The fractal dimension of Liouville quantum gravity: monotonicity, universality, and bounds
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About this item
| Description: |
Gwynne, E
Thursday 19th July 2018 - 13:45 to 14:30 |
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| Created: | 2018-07-20 09:17 |
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| Collection: | RGM follow up |
| Publisher: | Isaac Newton Institute |
| Copyright: | Gwynne, E |
| Language: | eng (English) |
| Distribution: |
World
(downloadable)
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| Explicit content: | No |
| Aspect Ratio: | 16:9 |
| Screencast: | No |
| Bumper: | UCS Default |
| Trailer: | UCS Default |
| Abstract: | We show that for each γ∈(0,2)γ∈(0,2), there is an exponent dγ>2dγ>2, the ``fractal dimension of γγ-Liouville quantum gravity (LQG)", which describes the ball volume growth exponent for certain random planar maps in the γγ-LQG universality class, the graph-distance displacement exponent for random walk on these random planar maps, the exponent for the Liouville heat kernel, and exponents for various continuum approximations of γγ-LQG distances such as Liouville graph distance and Liouville first passage percolation. This builds on work of Ding-Zeitouni-Zhang (2018). We also show that dγdγ is a continuous, strictly increasing function of γγ and prove upper and lower bounds for dγdγ which in some cases greatly improve on previously known bounds for the aforementioned exponents. For example, for γ=2–√γ=2 (which corresponds to spanning-tree weighted planar maps) our bounds give 3.4641≤d2√≤3.632993.4641≤d2≤3.63299 and in the limiting case we get 4.77485≤limγ→2−dγ≤4.898984.77485≤limγ→2−dγ≤4.89898. Based on joint works with Jian Ding, Nina Holden, Tom Hutchcroft, Jason Miller, and Xin Sun. |
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