## Abstract

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite-range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space H^{n} or its supersymmetric counterpart H^{2|2}. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin-Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin-Wagner theorem applies even though the symmetry groups of H^{n} and H^{2|2} are nonamenable.

Original language | English (US) |
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Pages (from-to) | 3375-3396 |

Number of pages | 22 |

Journal | Annals of Probability |

Volume | 47 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2019 |

## Keywords

- Dynkin isomorphism
- Hyperbolic sigma models
- Mermin-wagner theorem
- Supersymmetry
- Vertex-reinforced jump process

## ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty