Percolation and Random Graphs

In this chapter, we define percolation and random graph models, and survey the features of these model

An expansion for self-interacting random walks

We derive a perturbation expansion for general self-interacting random walks, where steps are made on the basis of the history of the path. Examples of models where this expansion applies are reinforced random walk, excited random walk, the true (weakly) self-avoiding walk, loop-erased random walk, and annealed random walk in random environment. In this paper we show that the expansion gives rise to useful formulae for the speed and variance of the random walk, when these quantities are known to exist. The results and formulae of this paper have been used elsewhere by the authors to prove monotonicity properties for the speed (in high dimensions) of excited random walk and related models, and certain models of random walk in random environment. We also derive a law of large numbers and central limit theorem (with explicit error terms) directly from this expansion, under strong assumptions on the expansion coefficients. The assumptions are shown to be satisfied by excited random walk in high dimensions with small excitation parameter, a model of reinforced random walk with underlying drift and small reinforcement parameter, and certain models of random walk in random environment under strong ellipticity conditions

Universality for first passage percolation on sparse uniform and rank-1 random graphs

In [3], we considered first passage percolation on the configuration model equipped with general independent and identically distributed edge weights, where the common distribution function admits a density. Assuming that the degree distribution satisfies a uniform X^2 log X - condition, we analyzed the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well as the asymptotic distribution of the number of edges on this path. Given the interest in understanding such questions for various other random graph models, the aim of this paper is to show how these results extend to uniform random graphs with a given degree sequence and rank-one inhomogeneous random graphs

Universality for first passage percolation on sparse random graphs

We consider first passage percolation on the conguration model with n vertices, and general independent and identically distributed edge weights assumed to have a density. Assuming that the degree distribution satisfies a uniform X2 logX-condition, we analyze the asymptotic distribution for the minimal weight path between a pair of typical vertices, as well the number of edges on this path namely the hopcount. The hopcount satisfies a central limit theorem (CLT). Furthermore, writing Ln for the weight of this optimal path, then we shown that Ln(log n)= n converges to a limiting random variable, for some sequence n. This sequence n and the norming constants for the CLT are expressible in terms of the parameters of an associated continuous-time branching process that describes the growth of neighborhoods around a uniformly chosen vertex in the random graph. The limit of Ln(log n)= n equals the sum of the logarithm of the product of two independent martingale limits, and a Gumbel random variable. Till date, for sparse random graph models, such results have been shown only for the special case where the edge weights have an exponential distribution, wherein the Markov property of this distribution plays a crucial role in the technical analysis of the problem. The proofs in the paper rely on a refined coupling between shortest path trees and continuous- time branching processes, and on a Poisson point process limit for the potential closing edges of shortest-weight paths between the source and destination

Functionals of Brownian bridges arising in the current mismatch in D/A-converters

Digital-to-analog converters (DAC) transform signals from the abstract digital domain to the real analog world. In many applications, DAC’s play a crucial role. Due to variability in the production, various errors arise that influence the performance of the DAC. We focus on the current errors, which describe the fluctuations in the currents of the various unit current elements in the DAC. A key performance measure of the DAC is the Integrated Non-linearity (INL), which we study in this paper. There are several DAC architectures. The most widely used architectures are the ther-mometer, the binary and the segmented architectures. We study the two extreme architec-tures, namely, the thermometer and the binary architectures. We assume that the current errors are i.i.d. normally distributed, and reformulate the INL as a functional of a Brownian bridge. We then proceed by investigating these functionals. For the thermometer case, the functional is the maximal absolute value of the Brownian bridge, which has been investi-gated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance and density

Scale-free percolation

Abstract We formulate and study a model for inhomogeneous long-range percolation on Zd. Each vertex x¿Zd is assigned a non-negative weight Wx, where (Wx)x¿Zd are i.i.d. random variables. Conditionally on the weights, and given two parameters a,¿>0, the edges are independent and the probability that there is an edge between x and y is given by pxy=1-exp{-¿WxWy/|x-y|a}. The parameter ¿ is the percolation parameter, while a describes the long-range nature of the model. We focus on the degree distribution in the resulting graph, on whether there exists an infinite component and on graph distance between remote pairs of vertices. First, we show that the tail behavior of the degree distribution is related to the tail behavior of the weight distribution. When the tail of the distribution of Wx is regularly varying with exponent t-1, then the tail of the degree distribution is regularly varying with exponent ¿=a(t-1)/d. The parameter ¿ turns out to be crucial for the behavior of the model. Conditions on the weight distribution and ¿ are formulated for the existence of a critical value ¿c¿(0,8) such that the graph contains an infinite component when ¿>¿c and no infinite component when ¿0, les arêtes sont indépendantes et la probabilité qu’il existe un lien entre x et y est pxy=1-exp{-¿WxWy/|x-y|a}. Le paramètre ¿ est le paramètre de percolation tandis que a caractérise la portée des interactions. Nous étudierons la distribution des degrés dans le graphe résultant et l’existence éventuelle d’une composante infinie ainsi que la distance de graphe entre deux sites éloignés. Nous montrons d’abord que la queue de la distribution des degrés est liée à la queue de la distribution des poids. Quand la queue de la distribution de Wx est à variation régulière d’indice t-1, alors la queue de la distribution des degrés est à variation régulière d’indice ¿=a(t-1)/d. Le paramètre ¿ s’avère crucial pour décrire le modèle. Des conditions sur la distribution des poids et de ¿ sont formulées pour l’existence d’une valeur critique ¿c¿(0,8) telle que le graphe contienne une composante infinie quand ¿>¿c et aucune composante infinie quand

Construction of the incipient infinite cluster for spread-out oriented percolation above 4+1 dimensions

Analysis and Stochastic

Weak interaction limits for one-dimensional random polymers

Analysis and Stochastic