[Paper Review] Multilevel Dyson Brownian motions via Jack polynomials

Vadim Gorin, Mykhaylo Shkolnikov|arXiv (Cornell University)|Jan 22, 2014

Random Matrices and Applications32 references4 citations

TL;DR

This paper introduces multilevel Dyson Brownian motions for arbitrary $β > 0$ using Jack symmetric polynomials, generalizing Warren's $β=2$ interlacing processes. It establishes the existence and uniqueness of such processes, proves they are intertwined for $β \geq 1$, and derives their SDEs via approximation by discrete Markov chains and convergence arguments.

ABSTRACT

We introduce multilevel versions of Dyson Brownian motions of arbitrary parameter $β>0$, generalizing the interlacing reflected Brownian motions of Warren for $β=2$. Such processes unify $β$ corners processes and Dyson Brownian motions in a single object. Our approach is based on the approximation by certain multilevel discrete Markov chains of independent interest, which are defined by means of Jack symmetric polynomials. In particular, this approach allows to show that the levels in a multilevel Dyson Brownian motion are intertwined (at least for $β\ge 1$) and to give the corresponding link explicitly.

Motivation & Objective

• To construct multilevel Dyson Brownian motions for arbitrary $β > 0$, generalizing Warren's $β=2$ process.
• To unify $β$-corners processes and Dyson Brownian motions into a single multilevel diffusion framework.
• To establish the existence and uniqueness of such multilevel processes via approximation by discrete Markov chains based on Jack polynomials.
• To prove that the levels in the process are intertwined for $β \geq 1$, providing an explicit link between levels.

Proposed method

• Construct discrete multilevel Markov chains using probability measures derived from Jack symmetric polynomials.
• Use the Jack polynomial framework to define transition kernels and dynamics that approximate the continuous Dyson Brownian motion.
• Establish tightness and convergence rates for the discrete chains to their continuous limits.
• Apply martingale techniques and optional sampling to derive the SDEs for the limiting process.
• Use the Dixon-Anderson integral formula to justify the joint density of the Hermite $β$ corners process.
• Prove that the limiting process satisfies a system of SDEs with drift terms involving inverse distances between particles at the same and adjacent levels.

Experimental results

Research questions

• RQ1 Can multilevel Dyson Brownian motions be constructed for arbitrary $β > 0$, not just $β=1,2,4$?
• RQ2 Do the levels in such a multilevel process exhibit interlacing dynamics for $β \geq 1$, and if so, what is the explicit form of the drift?
• RQ3 How can the joint law of eigenvalues across nested submatrices (i.e., corners) be generalized to arbitrary $β > 0$?
• RQ4 What is the limiting SDE governing the multilevel process, and how does it relate to the dynamics of the discrete approximating chains?
• RQ5 Is the multilevel Dyson Brownian motion uniquely characterized by its initial law and the interlacing structure?

Key findings

• The multilevel Dyson Brownian motion for arbitrary $β > 0$ exists and is uniquely characterized by its initial condition and the SDE structure.
• For $β \geq 1$, the levels in the process are intertwined, meaning the particle positions satisfy the interlacing condition $x_i^k \leq x_i^{k-1} \leq x_{i+1}^k$ almost surely.
• The limiting process satisfies a system of SDEs with drift terms proportional to $\frac{1-\theta}{x_i^k - x_m^k}$ and $\frac{1-\theta}{x_i^k - x_m^{k-1}}$, where $\theta = \frac{2}{\beta}$.
• The discrete Markov chains based on Jack polynomials converge weakly to the continuous multilevel Dyson Brownian motion, with explicit rates of convergence established.
• The joint density of the Hermite $β$ corners process is derived using the Dixon-Anderson formula, confirming its consistency with the continuous limit.
• The process is constructed via a martingale problem approach, showing that the limiting process satisfies the SDEs with independent Brownian motions as driving noise.