Return Probability and Recurrence for the Random Walk Driven by Two-Dimensional Gaussian Free Field

• Published in: Communications in mathematical physics 2020, Vol.373 (1), p.45-106
• Main Authors: Biskup, Marek, Ding, Jian, Goswami, Subhajit
• Format: Article
• Language: English
• Subjects: Apexes; Classical and Quantum Gravitation; Complex Systems; Economic models; Euclidean geometry; Mathematical and Computational Physics; Mathematical Physics; Physics; Physics and Astronomy; Quantum Physics; Random walk; Random walk theory; Relativity Theory; Resistance; Theoretical
• ISSN: 0010-3616; 1432-0916
• DOI: 10.1007/s00220-019-03589-z

Given any γ > 0 and for η = { η v } v ∈ Z 2 denoting a sample of the two-dimensional discrete Gaussian free field on Z 2 pinned at the origin, we consider the random walk on  Z 2 among random conductances where the conductance of edge ( u ,  v ) is given by e γ ( η u + η v ) . We show that, for almost every  η , this random walk is recurrent and that, with probability tending to 1 as T → ∞ , the return probability at time 2 T decays as T - 1 + o ( 1 ) . In addition, we prove a version of subdiffusive behavior by showing that the expected exit time from a ball of radius  N scales as N ψ ( γ ) + o ( 1 ) with ψ ( γ ) > 2 for all  γ > 0 . Our results rely on delicate control of the effective resistance for this random network. In particular, we show that the effective resistance between two vertices at Euclidean distance  N behaves as  N o ( 1 ) .