Central limit theorem for spectral partial bergman kernels

Partial Bergman kernels Π_k,Eare kernels of orthogonal projections onto subspaces S_k⊂ H0(M,L^k) of holomorphic sections of the kth power of an ample line bundle over a Kähler manifold (M; Ω). The subspaces of this article are spectral subspaces f H_k≤ Eg of the Toeplitz quantization H_kof a smooth Hamiltonian H : M → R. It is shown that the relative partial density of states Π_k,E(z)/Π_k(z)→ 1A where A = fH < Eg. Moreover it is shown that this partial density of states exhibits 'Erf'-asymptotics along the interface ∂A, that is, the density profile asymptotically has a Gaussian error function shape interpolating between the values 1; 0 of 1_A. Such 'erf'-asymptotics are a universal edge effect. The different types of scaling asymptotics are reminiscent of the law of large numbers and central limit theorem.