[Paper Review] Critical values of Gaussian SU(2) random polynomials

Renjie Feng, Zhenan Wang|arXiv (Cornell University)|Oct 17, 2012

Geometry and complex manifolds7 references3 citations

TL;DR

This paper studies the expected distribution of critical values of Gaussian SU(2) random polynomials as the degree $ n \to \infty $. Using the Kac-Rice formula, it derives the limiting density of critical values of $ |p_n| $, showing they accumulate at infinity and converge to an exponential distribution $ e^{-x} $ for $ x > 0 $, with zero density for $ x < 0 $, revealing a universal scaling limit in the rescaled critical value space.

ABSTRACT

In this note, we will get the estimate of the expected distribution of critical values of Gaussian SU(2) random polynomials as the degree large enough. The result is a direct application of the Kac-Rice formula. The critical values will accumulate at infinity, we further study the rate of this convergence and its rescaling limit as $n o\infty$.

Motivation & Objective

• To understand the asymptotic distribution of nonvanishing critical values of $ |p_n| $ for Gaussian SU(2) random polynomials as $ n \to \infty $.
• To determine the rescaling limit of the expected density of critical values, particularly how they accumulate at infinity.
• To apply the Kac-Rice formula to compute the expected density of critical values of $ p_n $, and then derive the corresponding density for $ |p_n| $ via polar coordinates.
• To establish the convergence of the rescaled critical value density to an exponential law $ e^{-x} $ for $ x > 0 $, and zero for $ x < 0 $.

Proposed method

• Uses the Kac-Rice formula to compute the expected density $ \mathbb{D}_{p_n}(x) $ of critical values of $ p_n $, which are points where $ p_n'(z) = 0 $.
• Applies a change of variables to the critical value density to derive the expected density of $ |p_n| $, mapping complex critical values to nonnegative real values.
• Introduces a rescaling of the critical value variable by $ x \mapsto nx $, leading to a limiting measure $ R_n(x) $ that captures the asymptotic behavior.
• Employs dominated convergence and asymptotic analysis to evaluate $ \lim_{n \to \infty} R_n(x) $, showing convergence to $ e^{-x} $ for $ x > 0 $ and 0 for $ x < 0 $.
• Analyzes the behavior of $ y_n(t) = nt^{n-1} - (n-1)t^n $ on $ [0,1] $, which arises in the density integral, and uses its pointwise limits to justify convergence.
• Uses integration by parts and variable substitution to decompose the integral $ R_n(x) $ into components that can be bounded and shown to vanish in the limit.

Experimental results

Research questions

• RQ1 How does the expected density of critical values of $ |p_n| $ behave as $ n \to \infty $?
• RQ2 What is the rescaling limit of the critical value distribution, and does it converge to a universal law?
• RQ3 Why do critical values of $ |p_n| $ accumulate at infinity, and what is the rate of this accumulation?
• RQ4 How does the Kac-Rice formula apply to the critical values of SU(2) random polynomials, and what is the resulting density?
• RQ5 What is the limiting behavior of the rescaled critical value density $ R_n(x) $ for $ x > 0 $, $ x = 0 $, and $ x < 0 $?

Key findings

• The expected density of critical values of $ p_n $ is derived via the Kac-Rice formula, providing a foundation for analyzing $ |p_n| $.
• The rescaled expected density of nonvanishing critical values of $ |p_n| $ converges to $ e^{-x} $ for $ x > 0 $ as $ n \to \infty $, indicating a universal exponential tail.
• For $ x < 0 $, the limiting rescaled density $ \lim_{n \to \infty} R_n(x) = 0 $, showing no mass in the negative real axis.
• At $ x = 0 $, the limiting density is $ \pi \cdot \lim_{n \to \infty} \mathbb{D}_{p_n}(1) $, a finite constant derived from the leading term of the density at unit modulus.
• The convergence of $ \int_0^1 e^{-y_n(t)e^{nx}} dt \to e^{-x} $ is established via dominated convergence, relying on pointwise limits of $ y_n(t)e^{nx} $.
• The remainder term $ Q_n(x) \to 0 $ as $ n \to \infty $, which is crucial for proving the exponential limit, by splitting the integral and bounding each part using exponential decay.