In (supercritical) Bernoulli bond percolation on $\mathbb{Z}^d$, the proportion of vertices in the largest cluster restricted to a volume-$n$ box converges to $\theta$: the probability that the origin lies in an infinite cluster. The probability that this proportion is smaller than $\theta-\varepsilon$ decays stretched exponentially with exponent strictly smaller than one. The probability that the largest cluster is much larger than expected decays exponentially. Thus, the upper tail decays much faster than the lower tail.

In this talk, we will see that the discrepancy between the tails is reversed in supercritical spatial random graph models in which the degrees have heavy tails. In particular, we will focus on the soft heavy-tailed Poisson-Boolean model. The lower tail decays stretched exponentially, with an exponent that is determined by the strongest of three competing effects. In contrast, the upper tail decays now polynomially, and thus decays much slower than the lower tail. We will give intuition for the exponent of this polynomial, which is determined by the generating function of the finite cluster-size distribution.

Joint work with Júlia Komjáthy and Dieter Mitsche.