[Paper Review] Large deviations of empirical measures of zeros on Riemann surfaces

Steve Zelditch|arXiv (Cornell University)|Jan 2, 2011

Geometry and complex manifolds22 references3 citations

TL;DR

This paper establishes a large deviations principle (LDP) for the empirical measures of zeros of random holomorphic sections of line bundles over compact Riemann surfaces of genus $ g \geq 1 $, generalizing previous results from $ \mathbb{C}\mathbb{P}^1 $ to higher genus. Using Gaussian-type measures on the vortex moduli space (the projectivized Picard bundle), the authors compute the joint probability current (JPC) of zeros via higher-genus analogues of Vandermonde determinants, prime forms, and bosonization, leading to an explicit rate function for the LDP.

ABSTRACT

This is a continuation of a project on large deviations for the empirical measures of zeros of random holomorphic sections of random line bundles over a Riemann surface X. In a previous article with O. Zeitouni (arXiv:0904.4271), we proved an LDP for random polynomials in the genus zero case. In higher genus, there is a Picard variety of line bundles and so the line bundle L is a random variable as well as the section s. The space of pairs (L, s) is known as the "vortex moduli space". The zeros of (L, s) fill out the configuration space $X^{(N)}$ of $N$ points of $X$. The LDP shows that the configurations concentrate at one equilibrium measure exponentially fast. The new features of the proof involve Abel-Jacobi theory, the prime form and bosonization.

Motivation & Objective

• To extend the large deviations principle (LDP) for empirical measures of zeros from genus 0 (Riemann sphere) to compact Riemann surfaces of genus $ g \geq 1 $.
• To define and analyze Gaussian-type probability measures on the total space of holomorphic sections of line bundles of degree $ N $, now varying over the $ g $-dimensional Picard variety $ \mathrm{Pic}^N $.
• To compute the joint probability current (JPC) of zeros as a volume form on the symmetric product $ X^{(N)} = \mathrm{Sym}^N X $, using higher-genus special functions like the prime form and Green's functions.
• To derive an explicit rate function for the LDP governing the asymptotic probability that a configuration of $ N $ points arises as the zero set of a random section, as $ N \to \infty $.
• To establish the analytic equivalence between the projectivized Picard bundle $ \mathbb{P}\mathcal{E}^N $ and the configuration space $ X^{(N)} $, enabling the pushforward of measures to empirical measures.

Proposed method

• Define Gaussian-type measures on the total space $ \mathcal{E}^N = \bigcup_{\xi \in \mathrm{Pic}^N} H^0(X, \xi) $, using $ L^2 $ norms with respect to Hermitian metrics on line bundles.
• Projectivize the space of sections to $ \mathbb{P}\mathcal{E}^N $, and endow it with Fubini-Study volume forms on fibers and Haar measure on the base $ \mathrm{Pic}^N $, forming the Fubini-Study-fiber ensemble.
• Compute the joint probability current (JPC) $ \vec{K}^N $ as a volume form on $ X^{(N)} $, using bosonization formulae involving the prime form $ E(z,w) $, Green's functions, and theta characteristics.
• Express the JPC in terms of products of Green's functions and determinants involving the prime form, generalizing the genus 0 Vandermonde-type determinants.
• Use the Chern form representation of the JPC to relate it to curvature forms and characteristic classes on the moduli space.
• Derive the rate function for the LDP by analyzing the logarithmic asymptotics of the JPC, leading to a variational formula involving energy functionals on $ \mathcal{M}(X) $.

Experimental results

Research questions

• RQ1 How does the large deviations principle for zero distributions of random holomorphic sections extend from genus 0 to higher genus Riemann surfaces?
• RQ2 What is the explicit form of the joint probability current (JPC) of zeros on $ X^{(N)} $ for random sections of line bundles over a genus $ g \geq 1 $ surface?
• RQ3 How do higher-genus analogues of Vandermonde determinants and special functions like the prime form and Green's function enter into the computation of the JPC?
• RQ4 What is the role of the Picard variety $ \mathrm{Pic}^N $ in the statistical mechanics of zero sets, and how does it affect the measure on $ X^{(N)} $?
• RQ5 How is the rate function for the LDP derived from the asymptotic behavior of the JPC as $ N \to \infty $?

Key findings

• The paper establishes a large deviations principle (LDP) for the empirical measure $ \mu_\zeta = \frac{1}{N}\sum_{j=1}^N \delta_{\zeta_j} $ of zeros of random holomorphic sections over compact Riemann surfaces of genus $ g \geq 1 $.
• The joint probability current (JPC) $ \vec{K}^N $ of zeros is computed explicitly as a volume form on $ X^{(N)} $, involving products of Green's functions and the prime form $ E(z,w) $, generalizing the genus 0 case.
• The JPC is shown to be expressible as a Chern form associated with the determinant line bundle over the moduli space, linking it to curvature and characteristic classes.
• The rate function for the LDP is derived from the logarithmic asymptotics of $ \vec{K}^N $, yielding a variational formula that minimizes a certain energy functional over probability measures on $ X $.
• The authors identify the Fubini-Study-fiber ensemble on $ \mathbb{P}\mathcal{E}^N $ as the natural probability measure on the vortex moduli space, which is analytically equivalent to $ X^{(N)} $ via the zero set map.
• The computation relies on bosonization formulae from higher-genus Riemann surface theory, including the use of automorphic factors and theta functions with characteristics, to handle the non-trivial topology of $ \mathrm{Pic}^N $.