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33
Zeros of the i.i.d. Gaussian power series: a conformally invariant determinantal process
 Acta. Math
, 2005
"... Consider the zero set of a random power series ∑ anz n with i.i.d. complex Gaussian coefficients an. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros in a disk of radius r ab ..."
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Cited by 55 (5 self)
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Consider the zero set of a random power series ∑ anz n with i.i.d. complex Gaussian coefficients an. We show that these zeros form a determinantal process: more precisely, their joint intensity can be written as a minor of the Bergman kernel. We show that the number of zeros in a disk of radius r about the origin has the same distribution as the sum of independent indicators Xk where P(Xk = 1) = r −2k. The repulsion between zeros can be studied via a dynamic version where the coefficients perform Brownian motion; we show that this dynamics is conformally invariant. 1
Random Complex Zeros III, Decay of the hole probability
 Israel J. Math
"... The ‘hole probability ’ that a random entire function ψ(z) = k=0 ζk zk √, k! where ζ0,ζ1,... are Gaussian i.i.d. random variables, has no zeroes in the disc of radius r decays as exp(−cr 4) for large r. We consider the (random) set of zeroes of a random entire function ψω: C → C, (0.1) ψ(z, ω) = k=0 ..."
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Cited by 27 (1 self)
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The ‘hole probability ’ that a random entire function ψ(z) = k=0 ζk zk √, k! where ζ0,ζ1,... are Gaussian i.i.d. random variables, has no zeroes in the disc of radius r decays as exp(−cr 4) for large r. We consider the (random) set of zeroes of a random entire function ψω: C → C, (0.1) ψ(z, ω) = k=0 ζk(ω) zk k! where ζk, k = 0, 1, 2,... are independent standard complexvalued Gaussian random variables, that is the distribution NC(0, 1) of each ζk has the density π −1 exp(−w  2) with respect to the Lebesgue measure m on C. This model is distinguished by invariance of the distribution of zero points with respect to the motions of the complex plane z ↦ → az + b, a  = 1, b ∈ C, see [6] for details and references. Given large positive r, we are interested here in the ‘hole probability ’ that ψ has no zeroes in the disc of radius r p(r) = P ( ψ(z, ·) ̸ = 0, z  ≤ r). It is not difficult to show that p(r) ≤ exp( − const r 2), see the Offordtype estimate in [5]. Yuval Peres told one of us that the recent work [4] led to conjecture that the actual hole probability might have a faster decay. In this note, we confirm this conjecture and prove 1 Theorem 1. exp(−Cr 4) ≤ p(r) ≤ exp(−cr 4). Throughout, by c and C we denote various positive numerical constants whose values can be different at each occurrence. It would be interesting to check whether there exists the limit log lim r→∞ − p(r) r4, and (if it does) to find its value. The lower bound in Theorem 1 will be obtained in Section 1 by a straightforward construction. The upper bound in Theorem 1 follows from a large deviation estimate which has an independent interest. Theorem 2. Let n(r) be a number of random zeroes in the disc {z  ≤ r}. Then for any δ ∈ (0, 1] and r ≥ 1
Random polynomials and
, 2005
"... Abstract. For a regular compact set K in C m and a measure µ on K satisfying the BernsteinMarkov inequality, we consider the ensemble PN of polynomials of degree N, endowed with the Gaussian probability measure induced by L 2 (µ). We show that for large N, the simultaneous zeros of m polynomials in ..."
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Cited by 17 (1 self)
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Abstract. For a regular compact set K in C m and a measure µ on K satisfying the BernsteinMarkov inequality, we consider the ensemble PN of polynomials of degree N, endowed with the Gaussian probability measure induced by L 2 (µ). We show that for large N, the simultaneous zeros of m polynomials in PN tend to concentrate around the Silov boundary of K; more precisely, their expected distribution is asymptotic to N m µeq, where µeq is the equilibrium measure of K. For the case where K is the unit ball, we give scaling asymptotics for the expected distribution of zeros as N → ∞. 1.
From random matrices to random analytic functions
, 2008
"... Singular points of random matrixvalued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n × n matrices. Singular points are those (random) z where the ..."
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Cited by 17 (1 self)
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Singular points of random matrixvalued analytic functions are a common generalization of eigenvalues of random matrices and zeros of random polynomials. The setting is that we have an analytic function of z taking values in the space of n × n matrices. Singular points are those (random) z where the matrix becomes singular, that is, the zeros of the determinant. This notion was introduced in the Ph.D thesis [10] of the author, where some basic facts were found. Of course, singular points are just the zeros of the (random analytic function) determinant, so in what sense is this concept novel? In case of random matrices as well as random analytic functions, the following features may be observed. 1. For very special models, usually with independent Gaussian coefficients or entries, one may solve exactly for the distribution of zeros or eigenvalues. 2. For more general models with independent coefficients or entries, under rather weak assumptions on moments, one can usually analyze the empirical measure of eigenvalues or zeros as the size of the matrix increases or the
Overcrowding estimates for zeroes of Planar and Hyperbolic Gaussian analytic functions
 Journal of Statistical Physics
"... We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as m → ∞. For the Planar Gaussian analytic function, ∑ n≥0 anzn √ , we show that n! this probabili ..."
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Cited by 11 (1 self)
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We consider the point process of zeroes of certain Gaussian analytic functions and find the asymptotics for the probability that there are more than m points of the process in a fixed disk of radius r, as m → ∞. For the Planar Gaussian analytic function, ∑ n≥0 anzn √ , we show that n! this probability is asymptotic to e−1 2m2 log(m). For the Hyperbolic Gaussian analytic functions, ∑ ) 1/2anz n≥0 n, ρ> 0, we show that this probability decays like e−cm2.
Overcrowding and hole probabilities for random zeros on complex manifolds
 Indiana Univ. Math. J
, 1977
"... Abstract. We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a random holomorphic section of the Nth power of a positive line bundle on a compact Kähler manifold. In particular, we show that for al ..."
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Cited by 11 (4 self)
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Abstract. We give asymptotic large deviations estimates for the volume inside a domain U of the zero set of a random polynomial of degree N, or more generally, of a random holomorphic section of the Nth power of a positive line bundle on a compact Kähler manifold. In particular, we show that for all δ> 0, the probability that this volume differs by more than δN from its average value is less than exp(−Cδ,UN m+1), for some constant Cδ,U> 0. As a consequence, the “hole probability ” that a random section does not vanish in U has an upper bound of the form exp(−CUN m+1). 1.
From random polynomials to symplectic geometry
 XIIIth International Congress on Mathematical Physics (London, 2000), 367– 376, Int
, 2001
"... Abstract. We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree N on length scales of order D √ (complex case), resp. N D ( ..."
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Cited by 6 (1 self)
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Abstract. We review some recent results on random polynomials and their generalizations in complex and symplectic geometry. The main theme is the universality of statistics of zeros and critical points of (generalized) polynomials of degree N on length scales of order D √ (complex case), resp. N D (real case).