Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: VRRW on complete-like graphs: Almost sure behavior   By a theorem of Volkov (2001) we know that on most graphs with positive
probability the linearly vertex-reinforced random walk (VRRW) stays within a
finite "trapping" subgraph at all large times. The question of whether this
tail behavior occurs with probability one is open in general. In his thesis,
Pemantle (1988) proved, via a dynamical system approach, that for a VRRW on any
complete graph the asymptotic frequency of visits is uniform over vertices.
These techniques do not easily extend even to the setting of complete-like
graphs, that is, complete graphs ornamented with finitely many leaves at each
vertex. In this work we combine martingale and large deviation techniques to
prove that almost surely the VRRW on any such graph spends positive (and equal)
proportions of time on each of its nonleaf vertices. This behavior was
previously shown to occur only up to event of positive probability (cf. Volkov
(2001)). We believe that our approach can be used as a building block in
studying related questions on more general graphs. The same set of techniques
is used to obtain explicit bounds on the speed of convergence of the empirical
occupation measure.

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The categories of this paper are: cmp-lg
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The categories of this paper are: chem-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: quant-ph
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The categories of this paper are: math-ph
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The categories of this paper are: adap-org
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The categories of this paper are: cs
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The categories of this paper are: eess
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The categories of this paper are: cond-mat