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

For each $N\geq 1$, let $G_N$ be a simple random graph on the set of vertices$[N]=\{1,2, ..., N\}$, which is invariant by relabeling of the vertices. Theasymptotic behavior as $N$ goes to infinity of correlation functions:  $$ \mathfrak C_N(T)= \mathbb E\bigg[ \prod_{(i,j) \in T} \Big(\mathbf1_{\big(\{i,j\} \in G_N \big)} - \mathbb P(\{i,j\} \in G_N) \Big)\bigg], \ T\subset [N]^2 \textrm{finite}$$ furnishes informations on the asymptoticspectral properties of the adjacency matrix $A_N$ of $G_N$. Denote by $d_N =N\times \mathbb P(\{i,j\} \in G_N) $ and assume $d_N, N-d_N\underset{N\rightarrow \infty}{\longrightarrow} \infty$. If $\mathfrak C_N(T)=\big(\frac{d_N}N\big)^{|T|} \times O\big(d_N^{-\frac {|T|}2}\big)$ for any$T$, the standardized empirical eigenvalue distribution of $A_N$ converges inexpectation to the semicircular law and the matrix satisfies asymptoticfreeness properties in the sense of free probability theory. We provide suchestimates for uniform $d_N$-regular graphs $G_{N,d_N}$, under the additionalassumption that $|\frac N 2 - d_N- \eta \sqrt{d_N}| \underset{N \rightarrow\infty}{\longrightarrow} \infty$ for some $\eta>0$. Our method applies also forsimple graphs whose edges are labelled by i.i.d. random variables.

Uniform regular weighted graphs with large degree: Wigner's law, asymptotic freeness and graphons limit