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[Lecture] Probability Seminar-Natural measures on random fractals

DEC . 24 2018
Title: Probability Seminar-Natural measures on random fractals

Time: 15:00-16:00, Dec. 24, 2018, Monday

Venue: Room 1418, Sciences Building No. 1

Speaker: Xin SunColumbia University, USA

Abstract: Random fractals arise naturally as the scaling limit of large discrete models at criticality. These fractals usually exhibit strong self similarity and spacial independence. In this talk, we will explain how these additional properties should give the existence of a natural occupation measure on the fractal set, defined to be the limit of the properly rescaled Lebesgue measure restricted to small neighborhoods of the fractal. Moreover, the occupation measure is also the scaling limit of the normalized counting measure over the corresponding discrete set. In two dimension, when putting an independent Liouville quantum gravity background over such a planar fractal, the quantum version of the occupation measure still exists, where the scaling dimension is related to the Euclidean one via the famous KPZ relation due to Knizhnik-Polyakov-Zamolodchikov and Duplantier-Sheffield. The quantum occupation measure is supposed to be the scaling limit of the normalized counting measure of the corresponding discrete set on certain random planar maps. The picture described above is expected to be true in great generality yet it is only established for a few models to various extents. In this talk, we report a fairly complete picture for planar percolation on the regular and random triangular lattice.

Edited by: Guo Xinyu

Source: School of Mathematical Sciences