[Paper Review] The probability measure corresponding to 2-plane trees

Wojciech Młotkowski, K. A. Penson|arXiv (Cornell University)|Apr 24, 2013

Random Matrices and Applications8 references3 citations

TL;DR

This paper studies the probability measure μ₀ whose moments are given by the Fuss-Catalan-like sequence $\binom{3n}{n}\frac{1}{n+1}$, proving it is absolutely continuous and deriving its density via Mellin convolution of beta distributions. The key contribution is showing μ₀ is infinitely divisible under additive free convolution, decomposing as μ₁ ⊞ μ₂ where both components are free Poisson-like and infinitely divisible, with explicit density formulas and generating function representations using hypergeometric and trigonometric functions.

ABSTRACT

We study the probability measure $μ_{0}$ for which the moment sequence is $\binom{3n}{n}\frac{1}{n+1}$. We prove that $μ_{0}$ is absolutely continuous, find the density function and prove that $μ_{0}$ is infinitely divisible with respect to the additive free convolution.

Motivation & Objective

• To characterize the probability measure μ₀ whose moments are $\binom{3n}{n}\frac{1}{n+1}$, a sequence related to 2-plane trees and Fuss-Catalan numbers.
• To prove that μ₀ is absolutely continuous and to derive its explicit density function using Mellin convolution of beta distributions.
• To establish that μ₀ is infinitely divisible with respect to additive free convolution (⊞), by decomposing it into two measures μ₁ and μ₂ that are themselves ⊞-infinitely divisible.
• To connect the moment-generating function of μ₀ to hypergeometric and trigonometric representations, enabling exact analytic expressions.

Proposed method

• Derives the generating function G(z) for the moment sequence using Lambert’s formula and hypergeometric identities, leading to a closed-form expression involving inverse sine and cosine functions.
• Expresses μ₀ as a Mellin convolution of two beta distributions: $\mathrm{Beta}(1/3,1/6)$ and $\mathrm{Beta}(2/3,4/3)$, establishing absolute continuity.
• Applies free probability theory by computing the R-transform and S-transform of μ₀, enabling decomposition into free Poisson-type measures.
• Uses the R-transform decomposition $R_{\mu_0}(z) = R_1(z) + R_2(z)$ to identify μ₁ and μ₂ as free convolutions of Poisson-like measures and delta masses.
• Derives the density of μ₀ as the free convolution of the densities of μ₁ and μ₂, with explicit formulas involving square roots and rational functions.
• Verifies the decomposition using moment-generating function identities and compares with known free Poisson semigroups $\varpi_t$.

Experimental results

Research questions

• RQ1 What is the explicit form of the probability measure μ₀ whose moments are $\binom{3n}{n}\frac{1}{n+1}$?
• RQ2 Is μ₀ absolutely continuous, and if so, what is its density function?
• RQ3 Can μ₀ be decomposed as an additive free convolution of two measures, and are those components infinitely divisible under ⊞?
• RQ4 How is the moment-generating function of μ₀ related to hypergeometric and trigonometric functions?
• RQ5 What is the connection between μ₀ and the free Poisson semigroup $\varpi_t$?

Key findings

• The measure μ₀ is absolutely continuous and its density is given by the Mellin convolution of $\mathrm{Beta}(1/3,1/6)$ and $\mathrm{Beta}(2/3,4/3)$, yielding a closed-form expression involving algebraic and trigonometric functions.
• The generating function $G(z)$ for the moment sequence is expressed as $G(z) = \frac{12\cos^2\alpha + 6}{(4\cos^2\alpha - 1)^2}$ with $\alpha = \frac{1}{3}\arcsin(\sqrt{27z/4})$, providing a complete analytic representation.
• μ₀ decomposes as $\mu_0 = \mu_1 \boxplus \mu_2$, where $\mu_1 = \mathbf{D}_2\varpi_{1/2}$ and $\mu_2 = \frac{1}{2}\delta_0 + \frac{1}{2}\varpi_1$, both of which are ⊞-infinitely divisible.
• The R-transform of μ₀ splits as $R_{\mu_0}(z) = \frac{z}{1-2z} + \frac{1 - \sqrt{1-2z}}{2\sqrt{1-2z}}$, confirming the decomposition into free Poisson components.
• The density of μ₀ is supported on $[0, \infty)$, with explicit expressions derived from the free convolution of the densities of μ₁ and μ₂, which are supported on $[3 - \sqrt{8}, 3 + \sqrt{8}]$ and $(0,4)$, respectively.
• The moments of μ₀ match the sequence A007226 in the OEIS, and the generating function satisfies a cubic equation: $2 - z - (1+2z)G(z) + 2zG(z)^2 - z^2G(z)^3 = 0$.