[Paper Review] Correlation functions of the shifted Schur measure

Sho Matsumoto|ArXiv.org|Dec 19, 2003

Random Matrices and Applications13 references4 citations

TL;DR

This paper derives the correlation function of the shifted Schur measure—defined via Schur Q-functions on strict partitions—as a Pfaffian using operators on the exterior algebra, generalizing Okounkov's work on the Schur measure. A key result establishes that the limit distribution of the length of the longest ascent pair in a random permutation follows the Tracy-Widom distribution, extending known results for longest increasing subsequences under Plancherel measures.

ABSTRACT

The shifted Schur measure introduced by Tracy and Widom is a measure on the set of all strict partitions, which is defined by Schur $Q$-functions. The main aim of this paper is to calculate the correlation function of this measure, which is given by a pfaffian. As an application, we prove that a limit distribution of $λ_j$'s with respect to a shifted version of the Plancherel measure for symmetric groups is identical with the corresponding distribution of the original Plancherel measure. Further we give expressions of the mean value and the variance of the size of a partition with respect to the measure defined by Hall-Littlewood functions.

Motivation & Objective

• To compute the correlation functions of the shifted Schur measure, a measure on strict partitions defined by Schur Q-functions.
• To extend the asymptotic analysis of Plancherel measures for symmetric groups to the shifted Schur measure framework.
• To determine the limit distribution of the length of the longest ascent pair in a random permutation.
• To derive explicit expressions for the mean and variance of the size |λ| under Hall-Littlewood measures.

Proposed method

• Uses operators on the exterior algebra to compute correlation functions, analogous to Okounkov’s infinite wedge method for the Schur measure.
• Applies Pfaffian determinants to express the correlation functions of the shifted Schur measure.
• Employs the poissonization technique to analyze limit distributions of row lengths λ_j under shifted Plancherel measures.
• Specializes the Hall-Littlewood measure via exponential or principal specializations to compute expectations and variances.
• Utilizes generating functions involving power-sum symmetric functions and q-Pochhammer symbols to analyze the measure's normalization and moments.
• Analyzes asymptotic behavior of E(λ₁) and compares it to the function M(t,X) to identify conditions under which the ratio converges to 1.

Experimental results

Research questions

• RQ1 What is the functional form of the correlation function for the shifted Schur measure, and how does it differ from the Schur measure case?
• RQ2 Does the limit distribution of the length of the longest ascent pair in a random permutation follow the Tracy-Widom distribution?
• RQ3 Under what conditions does the ratio E(λ₁)/M(t,X) converge to 1 for Hall-Littlewood measures?
• RQ4 How can the mean and variance of |λ| be explicitly expressed in terms of power-sum symmetric functions under Hall-Littlewood measures?
• RQ5 Is there a universal asymptotic relation between E(λ₁) and M(t,X) across different specializations of the Hall-Littlewood measure?

Key findings

• The correlation function of the shifted Schur measure is expressed as a Pfaffian, generalizing the determinant-based correlation functions of the Schur measure.
• The limit distribution of λ_j under the shifted Plancherel measure is identical to that under the original Plancherel measure, confirming universality in the asymptotic behavior.
• The length of the longest ascent pair in a random permutation converges in distribution to the Tracy-Widom distribution F₂(s) in the limit.
• For the Hall-Littlewood measure, the mean E(|λ|) and variance Var(|λ|) are explicitly given as sums of products of power-sum functions.
• In several special cases (e.g., Poissonized Plancherel, α-specialization), lim E(λ₁)/M(t,X) = 1 as the parameter tends to infinity.
• However, in the principal specialization with t→0, the ratio E(λ₁)/M(t,X) does not converge to 1, indicating that such convergence is not universal.