Separation capacity of linear reservoirs with random connectivity matrix

A natural hypothesis for the success of reservoir computing in generic tasks is the ability of the untrained reservoir to map distinct input time series to separable reservoir states, a property we term separation capacity. In this work, we develop a rigorous mathematical framework for analysing the separation capacity of random linear reservoirs. We show that the expected separation induced by a random reservoir is completely characterised by the spectral properties of a positive semi-definite matrix naturally associated with the connectivity matrix, which we call the generalised matrix of moments. Taking Gaussian connectivity matrices as an example, we investigate how separation depends on the reservoir dimension N, the scaling of the connectivity matrix, and structural assumptions such as symmetry. In the symmetric case, we show that, although the classical scaling N^{-1/2} yields the most balanced separation for large reservoirs, the quality of separation inevitably deteriorates as the length of the input time series increases. In contrast, for reservoirs with independent and identically distributed connectivity entries, we prove that the classical scaling N^{-1/2} is asymptotically optimal from the perspective of separation and derive quantitative bounds describing the evolution of separation with the time horizon. Finally, numerical experiments suggest a strong practical connection between balanced separation profiles and downstream learning performance. Beyond providing a theoretical justification for several common reservoir design choices, our results introduce separation capacity as a tractable and informative framework for the analysis of random reservoirs.