Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices
Phase Transitions and Edge Scaling of Number Variance in Gaussian Random Matrices
We consider $N\ifmmode\times\else\texttimes\fi{}N$ Gaussian random matrices, whose average density of eigenvalues has the Wigner semicircle form over $[\ensuremath{-}\sqrt{2},\sqrt{2}]$. For such matrices, using a Coulomb gas technique, we compute the large $N$ behavior of the probability ${\mathcal{P}}_{N,L}({N}_{L})$ that ${N}_{L}$ eigenvalues lie within the box $[\ensuremath{-}L,L]$. This probability scales as ${\mathcal{P}}_{N,L}({N}_{L}={\ensuremath{\kappa}}_{L}N)\ensuremath{\approx}\mathrm{exp}\mathbf{(}\ensuremath{-}\ensuremath{\beta}{N}^{2}{\ensuremath{\psi}}_{L}({\ensuremath{\kappa}}_{L})\mathbf{)}$, where …