Transience of Edge-Reinforced Random Walk

2014/03/24 by Disertori, Margherita, Sabot, Christophe, Tarrès, Pierre · 2 citations
#60K35 #81T60 #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #primary 60K37 #secondary 81T25

paper · doi:10.48550/arxiv.1403.6079

Abstract

We show transience of the edge-reinforced random walk (ERRW) for small reinforcement in dimension d greater than 2. This proves the existence of a phase transition between recurrent and transient behavior, thus solving an open problem stated by Diaconis in 1986. The argument adapts the proof of quasi-diffusive behavior of the SuSy hyperbolic model for fixed conductances by Disertori, Spencer and Zirnbauer [CMP 2010], using the representation of ERRW as a mixture of vertex-reinforced jump processes (VRJP) with independent gamma conductances, and the interpretation of the limit law of VRJP as a supersymmetric (SuSy) hyperbolic sigma model developed by Sabot and Tarrès in [JEMS 2014].

Cited by

• Once-Reinforced and Self-Interacting Random Walks beyond exchangeability
• Self-Reinforced Preferential Attachment

Related

• Edge-reinforced random walk, Vertex-Reinforced Jump Process and the supersymmetric hyperbolic sigma model
• Monotonicity and phase transition for the VRJP and the ERRW
• Step-reinforced random walks and one-half
• Variations on Reinforced Random Walks
• The *-Edge-Reinforced Random Walk