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 similar to b(2)vertical bar i - j vertical bar(-2) small compared to diagonal ones similar to 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 proportional to b(2) << 1 in the level compressibility chi(b). We demonstrate that only the leading terms in X (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 similar to b(2) being different. Numerical data confirm our analytical calculation.