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

We study the random geometry of first passage percolation on the completegraph equipped with independent and identically distributed edge weights,continuing the program initiated by Bhamidi and van der Hofstad [6]. Wedescribe our results in terms of a sequence of parameters $(s_n)_{n\geq 1}$that quantifies the extreme-value behavior of small weights, and that describesdifferent universality classes for first passage percolation on the completegraph. We consider both $n$-independent as well as $n$-dependent edge weights.The simplest example consists of edge weights of the form $E^{s_n}$, where $E$is an exponential random variable with mean 1.  In this paper, we focus on the case where $s_n\rightarrow \infty$ with$s_n=o(n^{1/3})$. Under mild regularity conditions, we identify the scalinglimit of the weight of the smallest-weight path between two uniform vertices,and we prove that the number of edges in this path obeys a central limittheorem with asymptotic mean $s_n\log{(n/s_n^3)}$ and variance$s_n^2\log{(n/s_n^3)}$. This settles a conjecture of Bhamidi and van derHofstad [6]. The proof relies on a decomposition of the smallest-weight treeinto an initial part following invasion percolation dynamics, and a main partfollowing branching process dynamics. The initial part has been studied in[14]; the current article focuses on the global branching dynamics.

Long paths in first passage percolation on the complete graph II. Global branching dynamics