Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Self-avoiding walks and amenability   The connective constant $\mu(G)$ of an infinite transitive graph $G$ is the
exponential growth rate of the number of self-avoiding walks from a given
origin. The relationship between connective constants and amenability is
explored in the current work.
  Various properties of connective constants depend on the existence of
so-called 'graph height functions', namely: (i) whether $\mu(G)$ is a local
function on certain graphs derived from $G$, (ii) the equality of $\mu(G)$ and
the asymptotic growth rate of bridges, and (iii) whether there exists a
terminating algorithm for approximating $\mu(G)$ to a given degree of accuracy.
  In the context of amenable groups, it is proved that the Cayley graphs of
infinite, finitely generated, elementary amenable groups support graph height
functions, which are in addition harmonic. In contrast, the Cayley graph of the
Grigorchuk group, which is amenable but not elementary amenable, does not have
a graph height function.
  In the context of non-amenable, transitive graphs, a lower bound is presented
for the connective constant in terms of the spectral bottom of the graph. This
is a strengthening of an earlier result of the same authors. Secondly, using a
percolation inequality of Benjamini, Nachmias, and Peres, it is explained that
the connective constant of a non-amenable, transitive graph with large girth is
close to that of a regular tree. Examples are given of non-amenable groups
without graph height functions, of which one is the Higman group.

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The categories of this paper are: quant-ph
27
The categories of this paper are: comp-gas
7
The categories of this paper are: hep-th
36
The categories of this paper are: cmp-lg
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The categories of this paper are: q-alg
33
The categories of this paper are: plasm-ph
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The categories of this paper are: eess
26
The categories of this paper are: bayes-an
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The categories of this paper are: adap-org
37
The categories of this paper are: supr-con
1
The categories of this paper are: cs
35
The categories of this paper are: mtrl-th
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The categories of this paper are: q-bio
0
The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
16
The categories of this paper are: stat