Abstract: The starting point is Bernoulli percolation on Z^d, where we keep each edge with probability p. If the p is larger than the critical value, the model contains an infinite component. We can then ask what is the probaiblity that the component of the origin is bigger than k but not infinite? This is known to decay stretched exponentially in k with stretch exponent (d-1)/d, which can be explained by "surface tension". 

We consider the same question in a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation/geometric inhomogeneous random graphs and the age-dependent random connection model. 

We identify the stretch-exponent zeta in (0,1) of the subexponential decay of the cluster-size distribution. That is, with C(0) denoting the number of vertices in the component of the vertex at the origin,

 
  P ( C is larger than k but finite ) = exp ( -k^zeta) as k tends to infinity
 
 The value of zeta undergoes several phase transitions with respect to three main model parameters: the Euclidean dimension d, the power-law tail exponent tau of the degree distribution and a long-range parameter alpha governing the presence of long edges in Euclidean space.

In this seminar I will describe the connection to the second largest component, and lower large deviations of the giant component, and present the key ideas of the proof. Based on joint work with Joost Jorritsma (Tu/e) and Dieter Mitsche (Lyon).