Level compressibility in a critical random matrix ensemble: the second virial coefficient

We study spectral statistics of a Gaussian unitary critical ensemble of almost diagonal Hermitian random matrices with off-diagonal entries (vertical bar Hij vertical bar2) ∼ b2 vertical bar |i - j vertical bar-2 small compared to diagonal ones (vertical bar Hii vertical bar2) ∼ 1. Using the recently suggested method of virial expansion in the number of interacting energy levels (Yevtushenko and Kravtsov 2003 J. Phys. A: Math. Gen. 36 8265), we calculate a coefficient ∼b2 << 1 in the level compressibility χ(b). We demonstrate that only the leading terms in χ(b) coincide for this model and for an exactly solvable model suggested by Moshe et al (1994 Phys. Rev. Lett. 73 1497), the sub-leading terms ∼b2 being different. Numerical data confirm our analytical calculation