Local neighbourhoods for first passage percolation on the configuration model

TL;DR

This paper studies first passage percolation on the configuration model, analyzing local graph structures around a recipient in different regimes, providing limit theorems for the coloured subgraph near the target node.

Contribution

It establishes local limit theorems for the coloured graph around the recipient in both explosive and Malthusian regimes, extending understanding of local structures in first passage percolation.

Findings

Proves local limit theorems in the explosive regime.

Establishes local limit theorems in the Malthusian regime.

Analyzes the structure of the coloured graph around the recipient.

Abstract

We consider first passage percolation on the configuration model. Once the network has been generated each edge is assigned an i.i.d. weight modeling the passage time of a message along this edge. Then independently two vertices are chosen uniformly at random, a sender and a recipient, and all edges along the geodesic connecting the two vertices are coloured in red (in the case that both vertices are in the same component). In this article we prove local limit theorems for the coloured graph around the recipient in the spirit of Benjamini and Schramm. We consider the explosive regime, in which case the random distances are of finite order, and the Malthusian regime, in which case the random distances are of logarithmic order.