SU(1, 1) Random Polynomials
Purchase on Springer.com
$39.95 / €34.95 / £29.95*
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.
We study statistical properties of zeros of random polynomials and random analytic functions associated with the pseudoeuclidean group of symmetries SU(1, 1), by utilizing both analytical and numerical techniques. We first show that zeros of the SU(1, 1) random polynomial of degree N are concentrated in a narrow annulus of the order of N −1 around the unit circle on the complex plane, and we find an explicit formula for the scaled density of the zeros distribution along the radius in the limit N→∞. Our results are supported through various numerical simulations. We then extend results of Hannay(1) and Bleher et al. (2) to derive different formulae for correlations between zeros of the SU(1, 1) random analytic functions, by applying the generalized Kac–Rice formula. We express the correlation functions in terms of some Gaussian integrals, which can be evaluated combinatorially as a finite sum over Feynman diagrams or as a supersymmetric integral. Due to the SU(1, 1) symmetry, the correlation functions depend only on the hyperbolic distances between the points on the unit disk, and we obtain an explicit formula for the two point correlation function. It displays quadratic repulsion at small distances and fast decay of correlations at infinity. In an appendix to the paper we evaluate correlations between the outer zeros |z j |>1 of the SU(1, 1) random polynomial, and we prove that the inner and outer zeros are independent in the limit when the degree of the polynomial goes to infinity.
- J. H. Hannay, Chaotic analytic zero points: exact statistics for those of a random spin state, J. Phys. A 29:101-105 (1996).
- P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142:351-395 (2000).
- E. Bogomolny, O. Bohigas, and P. Leboeuf, Quantum chaotic dynamics and random polynomials, J. Statist. Phys. 85:639-679 (1996).
- P. Leboeuf and P. Shukla, Universal fluctuations of zeros of chaotic wave functions, J. Phys. A 29:4827-4835 (1996).
- H. J. Korsch, C. Miller, and H. Wiescher, On the zeros of the Husimi distribution, J. Phys. A 30:L677-L684 (1997).
- S. Nonnenmacher and A. Voros, Chaotic eigen functions in phase space, J. Statist. Phys. 92:431-518 (1998).
- P. J. Forrester and G. Honner, Exact statistical properties of the zeros of complex random polynomials, J. Phys. A 32:2961-2981 (1999).
- P. Leboeuf, Random analytic chaotic eigenstates, J. Statist. Phys. 95:651-664 (1999).
- B. Shiffman and S. Zelditch, Distribution of zeros of random and quantum chaotic sections of positive line bundles, Commun. Math. Phys. 200:661-683 (1999).
- M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, Boston, 1991).
- M. Shub and S. Smale, Complexity of Bezout's theorem II: Volumes and probabilities, in Computational Algebraic Geometry (Nice, 1992), Progr. Math. 109, Birkhäuser, Boston (1993), pp. 267-285.
- P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of zeros on symplectic manifolds, in Random Matrix Models and Their Applications, Vol. 40, P. Blehr and A. Its, eds. (Cambridge University Press, Cambridge, 2001), pp. 31-70, http://xxx.lanl.gov/abs/ math-ph/0002039.
- P. Bleher, B. Shiffman, and S. Zelditch, Poincaré-Lelong approach to universality and scaling of correlations between zeros, Commun. Math. Phys. 208:771-785 (2000).
- P. Bleher, B. Shiffman, and S. Zelditch, Correlations between zeros and supersymmetry, http://xxx.lanl.gov/abs/math-ph/0011016 (to appear in Commun. Math. Phys.).
- B. Shiffman and S. Zelditch, Random almost holomorphic sections of ample line bundles on symplectic manifolds, e-print (2000), http://xxx.lanl.gov/abs/math.SG/0001102.
- B. Jancovici and G. Téllez, Two-dimensional Coulomb systems on a surface of constant negative curvature, J. Statist. Phys. 91:953-977 (1998).
- K. Zyczkowski and H.-J. Sommers, Truncations of random unitary matrices, J. Phys. A 60:2045-2057 (2000).
- P. Bleher and X. Di, Correlations between zeros of a random polynomial, J. Statist. Phys. 88:269-305 (1997).
- A. Bharucha-Reid and M. Sambadham, Random Polynomials (Academic Press, New York, 1986).
- M. Kac, On the average number of real roots of a random algebraic equation, Bull. Amer. Math. Soc. 49:314-320 (1943).
- S. O. Rice, Mathematical analysis of random noise, Bell System Tech. J. 23:282-332 (1944); 24:46-156 (1945); reprinted in Selected Papers on Noise and Stochastic Processes, (Dover, New York, 1954), pp. 133-294.
- SU(1, 1) Random Polynomials
Journal of Statistical Physics
Volume 106, Issue 1-2 , pp 147-171
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers-Plenum Publishers
- Additional Links
- random polynomial
- correlations between zeros
- Industry Sectors