition at alpha=1/2 distinguishes the types of sequences.

## Abstract

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