Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Inverting the cut-tree transform   We consider fragmentations of an R-tree $T$ driven by cuts arriving according
to a Poisson process on $T \times [0, \infty)$, where the first co-ordinate
specifies the location of the cut and the second the time at which it occurs.
The genealogy of such a fragmentation is encoded by the so-called cut-tree,
which was introduced by Bertoin and Miermont for a fragmentation of the
Brownian continuum random tree. The cut-tree was generalised by Dieuleveut to a
fragmentation of the $\alpha$-stable trees, $\alpha \in (1, 2)$, and by Broutin
and Wang to the inhomogeneous continuum random trees of Aldous and Pitman.
Remarkably, in all of these cases, the law of the cut-tree is the same as that
of the original R-tree. In this paper, we develop a clean general framework for
the study of cut-trees of R-trees. We then focus particularly on the problem of
reconstruction: how to recover the original R-tree from its cut-tree. This has
been studied in the setting of the Brownian CRT by Broutin and Wang, where they
prove that it is possible to reconstruct the original tree in distribution. We
describe an enrichment of the cut-tree transformation, which endows the cut
tree with information we call a consistent collection of routings. We show this
procedure is well-defined under minimal conditions on the R-trees. We then show
that, for the case of the Brownian CRT and the $\alpha$-stable trees with
$\alpha \in (1, 2)$, the original tree and the Poisson process of cuts thereon
can both be almost surely reconstructed from the enriched cut-trees. For the
latter results, our methods make essential use of the self-similarity and
re-rooting invariance of these trees.

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The categories of this paper are: quant-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: ao-sci
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The categories of this paper are: funct-an
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: q-bio
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The categories of this paper are: astro-ph
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The categories of this paper are: cond-mat