Abstract
We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [9]. We describe 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 describes different universality classes for first passage percolation on the complete graph. 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 investigate the case where $s_n\rightarrow \infty$, and focus on the local neighborhood of a vertex. We establish that the smallest-weight tree of a vertex locally converges to the invasion percolation cluster on the Poisson weighted infinite tree. In addition, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices.
Highlights
We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights
In this paper our motivation is threefold: First, we aim to establish that the smallestweight tree process locally converges to the invasion percolation cluster on the Poissonweighted infinite tree (PWIT) and we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices
first passage percolation (FPP) on the Poisson-weighted infinite tree (PWIT) is closely related to invasion percolation (IP) on the PWIT which is defined as follows
Summary
We study first passage percolation on the complete graph equipped with independent and identically distributed positive and continuous edge weights. In this paper our motivation is threefold: First, we aim to establish that the smallestweight tree process locally converges to the invasion percolation cluster on the Poissonweighted infinite tree (PWIT) and we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices. The cost regime introduced in (1.1) uses the information from all edges along the path and is known as the weak disorder regime. We establish a firm connection between the weak and strong disorder regimes in first passage percolation This connection establishes a strong relation to invasion percolation (IP) on the PWIT, which is the scaling limit of IP on the complete graph. E denotes an exponentially distributed random variable with mean 1, and U denotes a random variable uniformly distributed on [0, 1]
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