2004/01/31 by P. J. Forrester, P. J Forrester, J. P. Keating +1 · 28 citations
Mathematics · Physics and Astronomy · #Characteristic polynomial #Complex plane #Conjecture #Eigenvalues and eigenvectors #Mathematical functions and polynomials #Matrix (chemical analysis) #Polynomial #Random Matrices and Applications #Random matrix #Real line #Singularity #Spectral Theory in Mathematical Physics #Unit circle #math-ph #math.MP
paper · pdf · doi:10.1007/s00220-004-1121-8
published in Communications in Mathematical Physics 250(1), 119-131 (Springer Science+Business Media) · 11 pages, to appear Commun. Math. Phys
arxiv created 2004/01/31 · openalex publication_date 2004/06/10 · arxiv updated 2015/06/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
The circular and Jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval (0,1) of the real line respectively. The averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 μ diverges, for 2μ≤ -1, at points approaching the eigenvalue support. Using the theory of generalized hypergeometric functions based on Jack polynomials, the functional form of the leading asymptotic behaviour is established rigorously. In the circular ensemble case this confirms a conjecture of Berry and Keating.