A central limit theorem for weighted averages of spins in the high temperature region of the Sherrington-Kirkpatrick model

In this paper we prove that in the high temperature region of the Sherrington-Kirkpatrick model for a typical realization of the disorder the weighted average of spins ${\sum }_{i\le N}{t}_i{\sigma }_i$ will be approximately Gaussian provided that ${\max }_{i\le N}\mathrm{\mid }{t}_i\mathrm{\mid }\mathrm{/}{\sum }_{i\le N}{t}_i^2$ is small.