We consider the random connection model for three versions of the connection function ϕ: A finite-variance version (including the Boolean model), a spread-out version, and a long-range version. We adapt the lace expansion to fit the framework of the underlying continuum-space Poisson point process to derive the triangle condition in sufficiently high dimension and furthermore to establish the infra-red bound. From this, mean-field behavior of the model can be deduced. As an example, we show that the critical exponent γ takes its mean-field value γ = 1 and that the percolation function is continuous.

We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

We use the lace expansion to prove an infra-red bound for site percolation on the hypercubic lattice in high dimension. This implies the triangle condition and allows us to derive several critical exponents that characterize mean-field behavior in high dimensions.

We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

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