An ensemble of random particle-hole matrices with collective eigenstates

An ensemble of random particle-hole matrices is studied as a function of the signs of the matrix elements. When all matrix elements have the same sign, a theorem due to Perron ensures the existence of a coherent state: the collective particle-hole state. The example of Gamow-Teller transitions between²⁰⁸Bi(1+) and²⁰⁸Pb(0+) is considered and the strength of the Gamow-Teller giant resonance as well as the fluctuations in the ensemble are studied. When a fractionp of matrix elements are of opposite sign, we find that a giant resonance may still exist providedp is small enough. Its magnitude and position in the spectrum are studied. The method proposed is valid for any particle hole excitation. The two body matrix elements of several interactions are analysed and are shown to fit into a unique valuep=0.2 for the 1 + states.