Abstract:

Given an infinite transitive graph (such as the standard lattice $Z^d$), build a random subgraph by independently including each edge with probability $p$. This model undergoes a phase transition as $p$ varies across the critical value $p_c$ marking the emergence of an infinite component. A fundamental result is that for any fixed $p < p_c$, the probability that a given vertex is in a component of size at least $n$ decays exponentially with $n$. We will prove the analogous result for the supercritical regime, $p > p_c$.

Based on a (very recent) joint work with Easo, Ramanan-Radhakrishnan, Sudakov, and Tassion.