[Paper Review] On the simplest split-merge operator on the infinite-dimensional simplex

Н. В. Цилевич|ArXiv.org|Jun 1, 2001

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TL;DR

This paper studies the simplest split-merge Markov operator $T$ on the infinite-dimensional simplex $\Sigma_1$, which stochastically merges or splits elements of a sequence based on size-biased sampling. It proves that the binomial means of the $T$-trajectory starting from the Dirac measure at $(1,0,0,\ldots)$ converge weakly to the Poisson–Dirichlet distribution $PD(1)$, providing a new dynamical characterization of this fundamental measure.

ABSTRACT

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).

Motivation & Objective

• To investigate the long-term behavior of the simplest split-merge Markov operator $T$ on the infinite-dimensional simplex $\Sigma_1$.
• To provide evidence for Vershik's conjecture that the only $T$-invariant distribution on $\Sigma_1$ is the Poisson–Dirichlet distribution $PD(1)$.
• To establish weak convergence of binomial means of $T$-trajectories starting from $\delta_{(1,0,\ldots)}$ to $PD(1)$.

Proposed method

• Define the split-merge operator $T$ on $\Sigma_1$: with probability proportional to $2x_ix_j$, merge two distinct elements $x_i, x_j$; with probability proportional to $x_i$, split a single element $x_i$ uniformly into two parts.
• Use the binomial mean operator $\left(\frac{E + T}{2}\right)^m$ to average the $m$-step trajectories of the $\delta$-measure at $(1,0,0,\ldots)$.
• Leverage connections to the representation theory of the infinite symmetric group $\mathfrak{S}_\infty$ and the space of virtual permutations.
• Apply techniques from previous work on central measures and irreducible characters of symmetric groups to analyze the spectral and convergence properties of $T$.
• Use the Murnaghan–Nakayama rule and character decomposition to compute coefficients of the operator's action on irreducible representations.
• Prove convergence by showing that the coefficients of non-Haar components vanish in the limit, leaving only the $PD(1)$-like structure.

Experimental results

Research questions

• RQ1 Does the trajectory of the $\delta$-measure at $(1,0,0,\ldots)$ under the split-merge operator $T$ converge weakly to $PD(1)$?
• RQ2 Can the binomial mean of the $T$-trajectory be used to characterize $PD(1)$ as the unique weak limit?
• RQ3 To what extent does the dynamics of $T$ on sequences with finitely many non-zero entries lead to $PD(1)$?
• RQ4 How do the spectral properties of $T$ relate to the invariant measures on $\Sigma_1$?
• RQ5 Can the representation-theoretic structure of $\mathfrak{S}_\infty$ be used to prove uniqueness of $PD(1)$ as the invariant measure for $T$?

Key findings

• The sequence $\left(\frac{E + T}{2}\right)^m \delta_{(1,0,\ldots)}$ converges weakly to the Poisson–Dirichlet distribution $PD(1)$ as $m \to \infty$.
• For almost all $x \in \Sigma_1$ with finitely many non-zero coordinates, the sequence $\left(\frac{E + T}{2}\right)^m \delta_x$ also converges weakly to $PD(1)$.
• The coefficients of non-Haar components in the character decomposition of the operator's action vanish as $m \to \infty$, confirming convergence to $PD(1)$.
• The proof relies on the Murnaghan–Nakayama rule and character theory of symmetric groups, particularly the decomposition of induced representations.
• The convergence is established via asymptotic analysis of binomial means, showing that only the $PD(1)$-like component survives in the limit.
• The result supports Vershik's conjecture that $PD(1)$ is the unique $T$-invariant measure on $\Sigma_1$, though full uniqueness remains open for general Borel measures.