Supersymmetric nonlinear sigma models arise in the theory of disordered systems and share key features with O(N)-type models. They also exhibit surprising connections with probabilistic models such as the vertex-reinforced jump process and the arboreal gas. In my talk, I will consider a family of nonlinear sigma models on ℤ^d whose target space is the hyperbolic supermanifold H^(2∣2n), n>1, and show the exponential decay of the two-point function in the high-temperature regime T⩾C n, with C>0 a universal constant, for any n>1 and d⩾1. The proof is based on the reduction to a marginal fermionic theory and combines a high-temperature cluster expansion with Grassmann norm bounds. Based on joint work with M. Disertori and J. Duràn Fernàndez.