Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Localization on 4 sites for Vertex-reinforced random walks on $\mathbb
  Z$   We characterize non-decreasing weight functions for which the associated
one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A
phase transition appears for weights of order $n\log \log n$: for weights
growing faster than this rate, the VRRW localizes almost surely on at most 4
sites whereas for weights growing slower, the VRRW cannot localize on less than
5 sites. When $w$ is of order $n\log \log n$, the VRRW localizes almost surely
on either 4 or 5 sites, both events happening with positive probability.

35
The categories of this paper are: mtrl-th
5
The categories of this paper are: gr-qc
17
The categories of this paper are: q-fin
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
0
The categories of this paper are: hep-ph
7
The categories of this paper are: hep-th
31
The categories of this paper are: ao-sci
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
19
The categories of this paper are: eess
28
The categories of this paper are: alg-geom
3
The categories of this paper are: physics
12
The categories of this paper are: nucl-th
16
The categories of this paper are: stat
21
The categories of this paper are: adap-org