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Macroscopic loops in the loop O(n) model

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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.

This talk is part of the Probability series.

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