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On the logarithmic energy of solutions to the polynomial eigenvalue problem

Diego Armentano, Federico Carrasco, Marcelo Fiori

TL;DR

The paper introduces a unifying PEVP-ensemble $F(z)=\det\left(\sum_{i=0}^d G_i{d\choose i}^{1/2}z^i\right)$ with $N=dr$ roots projected to $\mathbb{S}^2$, bridging the Shub–Smale polynomial ensemble and the spherical ensemble. It proves exact formulas for the expected logarithmic energy: $\mathbb{E}\big(V(\hat z_1,\dots,\hat z_N)\big)=\frac{N^2}{4}-\frac{N\log d}{4}-\frac{N}{4}\bigl(1+\psi(r+1)-\psi(2)\bigr)$ and $\mathbb{E}\big(V(x_1,\dots,x_N)\big)=\frac{\kappa}{2}N^2-\frac{N\log d}{4}-\frac{N}{4}\bigl(1+\psi(r+1)-\psi(2)-2\log 2\bigr)$, with $\kappa=\tfrac{1}{2}-\log 2$. The results show the energy lies between the two extremal cases and decreases linearly with $d$, a finding supported by numerical experiments that illustrate the transition between the ABS and spherical ensembles as $d$ varies. This advances understanding of how intermediate PEVP configurations distribute energy on the sphere and provides exact asymptotics for practical configurations. The approach leverages coarea/Kac–Rice techniques and random-matrix theory to obtain closed-form expressions for the expected energy.

Abstract

In this paper, we compute the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices. We generalize some known results for the Shub-Smale polynomials, and the spherical ensemble. These two processes are the two extremal particular cases of the polynomial eigenvalue problem, and we prove that the logarithmic energy lies between these two cases. In particular, the roots of the Shub-Smale polynomials are the ones with the lowest logarithmic energy of the family.

On the logarithmic energy of solutions to the polynomial eigenvalue problem

TL;DR

The paper introduces a unifying PEVP-ensemble with roots projected to , bridging the Shub–Smale polynomial ensemble and the spherical ensemble. It proves exact formulas for the expected logarithmic energy: and , with . The results show the energy lies between the two extremal cases and decreases linearly with , a finding supported by numerical experiments that illustrate the transition between the ABS and spherical ensembles as varies. This advances understanding of how intermediate PEVP configurations distribute energy on the sphere and provides exact asymptotics for practical configurations. The approach leverages coarea/Kac–Rice techniques and random-matrix theory to obtain closed-form expressions for the expected energy.

Abstract

In this paper, we compute the expected logarithmic energy of solutions to the polynomial eigenvalue problem for random matrices. We generalize some known results for the Shub-Smale polynomials, and the spherical ensemble. These two processes are the two extremal particular cases of the polynomial eigenvalue problem, and we prove that the logarithmic energy lies between these two cases. In particular, the roots of the Shub-Smale polynomials are the ones with the lowest logarithmic energy of the family.
Paper Structure (8 sections, 11 theorems, 73 equations, 3 figures)

This paper contains 8 sections, 11 theorems, 73 equations, 3 figures.

Table of Contents

  1. Introduction and Main Result
  2. Technical tools and proof of Main Theorem
  3. Technical tools
  4. Geometric and probabilistic tools
  5. Random matrices computations
  6. Three little terms
  7. Proof of Main Theorem
  8. Discussion and experimental examples

Key Result

Theorem 1.1

Let $F(z)$ be the random complex polynomial of degree $N$ defined as where $G_i$ are $r \times r$ matrices with i.i.d. entries following $\mathcal{N}_{\mathbb C}(0,1)$. Then, with the definitions above, we have where $\psi(n)=\frac{\Gamma'(n)}{\Gamma(n)}$ is the digamma function, i.e., the logarithmic derivative of the gamma function $\Gamma(n)$.

Figures (3)

  • Figure 1: Empirical logarithmic energies for PEVP ensambles. The violin plots are computed using $100.000$ repetitions for each pair $(r,d)$.
  • Figure 2: Dependence of the logarithmic energy on $d$. The empirical results are the same as in Fig. \ref{['fig:todojunto']}, but with $d$ now correctly scaled on the x-axis, and the expected value from Theorem \ref{['main']} included.
  • Figure 3: Difference between the empirical results and the expected value of Theorem \ref{['main']}. Notice how the difference is centered at zero for all $d$.

Theorems & Definitions (25)

  • Theorem 1.1: Main Theorem
  • Remark 1.2
  • Theorem : BCSS p. 241 The coarea formula
  • Lemma 2.1
  • proof
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • ...and 15 more