DYNKIN ISOMORPHISM AND MERMIN-WAGNER THEOREMS FOR HYPERBOLIC SIGMA MODELS AND RECURRENCE OF THE TWO-DIMENSIONAL VERTEX-REINFORCED JUMP PROCESS

• 成果类型: Article
• 署名作者: Bauerschmidt, Roland; Helmuth, Tyler; Swan, Andrew
• 署名单位: University of Cambridge; University of Bristol
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/19-AOP1343
• 发表日期: 2019
• 页码: 3375-3396
• 关键词: spontaneous symmetry-breaking; random-walk; statistical-mechanics; localization; absence; time

摘要

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 super-symmetric counterpart H-2 broken vertical bar 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 vertical bar 2 are nonamenable.