Title:   Connected components in supercritical percolation
            on finite graphs

Abstract:

Let G=(V,E) be a graph, and let 0<p<1. Form a random subgraph G_p
of G by retaining every edge e of E(G) with probability p and
independently.

For the most basic case of G being the complete graph K_n on n
vertices, Erd\H{o}s and R\'enyi, in their foundational work from
around 1960, described the so called phase transition phenomenon:
for p=c/n with c<1, typically all connected components of the so
obtained random graph (denoted usually by G(n,p)) are at most
logarithmic in size, whereas for c>1 with high probability there is
a giant connected component of size linear in n while all other
components are at most logarithmic. Further research uncovered many
typical combinatorial properties of the giant component.

Over the years, a similar behavior has been observed sporadically
for other host graphs G, such as the d-dimensional binary cube Q^d.

I will report on a recent line of work, where we aim to develop
general sufficient conditions on the host graph G guaranteeing
phase transition in its random subgraph G_p, similar quantitatively
to the classical Erd\H{o}s-R\'enyi phenomenon.

Based on joint works with Sahar Diskin, Joshua Erde and Mihyun Kang.

Livestream/Recording: https://huji.cloud.panopto.eu/Panopto/Pages/Viewer.aspx?id=16ef88aa-fb15-485e-b2c8-b1a9005c1050