Three theorems in discrete random geometry

TL;DR

This paper discusses three recent significant theorems in discrete random geometry, including proofs of critical points and universality across various lattice models, emphasizing ideas, connections, and detailed proofs.

Contribution

It presents the proofs of three important theorems in discrete random geometry, highlighting new results on critical points and universality in lattice models.

Findings

Proof of the hexagonal lattice's connective constant as rac{2+rac{2}{}}

Universality of inhomogeneous bond percolation across multiple lattices

Critical point of the random-cluster model on Z^2 is rac{rac{q}{}}{1+rac{q}{}}

Abstract

These notes are focused on three recent results in discrete random geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is \sqrt{2+\sqrt 2}; the proof by the author and Manolescu of the universality of inhomogeneous bond percolation on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on Z^2 is \sqrt q/(1+\sqrt q). Background information on the relevant random processes is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs.