arXiv subjects

Remco van der Hofstad

Publications and source records attributed to Remco van der Hofstad.

At least 181 records · Page 10Linked to original sources

arXiv2007

An extension of the inductive approach to the lace expansion

We extend the inductive approach to the lace expansion, previously developed to study models with critical dimension 4, to be applicable more generally. In particular, the result of this note has recently been used to prove Gaussian asymptotic behaviour for the Fourier transform of the two-point function for sufficiently spread-out lattice trees in dimensions d>8, and it is potentially also applicable to percolation in dimensions d>6.

math.PR↗

arXiv2007

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.

math.PR↗

arXiv2007

Extension of the generalised inductive approach to the lace expansion: Full proof

This paper extends the inductive approach to the lace expansion of van der Hofstad and Slade in order to prove Gaussian asymptotic behaviour for models with critical dimension other than 4. The results are applied by Holmes to study sufficiently spread-out lattice trees in dimensions d>8 and may also be applicable to percolation in dimensions d>6.

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arXiv2006

Random graphs with arbitrary i.i.d. degrees

In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree distribution is regularly varying with exponent $1-τ$. There are three distinct cases: (i) $τ>3$, where the degrees have finite variance, (ii) $τ\in (2,3)$, where the degrees have infinite variance, but finite mean, and (iii) $τ\in (1,2)$, where the degrees have infinite mean. These random graphs can serve as models for complex networks where degree power laws are observed. Our results are twofold. First, we give a criterion when there exists a unique largest connected component of size proportional to the size of the graph, and study sizes of the other connected components. Secondly, we establish a phase transition for the diameter when $τ\in (2,3)$. Indeed, we show that for $τ>2$ and when nodes with degree 2 are present with positive probability, the diameter of the random graph is, with high probability, bounded below by a constant times the logarithm of the size of the graph. On the other hand, assuming that all degrees are at least 3 or more, we show that, for $τ\in (2,3)$, the diameter of the graph is with high probability bounded from above by a constant times the $\log \log$ of the size of the graph.

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arXiv2006

Random graph asymptotics on high-dimensional tori

We investigate the scaling of the largest critical percolation cluster on a large d-dimensional torus, for nearest-neighbor percolation in high dimensions, or when d>6 for sufficient spread-out percolation. We use a relatively simple coupling argument to show that this largest critical cluster is, with high probability, bounded above by a large constant times $V^{2/3}$ and below by a small constant times $V^{2/3}(log V)^{-4/3}$, where V is the volume of the torus. We also give a simple criterion in terms of the subcritical percolation two-point function on Z^d under which the lower bound can be improved to small constant times $V^{2/3}$, i.e., we prove random graph asymptotics for the largest critical cluster on the high-dimensional torus. This establishes a conjecture by Aizenman (1997), apart from logarithmic corrections. We discuss implications of these results on the dependence on boundary conditions for high-dimensional percolation. Our method is crucially based on the results by Borgs, Chayes, van der Hofstad, Slade and Spencer (2005a, 2005b), where the $V^{2/3}$ scaling was proved subject to the assumption that a suitably defined critical window contains the percolation threshold on Z^d. We also strongly rely on mean-field results for percolation on Z^d proved by Hara (1990, 2005), Hara and Slade (1990) and Hara, van der Hofstad and Slade (2003).

math.PR↗

arXiv2005

Deviations of a random walk in a random scenery with stretched exponential tails

Let (Z_n)_{n\in\N_0} be a d-dimensional random walk in random scenery, i.e., Z_n=\sum_{k=0}^{n-1}Y_{S_k} with (S_k)_{k\in\N_0} a random walk in Z^d and (Y_z)_{z\in Z^d} an i.i.d. scenery, independent of the walk. We assume that the random variables Y_z have a stretched exponential tail. In particular, they do not possess exponential moments. We identify the speed and the rate of the logarithmic decay of Pr(Z_n>t_n n) for all sequences (t_n)_{n\in\N} satisfying a certain lower bound. This complements previous results, where it was assumed that Y_z has exponential moments of all orders. In contrast to the previous situation,the event \{Z_n>t_nn\} is not realized by a homogeneous behavior of the walk's local times and the scenery, but by many visits of the walker to a particular site and a large value of the scenery at that site. This reflects a well-known extreme behavior typical for random variables having no exponential moments.

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arXiv2005

The universality classes in the parabolic Anderson model

We discuss the long time behaviour of the parabolic Anderson model, the Cauchy problem for the heat equation with random potential on $\Z^d$. We consider general i.i.d. potentials and show that exactly \emph{four} qualitatively different types of intermittent behaviour can occur. These four universality classes depend on the upper tail of the potential distribution: (1) tails at $\infty$ that are thicker than the double-exponential tails, (2) double-exponential tails at $\infty$ studied by Gärtner and Molchanov, (3) a new class called \emph{almost bounded potentials}, and (4) potentials bounded from above studied by Biskup and König. The new class (3), which contains both unbounded and bounded potentials, is studied in both the annealed and the quenched setting. We show that intermittency occurs on unboundedly increasing islands whose diameter is slowly varying in time. The characteristic variational formulas describing the optimal profiles of the potential and of the solution are solved explicitly by parabolas, respectively, Gaussian densities.

math.PR↗

arXiv2005

Counting Connected Graphs Asymptotically

We find the asymptotic number of connected graphs with $k$ vertices and $k-1+l$ edges when $k,l$ approach infinity, reproving a result of Bender, Canfield and McKay. We use the {\em probabilistic method}, analyzing breadth-first search on the random graph $G(k,p)$ for an appropriate edge probability $p$. Central is analysis of a random walk with fixed beginning and end which is tilted to the left.

math.CO↗

arXiv2005

Distances in random graphs with finite mean and infinite variance degrees

In this paper we study random graphs with independent and identically distributed degrees of which the tail of the distribution function is regularly varying with exponent $τ\in (2,3)$. The number of edges between two arbitrary nodes, also called the graph distance or hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. When $τ\in (2,3)$, this graph distance grows like $2\frac{\log\log N}{|\log(τ-2)|}$. In different papers, the cases $τ>3$ and $τ\in (1,2)$ have been studied. We also study the fluctuations around these asymptotic means, and describe their distributions. The results presented here improve upon results of Reittu and Norros, who prove an upper bound only.

math.PR↗

arXiv2004

Gaussian scaling for the critical spread-out contact process above the upper critical dimension

We consider the critical spread-out contact process in $\Zd$ with $d\geq 1$, whose infection range is denoted by $L\geq1$. The two-point function $τ_t(x)$ is the probability that $x\in\Zd$ is infected at time $t$ by the infected individual located at the origin $o\in\Zd$ at time 0. We prove Gaussian behavior for the two-point function with $L\geq L_0$ for some finite $L_0=L_0(d)$ for $d>4$. When $d\leq 4$, we also perform a local mean-field limit to obtain Gaussian behaviour for $τ_{tT}$ with $t>0$ fixed and $T\to \infty$ when the infection range depends on $T$ such that $L_T=LT^b$ for any $b>(4-d)/2d$. The proof is based on the lace expansion and an adaptation of the inductive approach applied to the discretized contact process. We prove the existence of several critical exponents and show that they take on mean-field values. The results in this paper provide crucial ingredients to prove convergence of the finite-dimensional distributions for the contact process towards the canonical measure of super-Brownian motion, which we defer to a sequel of this paper.

math.PR↗

arXiv2004

Critical points for spread-out self-avoiding walk, percolation and the contact process above the upper critical dimensions

We consider self-avoiding walk and percolation in $\Zd$, oriented percolation in $\Zd\times\Zp$, and the contact process in $\Zd$, with $p D(\cdot)$ being the coupling function whose range is denoted by $L<\infty$. For percolation, for example, each bond $\{x,y\}$ is occupied with probability $p D(y-x)$. The above models are known to exhibit a phase transition when the parameter $p$ varies around a model-dependent critical point $\pc$. We investigate the value of $\pc$ when $d>6$ for percolation and $d>4$ for the other models, and $L\gg1$. We prove in a unified way that $\pc=1+C(D)+O(L^{-2d})$, where the universal term 1 is the mean-field critical value, and the model-dependent term $C(D)=O(L^{-d})$ is written explicitly in terms of the function $D$. Our proof is based on the lace expansion for each of these models.

math.PR↗

arXiv2004

Distances in random graphs with infinite mean degrees

We study random graphs with an i.i.d. degree sequence of which the tail of the distribution function $F$ is regularly varying with exponent $τ\in (1,2)$. Thus, the degrees have infinite mean. Such random graphs can serve as models for complex networks where degree power laws are observed. The minimal number of edges between two arbitrary nodes, also called the graph distance or the hopcount, in a graph with $N$ nodes is investigated when $N\to \infty$. The paper is part of a sequel of three papers. The other two papers study the case where $τ\in (2,3)$, and $τ\in (3,\infty),$ respectively. The main result of this paper is that the graph distance converges for $τ\in (1,2)$ to a limit random variable with probability mass exclusively on the points 2 and 3. We also consider the case where we condition the degrees to be at most $N^α$ for some $α>0.$ For $τ^{-1}<α<(τ-1)^{-1}$, the hopcount converges to 3 in probability, while for $α>(τ-1)^{-1}$, the hopcount converges to the same limit as for the unconditioned degrees. Our results give convincing asymptotics for the hopcount when the mean degree is infinite, using extreme value theory.

math.PR↗

arXiv2004

Distances in random graphs with finite variance degrees

In this paper we study a random graph with $N$ nodes, where node $j$ has degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with $\prob(D_j\leq x)=F(x)$. We assume that $1-F(x)\leq c x^{-τ+1}$ for some $τ>3$ and some constant $c>0$. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when $N\to \infty$. We prove that the graph distance grows like $\log_νN$, when the base of the logarithm equals $ν=\expec[D_j(D_j -1)]/\expec[D_j]>1$. This confirms the heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the random fluctuations around this asymptotic mean $\log_ν{N}$ are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.

math.PR↗

arXiv2004

Infinite canonical super-Brownian motion and scaling limits

We construct a measure valued Markov process which we call infinite canonical super-Brownian motion, and which corresponds to the canonical measure of super-Brownian motion conditioned on non-extinction. Infinite canonical super-Brownian motion is a natural candidate for the scaling limit of various random branching objects on $\Z^d$ when these objects are (a) critical; (b) mean-field and (c) infinite. We prove that ICSBM is the scaling limit of the spread-out oriented percolation incipient infinite cluster above 4 dimensions and of incipient infinite branching random walk in any dimension. We conjecture that it also arises as the scaling limit in various other models above the upper-critical dimension, such as the incipient infinite lattice tree above 8 dimensions, the incipient infinite cluster for unoriented percolation, uniform spanning trees above 4 dimensions, and invasion percolation above 6 dimensions. This paper also serves as a survey of recent results linking super-Brownian to scaling limits in statistical mechanics.

math.PR↗

arXiv2004

Maximal clusters in non-critical percolation and related models

We investigate the maximal non-critical cluster in a big box in various percolation-type models. We investigate its typical size, and the fluctuations around this typical size. The limit law of these fluctuations are related to maxima of independent random variable with law described by a single cluster.

math.PR↗

arXiv2004

Random subgraphs of finite graphs: I. The scaling window under the triangle condition

We study random subgraphs of an arbitrary finite connected transitive graph $\mathbb G$ obtained by independently deleting edges with probability $1-p$. Let $V$ be the number of vertices in $\mathbb G$, and let $Ω$ be their degree. We define the critical threshold $p_c=p_c(\mathbb G,λ)$ to be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λV^{1/3}$, where $λ$ is fixed and positive. We show that for any such model, there is a phase transition at $p_c$ analogous to the phase transition for the random graph, provided that a quantity called the triangle diagram is sufficiently small at the threshold $p_c$. In particular, we show that the largest cluster inside a scaling window of size $|p-p_c|=Θ(\cn^{-1}V^{-1/3})$ is of size $Θ(V^{2/3})$, while below this scaling window, it is much smaller, of order $O(ε^{-2}\log(Vε^3))$, with $ε=\cn(p_c-p)$. We also obtain an upper bound $O(\cn(p-p_c)V)$ for the expected size of the largest cluster above the window. In addition, we define and analyze the percolation probability above the window and show that it is of order $Θ(\cn(p-p_c))$. Among the models for which the triangle diagram is small enough to allow us to draw these conclusions are the random graph, the $n$-cube and certain Hamming cubes, as well as the spread-out $n$-dimensional torus for $n>6$.

math.PR↗

arXiv2004

Random subgraphs of finite graphs: II. The lace expansion and the triangle condition

In a previous paper, we defined a version of the percolation triangle condition that is suitable for the analysis of bond percolation on a finite connected transitive graph, and showed that this triangle condition implies that the percolation phase transition has many features in common with the phase transition on the complete graph. In this paper, we use a new and simplified approach to the lace expansion to prove quite generally that for finite graphs that are tori the triangle condition for percolation is implied by a certain triangle condition for simple random walks on the graph. The latter is readily verified for several graphs with vertex set $\{0,1,..., r-1\}^n$, including the Hamming cube on an alphabet of $r$ letters (the $n$-cube, for $r=2$), the $n$-dimensional torus with nearest-neighbor bonds and $n$ sufficiently large, and the $n$-dimensional torus with $n>6$ and sufficiently spread-out (long range) bonds. The conclusions of our previous paper thus apply to the percolation phase transition for each of the above examples.

math.PR↗

arXiv2004

Random subgraphs of finite graphs: III. The phase transition for the $n$-cube

We study random subgraphs of the $n$-cube $\{0,1\}^n$, where nearest-neighbor edges are occupied with probability $p$. Let $p_c(n)$ be the value of $p$ for which the expected cluster size of a fixed vertex attains the value $λ2^{n/3}$, where $λ$ is a small positive constant. Let $ε=n(p-p_c(n))$. In two previous papers, we showed that the largest cluster inside a scaling window given by $|ε|=Θ(2^{-n/3})$ is of size $Θ(2^{2n/3})$, below this scaling window it is at most $2(\log2) nε^{-2}$, and above this scaling window it is at most $O(ε2^n)$. In this paper, we prove that for $p - p_c(n) \geq e^{-cn^{1/3}}$ the size of the largest cluster is at least $Θ(ε2^n)$, which is of the same order as the upper bound. This provides an understanding of the phase transition that goes far beyond that obtained by previous authors. The proof is based on a method that has come to be known as ``sprinkling,'' and relies heavily on the specific geometry of the $n$-cube.

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