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

In this paper, we study the level-set of the zero-average Gaussian Free Fieldon a uniform random $d$-regular graph above an arbitrary level $h\in (-\infty,h_{\star})$, where $h_{\star}$ is the level-set percolation threshold of theGFF on the $d$-regular tree $\mathbb{T}_d$. We prove that w.h.p as the number$n$ of vertices diverges, the GFF has a unique giant connected component$\mathcal{C}_1^{(n)}$ of size $\eta(h) n+o(n)$, where $\eta(h)$ is theprobability that the root percolates in the corresponding GFF level-set on$\mathbb{T}_d$. This gives a positive answer to the conjecture of\cite{ACregulgraphs} for most regular graphs. We also prove that the secondlargest component has size $\Theta(\log n)$. Moreover, we show that$\mathcal{C}_1^{(n)}$ shares the following similarities with the giantcomponent of the supercritical Erd\H{o}s-R\'enyi random graph. First, thediameter and the typical distance between vertices are $\Theta(\log n)$.Second, the $2$-core and the kernel encompass a given positive proportion ofthe vertices. Third, the local structure is a branching process conditioned tosurvive, namely the level-set percolation cluster of the root in $\mathbb{T}_d$(in the Erd\H{o}s-R\'enyi case, it is known to be a Galton-Watson tree with aPoisson distribution for the offspring).

Anatomy of a gaussian giant: supercritical level-sets of the free field on random regular graphs