English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We prove that the Tutte embeddings (a.k.a. harmonic/barycentric embeddings)of certain random planar maps converge to $\gamma$-Liouville quantum gravity($\gamma$-LQG). Specifically, we treat mated-CRT maps, which are discretizedmatings of correlated continuum random trees, and $\gamma$ ranges from $0$ to$2$ as one varies the correlation parameter. We also show that the associatedspace-filling path on the embedded map converges to space-fillingSLE$_{\kappa}$ for $\kappa =16/\gamma^2$ (in the annealed sense) and thatsimple random walk on the embedded map converges to Brownian motion (in thequenched sense).  This work constitutes the first proof that a discrete conformal embedding ofa random planar map converges to LQG. Many more such statements have beenconjectured. Since the mated-CRT map can be viewed as a coarse-grainedapproximation to other random planar maps (the UIPT, tree-weighted maps,bipolar-oriented maps, etc.), our results indicate a potential approach forproving that embeddings of these maps converge to LQG as well.  To prove the main result, we establish several (independently interesting)theorems about LQG surfaces decorated by space-filling SLE. There is a naturalway to use the SLE curve to divide the plane into "cells" corresponding tovertices of the mated-CRT map. We study the law of the shape of theorigin-containing cell, in particular proving moments for the ratio of itssquared diameter to its area. We also give bounds on the degree of theorigin-containing cell and establish a form of ergodicity for the entireconfiguration. Ultimately, we use these properties to show (with the help of ageneral theorem proved in a separate paper) that random walk on these cellsconverges to a time change of Brownian motion, which in turn leads to the Tutteembedding result.

The Tutte embedding of the mated-CRT map converges to Liouville quantum gravity