Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Spectral atoms of unimodular random trees   We use the Mass Transport Principle to analyze the local recursion governing
the resolvent $(A-z)^{-1}$ of the adjacency operator of unimodular random
trees. In the limit where the complex parameter $z$ approaches a given location
$\lambda$ on the real axis, we show that this recursion induces a decomposition
of the tree into finite blocks whose geometry directly determines the spectral
mass at $\lambda$. We then exploit this correspondence to obtain precise
information on the pure-point support of the spectrum, in terms of expansion
properties of the tree. In particular, we deduce that the pure-point support of
the spectrum of any unimodular random tree with minimum degree $\delta\ge 3$
and maximum degree $\Delta$ is restricted to finitely many points, namely the
eigenvalues of trees of size less than $\frac{\Delta-2}{\delta-2}$. More
generally, we show that the restriction $\delta\ge 3$ can be weakened to
$\delta\ge 2$, as long as the anchored isoperimetric constant of the tree
remains bounded away from $0$. This applies in particular to any unimodular
Galton-Watson tree without leaves, allowing us to settle a conjecture of
Bordenave, Sen and Vir\'ag (2013).

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The categories of this paper are: hep-ex
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The categories of this paper are: acc-phys
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
32
The categories of this paper are: atom-ph
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The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
14
The categories of this paper are: math-ph
29
The categories of this paper are: funct-an
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The categories of this paper are: q-fin
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The categories of this paper are: physics
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The categories of this paper are: nucl-ex
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The categories of this paper are: cs
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The categories of this paper are: cmp-lg
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The categories of this paper are: bayes-an