Stationary random metrics on hierarchical graphs via (min,+)-type recursive distributional equations

Abstract : This paper is inspired by the problem of understanding in a mathematical sense the Liouville quantum gravity on surfaces. Here we show how it is possible to define a 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 a given 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, that allows to show that it is possible to renormalize the sequence of distances by an exponential factor in such a way that they converge in law to a non-trivial distance on the limit space. Such limit law is stationary, in the sense that gluing together a certain number of copies of the random limit space, according to the combinatorics of the brick graph, the obtained new random distance has the same law. In other words, the stationary measure is the solution of a recursive distributional equation. Moreover, when the measure $m$ has a continuous positive density on $\mathbf{R}_+$, we show that the stationary law is unique up to rescaling and that any other distribution tends to a rescaled stationary law under the iterations of the hierarchical transformation. We hope that the technique could be modified in order to deal with genuine 2D problems.
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Contributor : Marie-Annick Guillemer <>
Submitted on : Monday, March 3, 2014 - 5:06:52 PM
Last modification on : Saturday, March 30, 2019 - 1:33:09 AM

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Mikhail Khristoforov, Victor Kleptsyn, Michele Triestino. Stationary random metrics on hierarchical graphs via (min,+)-type recursive distributional equations. Communications in Mathematical Physics, Springer Verlag, 2016, 345 (1), pp.1-76. ⟨10.1007/s00220-016-2650-7⟩. ⟨hal-00954977⟩

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