[Paper Review] Law of Large Numbers and Central Limit Theorems by Jack Generating Functions

Jiaoyang Huang|arXiv (Cornell University)|Jul 26, 2018

Random Matrices and Applications31 references4 citations

TL;DR

This paper establishes that the law of large numbers and central limit theorems for N-particle systems on the positive integers are determined by the germ at unity of their Jack generating functions, using Nazarov-Sklyanin operators and Jack symmetric functions as eigenfunctions. The key result is that asymptotic behavior—such as convergence to a deterministic limit or Gaussian fluctuations—depends precisely on the convergence of partial derivatives of the logarithm of the Jack generating function at unity.

ABSTRACT

In a series of papers [22-24] by Bufetov and Gorin, Schur generating functions as the Fourier transforms on the unitary group $U(N)$, are introduced to study the asymptotic behaviors of random $N$-particle systems. We introduce and study the Jack generating functions of random $N$-particle systems. In special cases, this can be viewed as the Fourier transforms on the Gelfand pairs $(GL_N(\mathbb R), O(N))$, $(GL_N(\mathbb C), U(N))$ and $(GL_N(\mathbb H), Sp(N))$. Our main results state that the law of large numbers and the central limit theorems for such particle systems, is equivalent to certain conditions on the germ at unity of their Jack generating functions. Our main tool is the Nazarov-Sklyanin operators [50], which have Jack symmetric functions as their eigenfunctions. As applications, we derive asymptotics of Jack characters, prove law of large numbers and central limit theorems for the Littlewood-Richardson coefficients of zonal polynomials, and show that the fluctuations of the height functions of a general family of nonintersecting random walks are asymptotically equal to those of the pullback of the Gaussian free field on the upper half plane.

Motivation & Objective

• To extend the Fourier-analytic approach to central limit theorems from classical probability to the setting of Gelfand pairs involving GL_N(R), GL_N(C), and GL_N(H) with their respective compact subgroups.
• To develop a framework using Jack generating functions as generalized Fourier transforms for random N-particle systems on Z_{>0}.
• To characterize the law of large numbers and central limit theorems in terms of the asymptotic behavior of the Jack generating function near unity.
• To unify and generalize previous results on Schur generating functions (e.g., Bufetov and Gorin) to the broader class of Jack polynomials with parameter θ.
• To apply the framework to derive asymptotics of Jack characters, Littlewood-Richardson coefficients for zonal polynomials, and fluctuations of height functions in nonintersecting random walks.

Proposed method

• Introduces Jack generating functions as Fourier transforms on the Gelfand pairs (GL_N(R), O(N)), (GL_N(C), U(N)), and (GL_N(H), Sp(N)), generalizing Schur generating functions.
• Uses Nazarov-Sklyanin operators, which have Jack symmetric functions as eigenfunctions, to relate moment generating functions to derivatives of the logarithm of the Jack generating function.
• Applies the Cauchy identity and skew Jack symmetric functions to express generating functions in terms of power sum symmetric functions.
• Analyzes the germ at unity of the Jack generating function by studying the convergence of partial derivatives of log F_N at p = 1^N.
• Employs combinatorial identities involving partitions and multinomial coefficients to relate moments of particle systems to derivatives of the generating function.
• Establishes equivalence between the law of large numbers and CLT and the convergence of the first and second-order derivatives of log F_N at unity, respectively.

Experimental results

Research questions

• RQ1 Under what conditions on the Jack generating function does the law of large numbers hold for N-particle systems?
• RQ2 How can the central limit theorem for particle systems be characterized in terms of the asymptotic behavior of the Jack generating function near unity?
• RQ3 What is the relationship between the germ of the Jack generating function at unity and the limiting distribution of particle systems?
• RQ4 How do the asymptotics of Jack characters and Littlewood-Richardson coefficients for zonal polynomials relate to the generating function's derivatives?
• RQ5 Can the fluctuations of height functions in nonintersecting random walks be shown to converge to those of the Gaussian free field via this framework?

Key findings

• The law of large numbers for N-particle systems is equivalent to the convergence of the first-order partial derivatives of log F_N at p = 1^N.
• The central limit theorem for one-level systems holds if and only if the second-order partial derivatives of log F_N at unity converge to finite limits.
• For multi-level systems, the joint convergence of mixed second-order derivatives of log F_N at unity characterizes the joint Gaussian fluctuation of particle systems.
• The fluctuations of height functions in a general family of nonintersecting random walks converge to those of the pullback of the Gaussian free field on the upper half-plane.
• The asymptotic behavior of Jack characters is fully determined by the germ of the Jack generating function at unity, with explicit formulas derived via derivative analysis.
• The Littlewood-Richardson coefficients for the product of zonal polynomials exhibit universal asymptotic fluctuations governed by the same derivative conditions on the generating function.