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Dynamics of roots of randomized derivative polynomials

André Galligo, Joseph Najnudel

TL;DR

The paper addresses the macroscopic evolution of root distributions under randomized derivatives of real-rooted polynomials. It introduces a unified β-parameter framework that recovers both derivative dynamics (β=∞) and minor-based dynamics (β∈{1,2}) and proves that, for large degree and fixed time horizon $\tau$, the limiting empirical root distribution $\mu_{[\tau]}$ is independent of $β$ and matches the non-randomized case. The authors construct a triangular Markovian coupling of polynomials $P_{n,m}$ using Dirichlet weights and analyze fluctuations of $\log P_{n,m}(z)$ via Dirichlet-Gamma representations and large deviation estimates, enabling a transfer of known non-randomized results to the randomized setting. This results in a robust hydrodynamic-type limit for the root sets and connects random matrix theory, free probability, and PDE-based descriptions of root dynamics, with potential implications for spectral dynamics under randomized linear operations. The work provides explicit probabilistic control and a rigorous path from finite-n models to deterministic macroscopic laws, broadening the understanding of how randomness in derivative-like operations shapes root distributions.

Abstract

In this paper, we study the asymptotic macroscopic behavior of the root sets of iterated, randomized derivatives of polynomials. The randomization depend on a parameter of inverse temperature $β\in (0, \infty]$, the case $β= \infty$ corresponding to the situation where one considers the derivative of polynomials, without randomization. Our constructions can be connected to random matrix theory: in particular, as detailed in Section 2, for $β= 2$ and roots on the real line, we get the distribution of the eigenvalues of minors of unitarily invariant random matrices. We prove that the asymptotic macroscopic behavior of the roots, i.e. the hydrodynamic limit, does not depend on $β$, and coincides with what we obtain for the non-randomized iterated derivatives, i.e. for $β= \infty$. Since recent results obtained for iterated derivations show that the limiting dynamics is governed by a non-local and non-linear PDE, we can transfer this information to the macroscopic behavior of the randomized setting. Our proof is completely explicit and relies on the analysis of increments in a triangular bivariate Markov chain.

Dynamics of roots of randomized derivative polynomials

TL;DR

The paper addresses the macroscopic evolution of root distributions under randomized derivatives of real-rooted polynomials. It introduces a unified β-parameter framework that recovers both derivative dynamics (β=∞) and minor-based dynamics (β∈{1,2}) and proves that, for large degree and fixed time horizon , the limiting empirical root distribution is independent of and matches the non-randomized case. The authors construct a triangular Markovian coupling of polynomials using Dirichlet weights and analyze fluctuations of via Dirichlet-Gamma representations and large deviation estimates, enabling a transfer of known non-randomized results to the randomized setting. This results in a robust hydrodynamic-type limit for the root sets and connects random matrix theory, free probability, and PDE-based descriptions of root dynamics, with potential implications for spectral dynamics under randomized linear operations. The work provides explicit probabilistic control and a rigorous path from finite-n models to deterministic macroscopic laws, broadening the understanding of how randomness in derivative-like operations shapes root distributions.

Abstract

In this paper, we study the asymptotic macroscopic behavior of the root sets of iterated, randomized derivatives of polynomials. The randomization depend on a parameter of inverse temperature , the case corresponding to the situation where one considers the derivative of polynomials, without randomization. Our constructions can be connected to random matrix theory: in particular, as detailed in Section 2, for and roots on the real line, we get the distribution of the eigenvalues of minors of unitarily invariant random matrices. We prove that the asymptotic macroscopic behavior of the roots, i.e. the hydrodynamic limit, does not depend on , and coincides with what we obtain for the non-randomized iterated derivatives, i.e. for . Since recent results obtained for iterated derivations show that the limiting dynamics is governed by a non-local and non-linear PDE, we can transfer this information to the macroscopic behavior of the randomized setting. Our proof is completely explicit and relies on the analysis of increments in a triangular bivariate Markov chain.

Paper Structure

This paper contains 6 sections, 4 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Some recent results on dynamics induced by derivation
  3. Eigenvalues of minors of unitarily invariant matrices
  4. Unification
  5. The main setting
  6. Proof of theorem \ref{['th1']}

Key Result

Proposition 1

For $\beta = \infty$, $P_{n,m}$ is $(n-m)!/n!$ times the $m$-th derivative of $P_{n,0}$. For $\beta = 2$ (respectively $\beta = 1$), $(P_{n,m})_{0 \leq m \leq n-1}$ has the same joint distribution as the characteristic polynomials of the successive top-left $(n-m) \times (n-m)$ minors of the $n \tim $U_n$ being independent of $( \lambda_j^{(n,0)})_{1 \leq j \leq n} )$ and Haar-distributed on the u

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Theorem 1
  • Proposition 2
  • proof
  • Corollary 1