Up-down ordered Chinese restaurant processes with two-sided immigration, emigration and diffusion limits

We introduce a three-parameter family of up-down ordered Chinese restaurant processes ${\rm PCRP}^{(\alpha)}(\theta_1,\theta_2)$, $\alpha\in(0,1)$, $\theta_1,\theta_2\ge 0$, generalising the two-parameter family of Rogers and Winkel. Our main result establishes self-similar diffusion limits, ${\rm SSIP}^{(\alpha)}(\theta_1,\theta_2)$-evolutions generalising existing families of interval partition evolutions. We use the scaling limit approach to extend stationarity results to the full three-parameter family, identifying an extended family of Poisson--Dirichlet interval partitions. Their ranked sequence of interval lengths has Poisson--Dirichlet distribution with parameters $\alpha\in(0,1)$ and $\theta:=\theta_1+\theta_2-\alpha\ge-\alpha$, including for the first time the usual range of $\theta>-\alpha$ rather than being restricted to $\theta\ge 0$. This has applications to Fleming--Viot processes, nested interval partition evolutions and tree-valued Markov processes, notably relying on the extended parameter range.