The Poisson-Dirichlet law is the unique invariant distribution for uniform split-merge transformations

Two-parameter Poisson-Dirichlet measures and reversible exchangeable fragmentation-coalescence processes

We show that for $0<α<1$ and $θ>-α$, the Poisson-Dirichlet distribution with parameter $(α, θ)$ is the unique reversible distribution of a rather natural fragmentation-coalescence process. This completes earlier results in the literature for certain split and merge transformations and the parameter $α=0$.

On the simplest split-merge operator on the infinite-dimensional simplex

We consider the simplest split-merge Markov operator $T$ on the infinite-dimensional simplex $Σ_1$ of monotone non-negative sequences with unit sum. For a sequence $x\inΣ_1$, it picks a size-biased sample (with replacement) of two elements of $x$; if these elements are distinct, it merges them and reorders the sequence, and if the same element is picked twice, it splits this element uniformly into two parts and reorders the sequence. We prove that the means along the $T$-trajectory of the $\de$-measure at the vector $(1,0,0,{...})$ converge to the Poisson--Dirichlet distribution PD(1).

区间和点过程在实线上扩展的分裂合并动力学

Split merge dynamics for expanding intervals and point processes on the real line

研究了实线上扩展区间和点过程的分裂合并动力学，探讨了在区间长度正则变化时极限分布的绝对连续性，并扩展了该动力学至实线上的分割合并过程。

由区间分裂-合并动力学产生的交错序列及其诱导的概率测度

Interlacing sequences resulting from an interval split-merge dynamics and the induced probability measures

研究单位区间划分序列的生成机制，分析其断裂点的分布特性及极限行为，揭示确定性规则下序列的丰富结构。

Dirichlet approximation of equilibrium distributions in Cannings models with mutation

Consider a haploid population of fixed finite size with a finite number of allele types and having Cannings exchangeable genealogy with neutral mutation. The stationary distribution of the Markov chain of allele counts in each generation is an important quantity in population genetics but has no tractable description in general. We provide upper bounds on the distributional distance between the Dirichlet distribution and this finite population stationary distribution for the Wright-Fisher genealogy with general mutation structure and the Cannings exchangeable genealogy with parent independent …

Split-and-Merge in Stationary Random Stirring on Lattice Torus

We show that in any dimension $d\ge1$, the cycle-length process of stationary random stirring (or, random interchange) on the lattice torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson-Dirichlet law $\mathsf{PD(1)}$, as the size of the system grows to infinity. In the case of transient dimensions, $d\ge 3$, the problem is motivated by attempts to understand the onset of long range order in quantum Heisenberg models via random loop representations of the latter.