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 $<|H_{ij}|^{2} > \sim b^{2} |i-j|^{-2}$ small compared to diagonal ones $<|H_{ii}|^{2} > \sim 1$. Using the recently suggested method of {\it virial expansion} in the number of interacting energy levels (J.Phys.A {\bf 36},8265 (2003)), we calculate a coefficient $\propto b^{2}\ll 1$ in the level compressibility $\chi(b)$. We demonstrate that only the leading terms in $\chi(b)$ coincide for this model and for an exactly solvable model suggested by Moshe, Neuberger and Shapiro (Phys.Rev.Lett. {\bf 73}, 1497 (1994)), the sub-leading terms $\sim b^{2}$ being different. Numerical data confirms our analytical calculation.