2016/03/31 by B. Shapiro, B Shapiro, K. Zarembo +1 · 6 citations
Computer Science · Mathematics · Physics and Astronomy · #Fixed point #Level crossing #Matrix (chemical analysis) #Matrix Theory and Algorithms #Matrix function #Perturbation (astronomy) #Probability distribution #Random Matrices and Applications #Random matrix #Spectral Theory in Mathematical Physics #Symmetric matrix #hep-th #math-ph #math.MP
paper · pdf · doi:10.1088/1751-8121/aa5186
published in Journal of Physics A Mathematical and Theoretical 50(4), 045201 (Institute of Physics) · 35 pages, 16 figures; v2: Introduction and sec. 3.3 expanded, refs. added
openalex created_date 2016/06/24 · arxiv created 2016/12/10 · openalex publication_date 2016/12/23 · arxiv updated 2017/02/01 · openalex updated_date 2026/08/06
Abstract We consider level crossing in a matrix family <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>H</mml:mi> <mml:mo>=</mml:mo> <mml:mstyle displaystyle="false"> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mn>0</mml:mn> </mml:msub> </mml:mstyle> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>V</mml:mi> </mml:mstyle> </mml:math> where H 0 is a fixed <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mi>N</mml:mi> <mml:mo>×</mml:mo> <mml:mi>N</mml:mi> </mml:mstyle> </mml:math> matrix and V belongs to one of the standard Gaussian random matrix ensembles. We study the probability distribution of level crossing points in the complex plane of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>λ</mml:mi> </mml:math> , for which we obtain a number of exact, asymptotic and approximate formulas.