The authors derive analytical expressions for the connections of large perceptrons, by studying the fixed points of the perceptron learning rule. If the training set consists of all possible input vectors, they can calculate (for large systems) the connections as a series expansion in the system size. The leading term in this expansion turns out to be either the Hebb rule (for unbiased distributions) or the biased Hebb rule (for biased distributions). The performance of their asymptotic expressions (and finite-size corrections) on small systems is studied numerically. For the more realistic case of having an extensive training set (patterns learned with training noise) they derive a self-consistent set of coupled nonlinear equations for the connections. In the limit of zero training noise, the solution of these equations is shown to give the connections with maximal stability in the Gardner sense.