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Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 163 records · Page 9Linked to original sources
arXiv2010

First passage percolation on random graphs with finite mean degrees

We study first passage percolation on the configuration model. Assuming that each edge has an independent exponentially distributed edge weight, we derive explicit distributional asymptotics for the minimum weight between two randomly chosen connected vertices in the network, as well as for the number of edges on the least weight path, the so-called hopcount. We analyze the configuration model with degree power-law exponent $τ>2$, in which the degrees are assumed to be i.i.d. with a tail distribution which is either of power-law form with exponent $τ-1>1$, or has even thinner tails ($τ=\infty$). In this model, the degrees have a finite first moment, while the variance is finite for $τ>3$, but infinite for $τ\in(2,3)$. We prove a central limit theorem for the hopcount, with asymptotically equal means and variances equal to $α\log{n}$, where $α\in(0,1)$ for $τ\in(2,3)$, while $α>1$ for $τ>3$. Here $n$ denotes the size of the graph. For $τ\in (2,3)$, it is known that the graph distance between two randomly chosen connected vertices is proportional to $\log \log{n}$ [Electron. J. Probab. 12 (2007) 703--766], that is, distances are ultra small. Thus, the addition of edge weights causes a marked change in the geometry of the network. We further study the weight of the least weight path and prove convergence in distribution of an appropriately centered version. This study continues the program initiated in [J. Math. Phys. 49 (2008) 125218] of showing that $\log{n}$ is the correct scaling for the hopcount under i.i.d. edge disorder, even if the graph distance between two randomly chosen vertices is of much smaller order. The case of infinite mean degrees ($τ\in[1,2)$) is studied in [Extreme value theory, Poisson--Dirichlet distributions and first passage percolation on random networks (2009) Preprint] where it is proved that the hopcount remains uniformly bounded and converges in distribution.

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arXiv2010

Ising models on power-law random graphs

We study a ferromagnetic Ising model on random graphs with a power-law degree distribution and compute the thermodynamic limit of the pressure when the mean degree is finite (degree exponent $τ>2$), for which the random graph has a tree-like structure. For this, we adapt and simplify an analysis by Dembo and Montanari, which assumes finite variance degrees ($τ>3$). We further identify the thermodynamic limits of various physical quantities, such as the magnetization and the internal energy.

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arXiv2010

Critical behavior in inhomogeneous random graphs

We study the critical behavior of inhomogeneous random graphs where edges are present independently but with unequal edge occupation probabilities. The edge probabilities are moderated by vertex weights, and are such that the degree of vertex i is close in distribution to a Poisson random variable with parameter w_i, where w_i denotes the weight of vertex i. We choose the weights such that the weight of a uniformly chosen vertex converges in distribution to a limiting random variable W, in which case the proportion of vertices with degree k is close to the probability that a Poisson random variable with random parameter W takes the value k. We pay special attention to the power-law case, in which P(W\geq k) is proportional to k^{-(τ-1)} for some power-law exponent τ>3, a property which is then inherited by the asymptotic degree distribution. We show that the critical behavior depends sensitively on the properties of the asymptotic degree distribution moderated by the asymptotic weight distribution W. Indeed, when P(W\geq k) \leq ck^{-(τ-1)} for all k\geq 1 and some τ>4 and c>0, the largest critical connected component in a graph of size n is of order n^{2/3}, as on the Erdős-Rényi random graph. When, instead, P(W\geq k)=ck^{-(τ-1)}(1+o(1)) for k large and some τ\in (3,4) and c>0, the largest critical connected component is of the much smaller order n^{(τ-2)/(τ-1)}.

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arXiv2010

First passage percolation on the Erdős-Rényi random graph

In this paper we explore first passage percolation (FPP) on the Erdős-Rényi random graph $G_n(p_n)$, where each edge is given an independent exponential edge weight with rate 1. In the sparse regime, i.e., when $np_n\to λ>1,$ we find refined asymptotics both for the minimal weight of the path between uniformly chosen vertices in the giant component, as well as for the hopcount (i.e., the number of edges) on this minimal weight path. More precisely, we prove a central limit theorem for the hopcount, with asymptotic mean and variance both equal to $λ/(λ-1)\log{n}$. Furthermore, we prove that the minimal weight centered by $\log{n}/(λ-1)$ converges in distribution. We also investigate the dense regime, where $np_n \to \infty$. We find that although the base graph is a {\it ultra small} (meaning that graph distances between uniformly chosen vertices are $o(\log{n})$), attaching random edge weights changes the geometry of the network completely. Indeed, the hopcount $H_n$ satisfies the universality property that whatever be the value of $p_n$, \ $H_n/\log{n}\to 1$ in probability and, more precisely, $(H_n-β_n\log{n})/\sqrt{\log{n}}$, where $β_n=λ_n/(λ_n-1)$, has a limiting standard normal distribution. The constant $β_n$ can be replaced by 1 precisely when $λ_n\gg \sqrt{\log{n}}$, a case that has appeared in the literature (under stronger conditions on $λ_n$). We also find bounds for the maximal weight and maximal hopcount between vertices in the graph. This paper continues the investigation of FPP initiated by the authors. Compared to the setting on the configuration model studied in \cite{BHHS08}, the proofs presented here are much simpler due to a direct relation between FPP on the Erdős-Rényi random graph and thinned continuous-time branching processes.

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arXiv2010

Diameters in preferential attachment models

In this paper, we investigate the diameter in preferential attachment (PA-) models, thus quantifying the statement that these models are small worlds. The models studied here are such that edges are attached to older vertices proportional to the degree plus a constant, i.e., we consider affine PA-models. There is a substantial amount of literature proving that, quite generally, PA-graphs possess power-law degree sequences with a power-law exponent τ>2. We prove that the diameter of the PA-model is bounded above by a constant times \log{t}, where t is the size of the graph. When the power-law exponent τexceeds 3, then we prove that \log{t} is the right order, by proving a lower bound of this order, both for the diameter as well as for the typical distance. This shows that, for τ>3, distances are of the order \log{t}. For τ\in (2,3), we improve the upper bound to a constant times \log\log{t}, and prove a lower bound of the same order for the diameter. Unfortunately, this proof does not extend to typical distances. These results do show that the diameter is of order \log\log{t}. These bounds partially prove predictions by physicists that the typical distance in PA-graphs are similar to the ones in other scale-free random graphs, such as the configuration model and various inhomogeneous random graph models, where typical distances have been shown to be of order \log\log{t} when τ\in (2,3), and of order \log{t} when τ>3.

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arXiv2010

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. This is the extended version of the paper, where we provide all proofs.

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arXiv2009

Scaling limits for critical inhomogeneous random graphs with finite third moments

We identify the scaling limits for the sizes of the largest components at criticality for inhomogeneous random graphs when the degree exponent $τ$ satisfies $τ>4$. We see that the sizes of the (rescaled) components converge to the excursion lengths of an inhomogeneous Brownian motion, extending results of \cite{Aldo97}. We rely heavily on martingale convergence techniques, and concentration properties of (super)martingales. This paper is part of a programme to study the critical behavior in inhomogeneous random graphs of so-called rank-1 initiated in \cite{Hofs09a}.

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arXiv2009

Extreme value theory, Poisson-Dirichlet distributions and FPP on random networks

We study first passage percolation on the configuration model (CM) having power-law degrees with exponent $τ\in [1,2)$. To this end we equip the edges with exponential weights. We derive the distributional limit of the minimal weight of a path between typical vertices in the network and the number of edges on the minimal weight path, which can be computed in terms of the Poisson-Dirichlet distribution. We explicitly describe these limits via the construction of an infinite limiting object describing the FPP problem in the densely connected core of the network. We consider two separate cases, namely, the {\it original CM}, in which each edge, regardless of its multiplicity, receives an independent exponential weight, as well as the {\it erased CM}, for which there is an independent exponential weight between any pair of direct neighbors. While the results are qualitatively similar, surprisingly the limiting random variables are quite different. Our results imply that the flow carrying properties of the network are markedly different from either the mean-field setting or the locally tree-like setting, which occurs as $τ>2$, and for which the hopcount between typical vertices scales as $\log{n}$. In our setting the hopcount is tight and has an explicit limiting distribution, showing that one can transfer information remarkably quickly between different vertices in the network. This efficiency has a down side in that such networks are remarkably fragile to directed attacks. These results continue a general program by the authors to obtain a complete picture of how random disorder changes the inherent geometry of various random network models.

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arXiv2009

Weak and almost sure limits for the parabolic Anderson model with heavy tailed potentials

We study the parabolic Anderson problem, that is, the heat equation $\partial_tu=Δu+ξu$ on $(0,\infty)\times{\mathbb{Z}}^d$ with independent identically distributed random potential $\{ξ(z):z\in{\mathbb{Z}}^d\}$ and localized initial condition $u(0,x)={\mathbf{1}}_0(x)$. Our interest is in the long-term behavior of the random total mass $U(t)=\sum_zu(t,z)$ of the unique nonnegative solution in the case that the distribution of $ξ(0)$ is heavy tailed. For this, we study two paradigm cases of distributions with infinite moment generating functions: the case of polynomial or Pareto tails, and the case of stretched exponential or Weibull tails. In both cases we find asymptotic expansions for the logarithm of the total mass up to the first random term, which we describe in terms of weak limit theorems. In the case of polynomial tails, already the leading term in the expansion is random. For stretched exponential tails, we observe random fluctuations in the almost sure asymptotics of the second term of the expansion, but in the weak sense the fourth term is the first random term of the expansion. The main tool in our proofs is extreme value theory.

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arXiv2009

The second largest component in the supercritical 2D Hamming graph

The 2-dimensional Hamming graph H(2,n) consists of the $n^2$ vertices $(i,j)$, $1\leq i,j\leq n$, two vertices being adjacent when they share a common coordinate. We examine random subgraphs of H(2,n) in percolation with edge probability $p$, so that the average degree $2(n-1)p=1+ε$. Previous work by van der Hofstad and Luczak had shown that in the barely supercritical region $n^{-2/3}\ln^{1/3}n\ll ε\ll 1$ the largest component has size $\sim 2εn$. Here we show that the second largest component has size close to $ε^{-2}$, so that the dominant component has emerged. This result also suggests that a {\it discrete duality principle} might hold, whereby, after removing the largest connected component in the supercritical regime, the remaining random subgraphs behave as in the subcritical regime.

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arXiv2008

Random subgraphs of the 2D Hamming graph: the supercritical phase

We study random subgraphs of the 2-dimensional Hamming graph H(2,n), which is the Cartesian product of two complete graphs on $n$ vertices. Let $p$ be the edge probability, and write $p=\frac{1+\vep}{2(n-1)}$ for some $\vep\in \R$. In Borgs et al., Random subgraphs of finite graphs: I. The scaling window under the triangle condition, Rand. Struct. Alg. (2005), and in Borgs et al., Random subgraphs of finite graphs: II. The lace expansion and the triangle condition, Ann. Probab. (2005), the size of the largest connected component was estimated precisely for a large class of graphs including H(2,n) for $\vep\leq ΛV^{-1/3}$, where $Λ> 0$ is a constant and $V=n^2$ denotes the number of vertices in H(2,n). Until now, no matching lower bound on the size in the supercritical regime has been obtained. In this paper we prove that, when $\vep\gg (\log{V})^{1/3} V^{-1/3}$, then the largest connected component has size close to $2\vep V$ with high probability. We thus obtain a law of large numbers for the largest connected component size, and show that the corresponding values of $p$ are supercritical. Barring the factor $(\log{\chs{V}})^{1/3}$, this identifies the size of the largest connected component all the way down to the critical $p$ window.

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arXiv2008

Large deviations for eigenvalues of sample covariance matrices, with applications to mobile communication systems

We study sample covariance matrices of the form $W=\frac 1n C C^T$, where $C$ is a $k\times n$ matrix with i.i.d. mean zero entries. This is a generalization of so-called Wishart matrices, where the entries of $C$ are independent and identically distributed standard normal random variables. Such matrices arise in statistics as sample covariance matrices, and the high-dimensional case, when $k$ is large, arises in the analysis of DNA experiments. We investigate the large deviation properties of the largest and smallest eigenvalues of $W$ when either $k$ is fixed and $n\to \infty$, or $k_n\to \infty$ with $k_n=o(n/\log\log{n})$, in the case where the squares of the i.i.d. entries have finite exponential moments. Previous results, proving a.s. limits of the eigenvalues, only require finite fourth moments. Our most explicit results for $k$ large are for the case where the entries of $C$ are $\pm1$ with equal probability. We relate the large deviation rate functions of the smallest and largest eigenvalue to the rate functions for independent and identically distributed standard normal entries of $C$. This case is of particular interest, since it is related to the problem of the decoding of a signal in a code division multiple access system arising in mobile communication systems. In this example, $k$ plays the role of the number of users in the system, and $n$ is the length of the coding sequence of each of the users. Each user transmits at the same time and uses the same frequency, and the codes are used to distinguish the signals of the separate users. The results imply large deviation bounds for the probability of a bit error due to the interference of the various users.

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arXiv2008

Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions

We consider the critical spread-out contact process in Z^d with d\ge1, whose infection range is denoted by L\ge1. In this paper, we investigate the r-point function τ_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that, for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time t_i by the individual at the origin o\in Z^d at time 0. Together with the results of the 2-point function in [van der Hofstad and Sakai, Electron. J. Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially rely, we prove that the r-point functions converge to the moment measures of the canonical measure of super-Brownian motion above the upper-critical dimension 4. We also prove partial results for d\le4 in a local mean-field setting.

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arXiv2008

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 thermometer, the binary and the segmented architectures. We study the two extreme architectures, 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 investigated in the literature. For the binary case, we investigate properties of the functional, such as its mean, variance and density.

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arXiv2008

Mean-field behavior for long- and finite range Ising model, percolation and self-avoiding walk

We consider self-avoiding walk, percolation and the Ising model with long and finite range. By means of the lace expansion we prove mean-field behavior for these models if $d>2(α\wedge2)$ for self-avoiding walk and the Ising model, and $d>3(α\wedge2)$ for percolation, where $d$ denotes the dimension and $α$ the power-law decay exponent of the coupling function. We provide a simplified analysis of the lace expansion based on the trigonometric approach in Borgs et al. (2007)

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arXiv2008

Universality for the distance in finite variance random graphs: Extended version

We generalize the asymptotic behavior of the graph distance between two uniformly chosen nodes in the configuration model to a wide class of random graphs. Among others, this class contains the Poissonian random graph, the expected degree random graph and the generalized random graph (including the classical Erdos-Renyi graph). In the paper we assign to each node a deterministic capacity and the probability that there exists an edge between a pair of nodes is equal to a function of the product of the capacities of the pair divided by the total capacity of all the nodes. We consider capacities which are such that the degrees of a node has uniformly bounded moments of order strictly larger than two, so that, in particular, the degrees have finite variance. We prove that the graph distance grows like \log_νN, where the νdepends on the capacities. In addition, the random fluctuations around this asymptotic mean \log_νN are shown to be tight. We also consider the case where the capacities are independent copies of a positive random Λwith \prob{Λ> x}\leq cx^{1-τ}, for some constant c and τ>3, again resulting in graphs where the degrees have finite variance. The method of proof of these results is to couple each member of the class to the Poissonian random graph, for which we then give the complete proof by adapting the arguments of the corresponding result for the configuration model.

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arXiv2007

Joint density for the local times of continuous-time Markov chains

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of $\mathbb{Z}^d$ tending to $\mathbb{Z}^d$ as time diverges. We finally discuss the relation of our density formula to the Ray--Knight theorem for continuous-time simple random walk on $\mathbb{Z}$, which is analogous to the well-known Ray--Knight description of Brownian local times.

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