Distribution of roots of random real generalized polynomials |
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Authors: | G. Andrei Mezincescu Daniel Bessis Jean-Daniel Fournier Giorgio Mantica Francisc D. Aaron |
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Affiliation: | (1) Service de Physique Théorique, C.E. Saclay, F-91191 Gif-sur-Yvette Cedex, France;(2) Institul de Fizica i Tehnologia Materialelor, Bucureti-Mgurele, Romania;(3) CTSPS, Clark-Atlanta University, 30314 Atlanta, Georgia;(4) Laboratoire Cassini, Observatoire de Nice, F-06304 Nice Cedex 4, France;(5) Universita di Como, I-22100 Como, Italia;(6) Facultatea de Fizic, Universitatea Bucureti, Bucureti-Mgurele, Romania |
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Abstract: | The average density of zeros for monic generalized polynomials,, with real holomorphic ,fk and real Gaussian coefficients is expressed in terms of correlation functions of the values of the polynomial and its derivative. We obtain compact expressions for both the regular component (generated by the complex roots) and the singular one (real roots) of the average density of roots. The density of the regular component goes to zero in the vicinity of the real axis like |lmz|. We present the low- and high-disorder asymptotic behaviors. Then we particularize to the large-n limit of the average density of complex roots of monic algebraic polynomials of the form with real independent, identically distributed Gaussian coefficients having zero mean and dispersion. The average density tends to a simple,universal function of =2nlog|z| and in the domain coth(/2)n|sin arg(z)|, where nearly all the roots are located for largen. |
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Keywords: | Random polynomials density of roots universal asymptotic behavior |
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