Abstract
We prove correlation inequalities for linearly edge-reinforced random walk. These correlation inequalities concern the first entry tree, i.e. the tree of edges used to enter any vertex for the first time. They also involve the asymptotic fraction of time spent on particular edges. Basic ingredients are known FKG-type inequalities and known negative associations for determinantal processes.
Highlights
In each discrete time-step, the reinforced random walker jumps from its current position to a neighboring vertex with probability proportional to the weight of the connecting edge
We always use the version described by formula (1.14) of the conditional distribution of the first entry tree T first entry given the asymptotic ratios of visits xarv
The following theorem claims negative associations for the first entry tree given the asymptotic ratios of visits
Summary
Since linearly edge-reinforced random walk on a finite graph visits every vertex almost surely (see e.g. Proposition 1 in [KR00]), Ttfirst entry ∈ for all t large enough almost surely. We show the following: Theorem 1.1 (First entry tree and asymptotic ratios of visits) The random vector. We always use the version described by formula (1.14) of the conditional distribution of the first entry tree T first entry given the asymptotic ratios of visits xarv. Ea denotes expectation with respect to Pa. The following theorem claims negative associations for the first entry tree given the asymptotic ratios of visits. Theorem 1.2 (Correlation inequality given the asymptotic ratios of visits) Let F, G ⊆ E be disjoint sets of edges. The following theorem shows positive correlations for increasing functions of the asymptotic ratios of visits conditional on the first entry tree. The proofs rely on FKG-type inequalities as summarized in Keane and den Hollander [dHK86]; see Preston [Pre74]
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