  We study the Conjugate Kernel associated to a multi-layer linear-width
feed-forward neural network with random weights, biases and data. We show that
the empirical spectral distribution of the Conjugate Kernel converges to a
deterministic limit. More precisely we obtain a deterministic equivalent for
its Stieltjes transform and its resolvent, with quantitative bounds involving
both the dimension and the spectral parameter. The limiting equivalent objects
are described by iterating free convolution of measures and classical matrix
operations involving the parameters of the model.
