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 Plus annual

Presents the access of premium content as premium feature

Premium Content

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 annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

We show the existence of the scaling exponent $\chi\in (0,4[(1+\gamma^2/4)-\sqrt{1+\gamma^4/16}]/\gamma^2]$ of the graph distance associated withsubcritical two-dimensional Liouville quantum gravity of paramater $\gamma<2$on $\mathbb V =[0,1]^2 $. We also show that the Liouville heat kernelsatisfies, for any fixed $u,v\in \mathbb V^o$, the short time estimates $$\lim_{ t \to 0} \frac{\log |\log {\mathsf p}_t^\gamma(u,v)| }{|\logt|}=\frac{\chi}{2-\chi}, \ \mbox{\rm a.s.} $$

Heat kernel for Liouville Brownian motion and Liouville graph distance