Continuum percolation and stochastic epidemic models on Poisson and Ginibre point processes

Continuum percolation for Gibbs point processes ∗

We consider percolation properties of the Boolean model generated by a Gibbs point process and balls with deterministic radius. We show that for a large class of Gibbs point processes there exists a critical activity, such that percolation occurs a.s. above criticality.

Continuum Percolation for Gaussian zeroes and Ginibre eigenvalues

We study continuum percolation on certain negatively dependent point processes on R. Specifically, we study the Ginibre ensemble and the planar Gaussian zero process, which are the two main natural models of translation invariant point processes on the plane exhibiting local repulsion. For the Ginibre ensemble, we establish the uniqueness of infinite cluster in the supercritical phase. For the ...

Continuum Limits for Critical Percolation and Other Stochastic Geometric Models

The talk presented at ICMP 97 focused on the scaling limits of critical percolation models, and some other systems whose salient features can be described by collections of random lines. In the scaling limit we keep track of features seen on the macroscopic scale, in situations where the short–distance scale at which the system’s basic variables are defined is taken to zero. Among the challengi...

Continuum percolation for Gibbsian point processes with attractive interactions

We study the problem of continuum percolation in infinite volume Gibbs measures for particles with an attractive pair potential, with a focus on low temperatures (large β). The main results are bounds on percolation thresholds ρ±(β) in terms of the density rather than the chemical potential or activity. In addition, we prove a variational formula for a large deviations rate function for cluster...

Stochastic epidemic models with Poisson infection and carrier rates