The Probability Distribution of Spectral Moments for the Gaussian β-Ensembles |

T. Maciążek
^{a}, C.H. Joyner^{ b} and U. Smilansky^{ c}^{a}Center for Theoretical Physics, Polish Academy of Sciences, al. Lotników 32/46, 02-668 Warsaw, Poland
^{b}School of Mathematical Sciences, Queen Mary University of London, London, E1 4NS, UK
^{c}Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel |

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We derive the joint probability distribution of the first two spectral moments for the Gaussian β-ensemble random matrix ensembles in N dimensions for any N. This is achieved by making use of two complementary invariants of the domain in ℝ^{N} where the spectral moments are defined. Our approach is significantly different from those employed previously to answer related questions and potentially offers new insights. We also discuss the problems faced when attempting to include higher spectral moments. |

DOI: 10.12693/APhysPolA.128.983 PACS numbers: 02.10.Yn, 02.50.-r, 02.70.Rr, 05.10.Gg, 05.40.-a, 05.90.+m |