# セミナー

## 談話会

タイトル Obtaining stationary random metrics on hierarchical graphs by a cut-off method (joint work with M. Khristoforov & V. Kleptsyn) 2014年3月7日 16：10-17：00 Michele Triestino氏（ ENS Lyon, UMPA） 14棟631A.B Give a random metric on the square, it is possible to define a new random (pseudo-)metric taking four independent copies of it and gluing the four squares together. Is there any non-trivial stationary random metric? This is a question that has grown of interest very recently, due to its relations with Liouville Quantum Gravity and planar maps: it is indeed a geometric description of the multiplicative cascade producing the conformal « metric », whose density is the exponential of the Dyadic Gaussian Free Field. Such a « metric » makes sense only if we consider it as a distribution — the open challenge for mathematicians is to define it rigorously. Successful results have been achieved a few years ago by Le Gall and Miermont (independently), using planar maps. However, in the way we have formulate this problem, there is no definite result. In 2008, Benjamini and Schramm proved that multiplicative cascades on the interval produce well-defined metrics. Inspired by this problem, we study simpler models (following a suggestion of Benjamini): for self-similar objects like hierarchical graphs and the Sierpinski triangle, we get positive answers. This is done by introducing a cut-off process that could be useful in a more general setting. We will also discuss finer results about uniqueness of the stationary random metric and convergence.