Long Paths in First Passage Percolation on the Complete Graph II. Global Branching Dynamics

Abstract

AbstractWe study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed positive edge weights. We consider the case where the lower extreme values of the edge weights are highly separated. This model exhibits strong disorder and a crossover between local and global scales. Local neighborhoods are related to invasion percolation that display self-organised criticality. Globally, the edges with relevant edge weights form a barely supercritical Erdős–Rényi random graph that can be described by branching processes. This near-critical behaviour gives rise to optimal paths that are considerably longer than logarithmic in the number of vertices, interpolating between random graph and minimal spanning tree path lengths. Crucial to our approach is the quantification of the extreme-value behavior of small edge weights in terms of a sequence of parameters $$(s_n)_{n\ge 1}$$ ( s n ) n ≥ 1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where $$s_n\rightarrow \infty $$ s n → ∞ with $$s_n=o(n^{1/3})$$ s n = o ( n 1 / 3 ) , which corresponds to the barely supercritical setting. We identify the scaling limit of the weight of the optimal path between two vertices, and we prove that the number of edges in this path obeys a central limit theorem with mean approximately $$s_n\log {(n/s_n^3)}$$ s n log ( n / s n 3 ) and variance $$s_n^2\log {(n/s_n^3)}$$ s n 2 log ( n / s n 3 ) . Remarkably, our proof also applies to n-dependent edge weights of the form $$E^{s_n}$$ E s n , where E is an exponential random variable with mean 1, thus settling a conjecture of Bhamidi et al. (Weak disorder asymptotics in the stochastic meanfield model of distance. Ann Appl Probab 22(1):29–69, 2012). The proof relies on a decomposition of the smallest-weight tree into an initial part following invasion percolation dynamics, and a main part following branching process dynamics. The initial part has been studied in Eckhoff et al. (Long paths in first passage percolation on the complete graph I. Local PWIT dynamics. Electron. J. Probab. 25:1–45, 2020. 10.1214/20-EJP484); the current paper focuses on the global branching dynamics.