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 parameters $\alpha\in(0,1)$ 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 identified 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 the critical regime and, under an additional rate condition in the subcritical regime ($j_n\ll n^{\alpha/(1+\alpha)}/(\log\log n)^{1/(1+\alpha)}$), 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. We first establish these results for the more general Karlin infinite urn model and then adapt the analysis to the Chinese restaurant process. For the latter model, most of our limit theorems are established in the quenched sense.