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 Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Physics considerations suggest that a theory of Liouville quantum gravity(LQG) should exist for all values of matter central charge $\mathbf{c} \in(-\infty,25)$. Probabilists have rigorously defined LQG as a random metricmeasure space for $\mathbf{c} < 1$; however, they have only very recently begunto explore LQG in the $\mathbf{c} \in (1,25)$ phase. We define a random metricassociated to LQG for all $\mathbf{c} < 25$ by a collection of axioms analogousto axioms stated in the $\mathbf{c} < 1$ setting. We show that such a metricexists for each $\mathbf{c}$ by considering an approximating metric known asLiouville first passage percolation. Ding and Gwynne proved that theseapproximating metrics are tight in a suitably chosen topology; we show thatevery subsequential limit satisfies our axioms. In particular, our resultdefines a metric associated to LQG in the critical case $\mathbf{c} =1$.  The metrics we define for $\mathbf{c} \in (1,25)$ exhibit geometric behaviorthat sharply contrasts with the $\mathbf{c} < 1$ regime. We show that, for$\mathbf{c} \in (1,25)$, the metrics do not induce the Euclidean topology sincethey a.s. have a dense (measure zero) set of singular points, points atinfinite distance from all other points. We use this fact to prove that a.s.the metric ball is not compact and its boundary has infinite Hausdorffdimension. Despite these differences, we demonstrate that many properties ofLQG metrics for $\mathbf{c} < 1$ extend in some form to the entire range$(-\infty, 25)$. We show that the metrics are a.s. reverse H\"older continuouswith respect to the Euclidean metric, are a.s. complete and geodesic away fromthe set of singular points, and satisfy the bounds for set-to-set distancesthat hold in the $\mathbf{c} < 1$ phase. Finally, we prove that the metricssatisfy a version of the (geometric) Knizhnik-Polyakov-Zamolodchikov (KPZ)formula.

Weak Liouville quantum gravity metrics with matter central charge $\mathbf{c} \in (-\infty, 25)$