Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Stationary random metrics on hierarchical graphs via $(\min,+)$-type
  recursive distributional equations   This paper is inspired by the problem of understanding in a mathematical
sense the Liouville quantum gravity on surfaces. Here we show how to define a
stationary random metric on self-similar spaces which are the limit of nice
finite graphs: these are the so-called hierarchical graphs. They possess a
well-defined level structure and any level is built using a simple recursion.
Stopping the construction at any finite level, we have a discrete random metric
space when we set the edges to have random length (using a multiplicative
cascade with fixed law $m$). We introduce a tool, the cut-off process, by means
of which one finds that renormalizing the sequence of metrics by an exponential
factor, they converge in law to a non-trivial metric on the limit space. Such
limit law is stationary, in the sense that glueing together a certain number of
copies of the random limit space, according to the combinatorics of the brick
graph, the obtained random metric has the same law when rescaled by a random
factor of law $m$. In other words, the stationary random metric is the solution
of a distributional equation. When the measure $m$ has continuous positive
density on $\mathbf{R}_+$, the stationary law is unique up to rescaling and any
other distribution tends to a rescaled stationary law under the iterations of
the hierarchical transformation. We also investigate topological and geometric
properties of the random space when $m$ is $\log$-normal, detecting a phase
transition influenced by the branching random walk associated to the
multiplicative cascade.

19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
30
The categories of this paper are: q-alg
13
The categories of this paper are: quant-ph
4
The categories of this paper are: cond-mat
0
The categories of this paper are: hep-ph
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th