Second-order fluctuations for a phase transition in random partitions

In a recent paper, Banderier et al. (2024) investigated the limiting behavior of component counts of random partitions induced by the Chinese restaurant process with parameter $\alpha\in(0,1)$ and $\theta>-\alpha$. Let $C_j(n)$ denote the number of components of size $j$ of a partition of $\{1,\ldots,n\}$ and consider $j=j_n\to\infty$ as $n\to\infty$. They revealed a phase transition in the first-order limit behavior of $C_{j_n}(n)$, where the critical regime corresponds to $j_n\sim rn^{\alpha/(1+\alpha)}$ for some $r>0$. A natural next question is to understand the corresponding second-order fluctuations. We establish second-order limit theorems in both the subcritical ($j_n\ll n^{\alpha/(1+\alpha)}$) and critical regimes for the counting process $(C_{j_n}(n(1+t/j_n)_+))_{t\in\mathbb R}$. In the subcritical regime, after appropriate normalization, the limit is a stationary Ornstein--Uhlenbeck Gaussian process, whereas in the critical regime the limit is a stationary $M/M/\infty$ queue. We also establish a more refined point-process convergence in the critical regime. In fact, we establish second-order limit theorems for the more general Karlin infinite urn model, and then adapt the analysis to the Chinese restaurant process.