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Tridiagonal random matrices, an analytic approach

Lucas Babet, Ionel Popescu

TL;DR

This work analyzes the eigenvalue distributions of random tridiagonal matrices, extending Wigner-type results by leveraging the Stieltjes transform to relax moment assumptions. It first treats a simple tridiagonal model with i.i.d. off-diagonal entries, deriving a self-consistent description of the limiting spectral distribution, and then generalizes to deformed tridiagonal matrices with growth factors, showing the limit is a scale mixture governed by both the base off-diagonal law and the deformation law. A joint distribution result demonstrates asymptotic independence in mixed moments between diagonal/deformation components and the tridiagonal part. An algebraic structure based on colored paths and shift operators is developed to describe limiting moments and to discuss addition in a noncommutative framework, raising open questions about freeness analogues and broader algebraic principles for tridiagonal ensembles.

Abstract

In this paper, we study the limiting distribution of the eigenvalues for random tridiagonal matrix models. The limiting distribution is well described by its moments. Here, an analytical approach allows us, as in the case of Wigner matrices, to relax the assumptions on the random variables. With this method, we proved the convergence of the spectral distribution under an assumption on the second moment. We discuss also about an algebraic approach for the tridiagonal models, which are more complicated than the classic freeness.

Tridiagonal random matrices, an analytic approach

TL;DR

This work analyzes the eigenvalue distributions of random tridiagonal matrices, extending Wigner-type results by leveraging the Stieltjes transform to relax moment assumptions. It first treats a simple tridiagonal model with i.i.d. off-diagonal entries, deriving a self-consistent description of the limiting spectral distribution, and then generalizes to deformed tridiagonal matrices with growth factors, showing the limit is a scale mixture governed by both the base off-diagonal law and the deformation law. A joint distribution result demonstrates asymptotic independence in mixed moments between diagonal/deformation components and the tridiagonal part. An algebraic structure based on colored paths and shift operators is developed to describe limiting moments and to discuss addition in a noncommutative framework, raising open questions about freeness analogues and broader algebraic principles for tridiagonal ensembles.

Abstract

In this paper, we study the limiting distribution of the eigenvalues for random tridiagonal matrix models. The limiting distribution is well described by its moments. Here, an analytical approach allows us, as in the case of Wigner matrices, to relax the assumptions on the random variables. With this method, we proved the convergence of the spectral distribution under an assumption on the second moment. We discuss also about an algebraic approach for the tridiagonal models, which are more complicated than the classic freeness.

Paper Structure

This paper contains 16 sections, 23 theorems, 143 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Stieltjes Transform
  3. The Wasserstein space
  4. Main result
  5. Structure of the paper.
  6. The simple tridiagonal model
  7. A more general tridiagonal model
  8. Compact case
  9. The general tridiagonal model
  10. Examples
  11. Constant and classical laws
  12. Bernoulli case
  13. Heuristic approach
  14. A joint distribution result
  15. An algebraic structure
  16. ...and 1 more sections

Key Result

Theorem 1.3

Let $\alpha > 0$. Assume that all random variables $a_n$ and $b_n$ are independent and there exists a sequence $\{m_k\}_{k \geq 0}$ with $m_0 = 1$ such that Moreover, assume Denoting $X_n = \frac{1}{n^\alpha} A_n$, we have and almost surely, Moreover, $L_k$ is given by where $\Gamma_k$ is the set of one-step paths of length $k$ linking $0$ to $0$, i.e., and for any path $\gamma$, Notably, $

Figures (5)

  • Figure 1: Spectral distribution when $b_i$ follows a Pareto$(1,4)$ law for $N=10^4$.
  • Figure 2: Simulation of Ullmann's law for $\alpha = 0$, $\alpha = 0.2$, and $\alpha = 0.8$ with $N = 2000$.
  • Figure 3: Limiting distribution with parameters $p = 0.5$ (left) and $p = 0.9$ (right).
  • Figure 4: A 4-step path.
  • Figure 5: Example of a 2-colored path.

Theorems & Definitions (45)

  • Theorem 1.3: I. Popescu
  • Definition 1.4: Stieltjes Transform
  • Proposition 1.5
  • Example 1.6
  • Example 1.7
  • Definition 1.8
  • Lemma 1.9
  • Theorem
  • Lemma 2.2: Hoffman–Wielandt
  • Lemma 2.3
  • ...and 35 more