  Here we prove critical exponents for Random Connections Models (RCMs) with
random marks. The vertices are given by a marked Poisson point process on
$\mathbb{R}^d$ and an edge exists between any pair of vertices independently
with a probability depending upon their spatial displacement and on their
respective marks. Given conditions on the edge probabilities, we prove
mean-field lower bounds for the susceptibility and percolation functions. In
particular, we prove the equality of the susceptibility and percolation
critical intensities. If we assume that a form of the triangle condition holds,
then we also prove that the susceptibility, percolation and cluster tail
critical exponents exist and take their mean-field values. Our proof approach
adapts the differential inequality and magnetization function approaches that
have been previously applied to discrete homogeneous settings to our continuum
marked setting. This includes a proof of the analyticity of the magnetization
function in the required parameter regime.
