50 years of first passage percolation

TL;DR

This survey reviews 50 years of first passage percolation, highlighting classical results, recent advances, and open problems in the study of limit shapes, geodesics, and related probabilistic models.

Contribution

It provides self-contained proofs of key classical results and discusses recent developments and open questions in first passage percolation research.

Findings

Proof of classical limit shape results

Recent progress on sublinear variance under specific moment conditions

Connections between Busemann functions, geodesics, and growth models

Abstract

We celebrate the 50th anniversary of one the most classical models in probability theory. In this survey, we describe the main results of first passage percolation, paying special attention to the recent burst of advances of the past 5 years. The purpose of these notes is twofold. In the first chapters, we give self-contained proofs of seminal results obtained in the '80s and '90s on limit shapes and geodesics, while covering the state of the art of these questions. Second, aside from these classical results, we discuss recent perspectives and directions including (1) the connection between Busemann functions and geodesics, (2) the proof of sublinear variance under 2+log moments of passage times and (3) the role of growth and competition models. We also provide a collection of (old and new) open questions, hoping to solve them before the 100th birthday.