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

Let $d\ge 3$ be a fixed integer, $p\in (0,1)$, and let $n\geq 1$ be apositive integer such that $dn$ is even. Let $\mathbb{G}(n, d, p)$ be a(random) graph on $n$ vertices obtained by drawing uniformly at random a$d$-regular (simple) graph on $[n]$ and then performing independent $p$-bondpercolation on it, i.e. we independently retain each edge with probability $p$and delete it with probability $1-p$. Let $|\mathcal{C}_{\text{max}}|$ be thesize of the largest component in $\mathbb{G}(n, d, p)$. We show that, when $p$is of the form $p=(d-1)^{-1}(1+\lambda n^{-1/3})$ for $\lambda\in \mathbb{R}$,and $A$ is large, \begin{align*}\mathbb{P}(|\mathcal{C}_{\text{max}}|>An^{2/3})\asympA^{-3/2}e^{-\frac{A^3(d-1)(d-2)}{8d^2}+\frac{\lambdaA^2(d-2)^2}{2d(d-1)}-\frac{\lambda^2 A(d-1)}{2(d-2)}}. \end{align*} Thisimproves on a result of Nachmias and Peres. We also give an analogousasymptotic for the probability that a particular vertex is in a component ofsize larger than $An^{2/3}$.

The probability of unusually large components for critical percolation on random $d$-regular graphs