The Extreme Tail

Since their inception, random graphs have been a central topic in probability and combinatorics, offering a remarkably rich landscape for studying how complex structures emerge from randomness. We will discuss the following fundamental questions: What is the probability that the random graph fails to contain certain structures? For instance, what is the probability that the random graph is triangle-free?

Studying different regimes of such containment probability leads to the heart of different subjects with rich understanding and fundamental phenomena. While the study of thresholds occupies the typical regime of the containment probability, understanding precise asymptotics in the extreme tail is a central topic in the study of large deviations. I will discuss recent progress in this regime in joint work with Matthew Kwan, and describe connections with other cornerstone ideas and developments in modern combinatorics and probability.