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An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves

Turgay Bayraktar, Emel Karaca

TL;DR

This work addresses the rarity of maximal real components in real algebraic curves arising from random real holomorphic sections of an ample line bundle over a real projective surface. It extends Gayet–Welschinger’s Gaussian-result framework to sub-Gaussian coefficients on smooth metrics, leveraging mass asymptotics and concentration inequalities to obtain a sharper exponential decay, $\text{Prob}_n(M^{a}_{n})\le C n^{2} e^{-n^{2}}$. The analysis identifies the limiting complex zero distribution with the extremal current $dd^{c}\varphi_e$ and shows this current is nowhere weakly laminar in the bulk, linking probabilistic geometry with laminar current theory. These results broaden the probabilistic understanding of real zero loci, providing stronger decay rates under more general coefficient distributions and metric assumptions, with implications for the topology of random real algebraic sets.

Abstract

We prove that maximal real algebraic curves associated with sub-Gaussian random real holomorphic sections of a smoothly curved ample line bundle are exponentially rare. This generalizes the result of Gayet and Welschinger \cite{GW} proved in the Gaussian case for positively curved real holomorphic line bundles.

An Exponential Rarefaction Result for Sub-Gaussian Real Algebraic Maximal Curves

TL;DR

This work addresses the rarity of maximal real components in real algebraic curves arising from random real holomorphic sections of an ample line bundle over a real projective surface. It extends Gayet–Welschinger’s Gaussian-result framework to sub-Gaussian coefficients on smooth metrics, leveraging mass asymptotics and concentration inequalities to obtain a sharper exponential decay, . The analysis identifies the limiting complex zero distribution with the extremal current and shows this current is nowhere weakly laminar in the bulk, linking probabilistic geometry with laminar current theory. These results broaden the probabilistic understanding of real zero loci, providing stronger decay rates under more general coefficient distributions and metric assumptions, with implications for the topology of random real algebraic sets.

Abstract

We prove that maximal real algebraic curves associated with sub-Gaussian random real holomorphic sections of a smoothly curved ample line bundle are exponentially rare. This generalizes the result of Gayet and Welschinger \cite{GW} proved in the Gaussian case for positively curved real holomorphic line bundles.
Paper Structure (9 sections, 10 theorems, 58 equations)

This paper contains 9 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Laminar currents
  4. Quantum Ergodicity for Random Holomorphic Sections
  5. Subgaussian random variables
  6. Hanson-Wright Inequality
  7. Bergman Kernel Asymptotics
  8. Mass Asymptotics
  9. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Let $X$ be a real projective surface and $L\to X$ be a real holomorphic ample line bundle endowed with a real $\mathcal{C}^2$ metric $h=e^{-\varphi}$. Assume that $X_{h}^+$ is non-empty. Then for each $a>0$ there exists constants $C > 0$ and $n_0\in\Bbb{N}$ such that for $n\geq n_0$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Example 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6
  • Proposition 3.1
  • Theorem 3.2
  • ...and 6 more