## Macroscopic loops in the loop O(n) model

### With Yinon Spinka (Tel Aviv)

# Macroscopic loops in the loop O(n) model

A loop configuration on the hexagonal (honeycomb) lattice is a finite subgraph of the lattice in which every vertex has degree 0 or 2, so that every connected component is isomorphic to a cycle. The loop O(n) model on the hexagonal lattice is a random loop configuration, where the probability of a loop configuration is proportional to x^{ n}(#loops) and x,n>0 are parameters called the edge-weight and loop-weight. I will discuss the phase structure of the loop O(n) model for various parameters of n and x, focusing on recent results about the non-existence of macroscopic loops for large n, and about the existence of macroscopic loops on a critical line when n is between 1 and 2.

Based on joint works with Hugo Duminil-Copin, Alexander Glazman, Ron Peled and Wojciech Samotij.

- Speaker: Yinon Spinka (Tel Aviv)
- Tuesday 14 November 2017, 16:15–17:15
- Venue: MR12, CMS, Wilberforce Road, Cambridge, CB3 0WB.
- Series: Probability; organiser: Perla Sousi.