[Paper Review] The explicit formulae for scaling limits in the ergodic decomposition of infinite Pickrell measures

Alexander I. Bufetov, Yanqi Qiu|arXiv (Cornell University)|Feb 21, 2014

Random Matrices and Applications16 references3 citations

TL;DR

This paper derives explicit formulae for the correlation kernels of scaling limits in the ergodic decomposition of infinite Pickrell measures on infinite complex matrices. Using Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles, it identifies the limiting kernels as solutions to a generalized Bessel-type kernel via asymptotic analysis of orthogonal polynomials, providing explicit expressions in terms of confluent hypergeometric functions and Bessel functions for the resulting determinantal point processes.

ABSTRACT

The main result of this paper, Theorem 1.1, gives explicit formulae for the kernels of the ergodic decomposition measures for infinite Pickrell measures on spaces of infinite complex matrices. The kernels are obtained as the scaling limits of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles.

Motivation & Objective

• To derive explicit formulae for the kernels of ergodic decomposition measures in the scaling limit of infinite Pickrell measures.
• To characterize the limiting determinantal point processes arising from the ergodic decomposition of infinite Pickrell measures when $ s \leq -1 $.
• To establish a connection between the scaling limits of Christoffel-Uvarov deformations of Jacobi ensembles and the kernels of projection operators on $ L^2(0,\infty) $.
• To provide explicit representations of the limiting correlation kernels in terms of special functions, particularly confluent hypergeometric and Bessel functions.
• To extend the known finite $ s > -1 $ results to the infinite measure case $ s \leq -1 $, where the decomposition measure is infinite but still determinantal.

Proposed method

• The authors analyze the scaling limit of Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles under a specific asymptotic regime.
• They derive the limiting kernel $ \Phi_{\infty}^{(s,u)}(z_1,z_2) $ as the limit of normalized Christoffel-Darboux kernels associated with deformed weight functions.
• The limiting kernel is expressed in terms of solutions to a second-order linear differential equation related to the hypergeometric type, specifically the $ A_{III}^{(s,u)} $ and $ B_{III}^{(s,u)} $ functions.
• The method involves a change of variables to map the spectral parameter to the positive real line and analyze the asymptotic behavior of orthogonal polynomials under a scaling limit with $ n \to \infty $.
• The authors use integral representations involving Bessel functions and confluent hypergeometric functions to express the limiting kernel.
• They establish uniform convergence of the kernel expressions on compact subsets of $ \mathbb{C} \setminus \{0\} $, ensuring the validity of the limit.

Experimental results

Research questions

• RQ1 What is the explicit form of the correlation kernel in the scaling limit of the ergodic decomposition of infinite Pickrell measures for $ s \leq -1 $?
• RQ2 How do Christoffel-Uvarov deformations of Jacobi orthogonal polynomial ensembles lead to the emergence of Bessel-type kernels in the scaling limit?
• RQ3 Can the projection kernel associated with the infinite determinantal measure $ \mathbb{B}^{(s)} $ be explicitly computed via asymptotic analysis of orthogonal polynomials?
• RQ4 What is the role of the parameter $ u $ in the limiting kernel, and how does it affect the structure of the resulting point process?
• RQ5 How do the limiting kernels relate to known special functions such as Bessel functions and confluent hypergeometric functions?

Key findings

• The scaling limit of the Christoffel-Darboux kernel for the deformed Jacobi ensemble converges uniformly to a kernel $ \Phi_{\infty}^{(s,u)}(z_1,z_2) $, which is explicitly given in terms of confluent hypergeometric functions $ A_{III}^{(s,u)} $ and $ B_{III}^{(s,u)} $.
• For $ -1 < s < 1 $, the limiting kernel admits a decomposition into two terms: $ \mathscr{M}_0^{(s,u)}(z_1)\mathscr{M}_0^{(s,u)}(z_2) + \mathscr{M}_1^{(s,u)}(z_1)\mathscr{M}_1^{(s,u)}(z_2) $, where $ \mathscr{M}_0^{(s,u)} $ and $ \mathscr{M}_1^{(s,u)} $ are explicitly defined in terms of gamma functions and integrals.
• The normalization constants in the limiting kernel are derived via asymptotic evaluation of integrals involving $ t^s / (t+u)^2 $, yielding $ u^{s-1} \Gamma(1+s)\Gamma(1-s) $ in the limit.
• For $ -1 < s < 0 $, the second term $ \mathscr{M}_1^{(s,u)} $ is non-zero and explicitly given as a ratio involving a weighted $ L^2 $-norm of a linear function in $ z $, while for $ 0 \leq s < 1 $, it vanishes.
• The integral representation of the kernel includes a term involving $ \int_0^1 \frac{A_{III}^{(s,u)}(\kappa,z_1)A_{III}^{(s,u)}(\kappa,z_2)}{[C_{III}^{(s,u)}(\kappa)]^2} \kappa \, d\kappa $, which captures the continuous part of the spectral measure.
• The limiting kernel is shown to be the correlation kernel of a determinantal point process on $ (0,\infty) $, with the projection kernel $ \Pi^g $ explicitly identified via the asymptotic behavior of orthogonal polynomials.