Weighted distances in scale-free preferential attachment models

We study three preferential attachment models where the parameters are such that the asymptotic degree distribution has infinite variance. Every edge is equipped with a non-negative i.i.d. weight. We study the weighted distance between two vertices chosen uniformly at random, the typical weighted distance, and the number of edges on this path, the typical hopcount. We prove that there are precisely two universality classes of weight distributions, called the explosive and conservative class. In the explosive class, we show that the typical weighted distance converges in distribution to the sum of two i.i.d. finite random variables. In the conservative class, we prove that the typical weighted distance tends to infinity, and we give an explicit expression for the main growth term, as well as for the hopcount. Under a mild assumption on the weight distribution the fluctuations around the main term are tight.Comment: Revised version, results are unchanged. 30 pages, 1 figure. To appear in Random Structures and Algorithm

The constants in the CLT for the Edwards model

The Edwards model in one dimension is a transformed path measure for one-dimensional Brownian motion discouraging selfintersections In van der Hofstad den Hollander and Konig preprint a central limit theorem CLT is proved for the uctuations of the endpoint of the path around its linear asymptotics In the present paper we study the constants appearing in this CLT which represent the mean and the variance and the exponential rate of the normalizing constant We prove that the variance is strictly smaller than which shows that the weak interaction limit is singular Furthermore we give a relation between the normalizing constant in the Edwards model and the normalizing constant in the weakly interacting DombJoyce model The DombJoyce model is the discrete analogue of the Edwards model based on simple random walk and is studied in van der Hofstad den Hollander and Konig preprint The proofs are based on bounds for the eigenvalues of a certain one-parameter family of SturmLiouville dierential operators These bounds are obtained by using the monotonicity of the zeroes of the eigenfunctions in combination with computer plots of the power series approximation of the eigenfunctions and exact error estimates of the power series approximatio

Short paths for first passage percolation on the complete graph

We study the complete graph equipped with a topology induced by independent and identically distributed edge weights. The focus of our analysis is on the weight W_n and the number of edges H_n of the minimal weight path between two distinct vertices in the weak disorder regime. We establish novel and simple first and second moment methods using path counting to derive first order asymptotics for the considered quantities. Our results are stated in terms of a sequence of parameters (s_n) that quantifies the extreme-value behaviour of the edge weights, and that describes different universality classes for first passage percolation on the complete graph. These classes contain both n-independent and n-dependent edge weight distributions. The method is most effective for the universality class containing the edge weights E^{s_n}, where E is an exponential(1) random variable and s_n log n -> infty, s_n^2 log n -> 0. We discuss two types of examples from this class in detail. In addition, the class where s_n log n stays finite is studied. This article is a contribution to the program initiated in \cite{BhaHof12}.Comment: 31 pages, 4 figure

Pattern theorems, ratio limit theorems and Gumbel maximal clusters for random fields

We study occurrences of patterns on clusters of size n in random fields on Z^d. We prove that for a given pattern, there is a constant a>0 such that the probability that this pattern occurs at most an times on a cluster of size n is exponentially small. Moreover, for random fields obeying a certain Markov property, we show that the ratio between the numbers of occurrences of two distinct patterns on a cluster is concentrated around a constant value. This leads to an elegant and simple proof of the ratio limit theorem for these random fields, which states that the ratio of the probabilities that the cluster of the origin has sizes n+1 and n converges as n tends to infinity. Implications for the maximal cluster in a finite box are discussed.Comment: 23 pages, 2 figure

The survival probability for critical spread-out oriented percolation above 4+1 dimensions. II. Expansion

Article / Letter to editorMathematisch Instituu

A local limit theorem for the critical random graph

We consider the limit distribution of the orders of the k largest components in the Erd¿os-Rényi random graph inside the critical window for arbitrary k. We prove a local limit theorem for this joint distribution and derive an exact expression for the joint probability density function

Fluctuations in a general preferential attachment model via Stein's method

We consider a general preferential attachment model, where the probability that a newly arriving vertex connects to an older vertex is proportional to a sublinear function of the indegree of the older vertex at that time. It is well known that the distribution of a uniformly chosen vertex converges to a limiting distribution. Depending on the parameters, this model can show power law, but also stretched exponential behaviour. Using Stein's method we provide rates of convergence for the total variation distance. Our proof uses the fact that the limiting distribution is the stationary distribution of a Markov chain together with the generator method of Barbour

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

We 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 (sn)n≥1 that characterises the different universality classes for first passage percolation on the complete graph. We investigate the case where sn→ ∞ with sn= o(n1 / 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 snlog(n/sn3) and variance sn2log(n/sn3). Remarkably, our proof also applies to n-dependent edge weights of the form Esn, 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. https://doi.org/10.1214/20-EJP484); the current paper focuses on the global branching dynamics