EQUILIBRIUM DISTRIBUTION OF ZEROS OF RANDOM POLYNOMIALS

By Bernard Shiffman, Steve Zelditch. 2002

86 citations recorded by the source corpus.

We consider ensembles of random polynomials of the form p(z) = P N=1 ajPj where {aj} are independent complex normal random variables and where {Pj} are the orthonormal polynomials on the boundary of a bounded simply connected analytic plane domain ⊂ C relative to an analytic weight �(z)|dz| . In the simplest case where is the unit disk and � = 1, so that Pj(z) = z j , it is known that the average distribution of zeros is the uniform measure on S 1 . We show that for any analytic (,�), the zeros of random polynomials almost surely become equidistributed relative to the equilibrium measure on @ as N → ∞. We further show that on the length scale of 1/N, the correlations have a universal scaling limit independent of (,�).

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ec09e688796e2019a78a61582a514d5d1d062214

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Betti numbers of random real hypersurfaces and determinants of random symmetric matrices (2012, background): “th respect to which we are going to consider random sections, see 1 §3.1.1 of [12] for a discussion on this choice (previously considered in [14], [21], [19], [4]) and on other possible ones (compare [20]). For every generic section σ ∈ RH0(X,Ld), the real locus RC σ of its vanishing locus is a smoo”
Equidistribution of zeros of random holomorphic sections (2013, unknown): “Corresponding asymptotic distribution of zeros of Gaussian univariate polynomials are studied in [31] .”
Asymptotic distribution of complex zeros of random analytic functions (2014, background): “In the context of random polynomials, there were many results on the universal character of local correlations between close zeros [3, 19, 29, 30].”
Around the circular law (2011, result): “Further results were obtained by Shepp and Vanderbei [133], Zeitouni and Zelditch [157], Shiffman and Zelditch [134], and by Bloom and Shiffman [21].”
ZEROS OF RANDOM POLYNOMIALS ON C m (2006, background): “unit disk in the one variable case. In that case, detailed scaling limits are known (see, for example, [IZ]). For a more general compact set K⊂ Cwith an analytic boundary, scaling limits are found in [SZ2]. In this section, we consider the case where K= {z∈ Cm : kzk ≤ 1} is the unit ball and µis its”
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process (2003, background): “(12) (cf. (6) in Shiffman and Zelditch (2003)).”
Zeros of Gaussian Analytic Functions and Determinantal Point Processes (2009, background): “For instance see Shiffman, Zelditch (76) (and references therein) who study random sections of line bundles.”