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VERTEX-REINFORCED JUMP PROCESS ON THE INTEGERS WITH NONLINEAR REINFORCEMENT

Collevecchio, Andrea ; Nguyen, Tuan Minh LU and Volkov, Stanislav LU orcid (2022) In Annals of Applied Probability 32(4). p.2671-2705
Abstract

We consider a nonlinear vertex-reinforced jump process (VRJP(w)) on Z with an increasing measurable weight function w : [1, ∞) → [1, ∞) and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP(w) depending on the integrability of the reciprocal of w. In particular, we prove that if (Equation presented). then the process is recurrent, that is, it visits each vertex infinitely often and all local times are unbounded. On the other hand, if (Equation presented). and there exists a ρ > 0 such that (Equation presented). du is nonincreasing then the process will eventually get stuck on exactly three w(u) vertices, and there is only one vertex with unbounded local time. We also show that if the initial... (More)

We consider a nonlinear vertex-reinforced jump process (VRJP(w)) on Z with an increasing measurable weight function w : [1, ∞) → [1, ∞) and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP(w) depending on the integrability of the reciprocal of w. In particular, we prove that if (Equation presented). then the process is recurrent, that is, it visits each vertex infinitely often and all local times are unbounded. On the other hand, if (Equation presented). and there exists a ρ > 0 such that (Equation presented). du is nonincreasing then the process will eventually get stuck on exactly three w(u) vertices, and there is only one vertex with unbounded local time. We also show that if the initial weights are all the same, VRJP on Z cannot be transient, that is, there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov (Probab. Theory Related Fields 123 (2002) 281-300) who showed that VRJP with linear reinforcement on ℤ is recurrent.

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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
localization, random processes with reinforcement, Self-interacting processes, vertex-reinforced jump processes
in
Annals of Applied Probability
volume
32
issue
4
pages
35 pages
publisher
Institute of Mathematical Statistics
external identifiers
  • scopus:85138593432
ISSN
1050-5164
DOI
10.1214/21-AAP1743
language
English
LU publication?
yes
id
e607be13-3bd6-43d6-9418-99f299fbd806
date added to LUP
2022-12-14 09:42:45
date last changed
2022-12-14 09:42:45
@article{e607be13-3bd6-43d6-9418-99f299fbd806,
  abstract     = {{<p>We consider a nonlinear vertex-reinforced jump process (VRJP(w)) on Z with an increasing measurable weight function w : [1, ∞) → [1, ∞) and initial weights equal to one. Our main goal is to study the asymptotic behaviour of VRJP(w) depending on the integrability of the reciprocal of w. In particular, we prove that if (Equation presented). then the process is recurrent, that is, it visits each vertex infinitely often and all local times are unbounded. On the other hand, if (Equation presented). and there exists a ρ &gt; 0 such that (Equation presented). du is nonincreasing then the process will eventually get stuck on exactly three w(u) vertices, and there is only one vertex with unbounded local time. We also show that if the initial weights are all the same, VRJP on Z cannot be transient, that is, there exists at least one vertex that is visited infinitely often. Our results extend the ones previously obtained by Davis and Volkov (Probab. Theory Related Fields 123 (2002) 281-300) who showed that VRJP with linear reinforcement on ℤ is recurrent.</p>}},
  author       = {{Collevecchio, Andrea and Nguyen, Tuan Minh and Volkov, Stanislav}},
  issn         = {{1050-5164}},
  keywords     = {{localization; random processes with reinforcement; Self-interacting processes; vertex-reinforced jump processes}},
  language     = {{eng}},
  number       = {{4}},
  pages        = {{2671--2705}},
  publisher    = {{Institute of Mathematical Statistics}},
  series       = {{Annals of Applied Probability}},
  title        = {{VERTEX-REINFORCED JUMP PROCESS ON THE INTEGERS WITH NONLINEAR REINFORCEMENT}},
  url          = {{http://dx.doi.org/10.1214/21-AAP1743}},
  doi          = {{10.1214/21-AAP1743}},
  volume       = {{32}},
  year         = {{2022}},
}