[Paper Review] Markov property of determinantal processes with extended sine, Airy, and Bessel kernels

Makoto Katori, Hideki Tanemura|arXiv (Cornell University)|Jun 22, 2011

Random Matrices and Applications27 references21 citations

TL;DR

This paper establishes the Markov property for infinite-dimensional determinantal processes with extended sine, Airy, and Bessel kernels—key objects in random matrix theory—by constructing them as limits of finite-particle noncolliding diffusion processes in a new topology defined by entire functions. The key contribution is proving Markovianity for these processes, which are reversible with respect to bulk, soft-edge, and hard-edge scaling limits of Gaussian and chiral Gaussian unitary ensembles.

ABSTRACT

When the number of particles is finite, the noncolliding Brownian motion (the Dyson model) and the noncolliding squared Bessel process are determinantal diffusion processes for any deterministic initial configuration $ξ=\sum_{j \in Λ} δ_{x_j}$, in the sense that any multitime correlation function is given by a determinant associated with the correlation kernel, which is specified by an entire function $Φ$ having zeros in $\supp ξ$. Using such entire functions $Φ$, we define new topologies called the $Φ$-moderate topologies. Then we construct three infinite-dimensional determinantal processes, as the limits of sequences of determinantal diffusion processes with finite numbers of particles in the sense of finite dimensional distributions in the $Φ$-moderate topologies, so that the probability distributions are continuous with respect to initial configurations $ξ$ with $ξ(\R)=\infty$. We show that our three infinite particle systems are versions of the determinantal processes with the extended sine, Bessel, and Airy kernels, respectively, which are reversible with respect to the determinantal point processes obtained in the bulk scaling limit and the soft-edge scaling limit of the eigenvalue distributions of the Gaussian unitary ensemble, and the hard-edge scaling limit of that of the chiral Gaussian unitary ensemble studied in the random matrix theory. Then Markovianity is proved for the three infinite-dimensional determinantal processes.

Motivation & Objective

• To construct infinite-dimensional determinantal processes as limits of finite-particle noncolliding diffusion processes with continuous dependence on initial configurations.
• To define new topologies, called $Φ$-moderate topologies, using entire functions $Φ$ with zeros in the support of initial configurations.
• To prove that the resulting infinite systems are versions of determinantal processes with extended sine, Airy, and Bessel kernels.
• To establish the Markov property for these infinite systems, which are reversible with respect to bulk, soft-edge, and hard-edge scaling limits from random matrix theory.

Proposed method

• Constructs infinite particle systems as limits of finite-particle determinantal diffusion processes in the sense of finite-dimensional distributions.
• Introduces $Φ$-moderate topologies using entire functions $Φ$ that vanish on the support of initial configurations $ξ$, ensuring continuity in initial data.
• Uses correlation kernels derived from $Φ$ to define the multitime correlation functions via Fredholm determinants.
• Applies the Christoffel-Darboux formula and asymptotic analysis of Hermite polynomials to derive the extended kernels in the bulk, soft-edge, and hard-edge scaling limits.
• Establishes Markov property by verifying the strong Markov property and continuity of sample paths in the $Φ$-moderate topology.
• Relies on the convergence of finite-particle Dyson models to infinite systems under appropriate scaling and topology.

Experimental results

Research questions

• RQ1 Can infinite-dimensional determinantal processes with extended sine, Airy, and Bessel kernels be constructed as limits of finite-particle noncolliding diffusion processes with continuous dependence on initial configurations?
• RQ2 Do the resulting infinite systems satisfy the Markov property under the $Φ$-moderate topology defined by entire functions $Φ$?
• RQ3 Are these infinite systems equivalent to the determinantal processes with extended kernels known from random matrix theory scaling limits?
• RQ4 Is the Markov property preserved in the infinite particle limit, particularly in the bulk, soft-edge, and hard-edge regimes?
• RQ5 Can the Fredholm determinant representation of multitime correlation functions be preserved and extended to the infinite system under the new topology?

Key findings

• The infinite-dimensional determinantal processes with extended sine, Airy, and Bessel kernels are constructed as limits of finite-particle noncolliding diffusion processes in the $Φ$-moderate topology.
• The constructed processes are reversible with respect to the determinantal point processes arising in the bulk and soft-edge scaling limits of the Gaussian unitary ensemble and the hard-edge limit of the chiral Gaussian unitary ensemble.
• The Markov property is rigorously proven for all three infinite systems, establishing their time-homogeneous Markovian dynamics.
• The extended sine kernel arises as the scaling limit of the Dyson model in the bulk, with correlation kernel derived from the asymptotic behavior of Hermite polynomials.
• The Airy kernel is obtained in the soft-edge scaling limit, where the particle density vanishes at the edge, and the kernel is expressed via the Airy function and its derivatives.
• The Bessel kernel emerges in the hard-edge limit of the chiral Gaussian unitary ensemble, with the kernel derived from asymptotics of Laguerre polynomials and Bessel functions.