Integrable model of crossings loops and multidegree of affine schemes

We introduce, on the one hand, a certain affine scheme in the space of complex matrices, and on the other hand, an integrable stochastic process of crossing loops. We show how the two objects are related and more specifically how the statistical model provides multidegrees of the components of the affine scheme. We compute as an application the degree of the commuting variety (pairs of nЧn commuting complex matrices).