We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.
Summary.We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on 7/e or zdx N with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ;ge or on a slab ;ge x {0,..., K } and prove both that percolation occurs and that the infinite component is unique for V = ~2 x {0, 1} or larger.
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