Gardner and Derrida have introduced a natural version of the problem of the capacity of the binary perceptron “with temperature,”and they proposed (based on “physical” methods) remarkable formulas for this model. We give a complete rigorous proof that these formulas are correct at sufficiently high temperature for a much larger class of models.
"Intersecting random half-spaces: toward the Gardner-Derrida formula." Ann. Probab. 28 (2) 725 - 758, April 2000. https://doi.org/10.1214/aop/1019160259