The local limit of sublinear preferential attachment graphs with fixed out-degree

We consider the preferential attachment graph in which every new vertex sends $m\ge2$ edges to the existing vertices, the endpoints being chosen one after another with probability proportional to $f(\text{in-degree})$, where the attachment function $f$ is bounded away from zero, has bounded increments and satisfies $f(k)/k\to0$. No monotonicity or concavity is assumed. We prove that the total weight of the first $n$ vertices divided by $n$ converges almost surely to the unique solution $c_f$ of an explicit scalar equation, that the empirical in-degree distribution converges almost surely to the law of a pure birth process with rates $mf(k)/c_f$ observed at an independent exponential time, and that the graph converges locally in probability to an infinite random tree. The probability of every finite ball in the limit is given by an explicit integral over the ages of its vertices. This answers, for this class of attachment functions, a question of Deijfen and van der Hofstad on local limits beyond the affine case when the out-degree is fixed.