Spin fluctuations in quantum dots

We explore the static longitudinal and dynamic transverse spin susceptibilities in quantum dots and nanoparticles within the framework of the Hamiltonian that extends the universal Hamiltonian to the case of uniaxial anisotropic exchange. For the limiting cases of Ising and Heisenberg exchange interactions, we ascertain how fluctuations of single-particle levels affect the Stoner instability in quantum dots. We reduce the problem to the statistics of extrema of a certain Gaussian process. We prove that, despite possible strong randomness of the single-particle levels, the spin susceptibility and all its moments diverge simultaneously at the point which is determined by the standard criterion of the Stoner instability involving the mean level spacing only.