arXiv subjects

Ronald Meester

Publications and source records attributed to Ronald Meester.

34 records · Page 2Linked to original sources

arXiv2012

Dimension (in)equalities and Hölder continuous curves in fractal percolation

We relate various concepts of fractal dimension of the limiting set C in fractal percolation to the dimensions of the set consisting of connected components larger than one point and its complement in C (the "dust"). In two dimensions, we also show that the set consisting of connected components larger than one point is a.s. the union of non-trivial Hölder continuous curves, all with the same exponent. Finally, we give a short proof of the fact that in two dimensions, any curve in the limiting set must have Hausdorff dimension strictly larger than 1.

math.PR↗

arXiv2012

Forensic identification: the Island Problem and its generalisations

In forensics it is a classical problem to determine, when a suspect $S$ shares a property $Γ$ with a criminal $C$, the probability that $S=C$. In this paper we give a detailed account of this problem in various degrees of generality. We start with the classical case where the probability of having $Γ$, as well as the a priori probability of being the criminal, is the same for all individuals. We then generalize the solution to deal with heterogeneous populations, biased search procedures for the suspect, $Γ$-correlations, uncertainty about the subpopulation of the criminal and the suspect, and uncertainty about the $Γ$-frequencies. We also consider the effect of the way the search for $S$ is conducted, in particular when this is done by a database search. A returning theme is that we show that conditioning is of importance when one wants to quantify the "weight" of the evidence by a likelihood ratio. Apart from these mathematical issues, we also discuss the practical problems in applying these issues to the legal process. The posterior probabilities of $C=S$ are typically the same for all reasonable choices of the hypotheses, but this is not the whole story. The legal process might force one to dismiss certain hypotheses, for instance when the relevant likelihood ratio depends on prior probabilities. We discuss this and related issues as well. As such, the paper is relevant both from a theoretical and from an applied point of view.

math.PR↗

arXiv2011

Survival of inhomogeneous Galton-Watson processes

We study survival properties of inhomogeneous Galton-Watson processes. We determine the so-called branching number (which is the reciprocal of the critical value for percolation) for these random trees (conditioned on being infinite), which turns out to be an a.s.\ constant. We also shed some light on the way the survival probability varies between the generations. When we perform independent percolation on the family tree of an inhomogeneous Galton-Watson process, the result is essentially a family of inhomogeneous Galton-Watson processes, parametrized by the retention probability $p$. We provide growth rates, uniformly in $p$, of the percolation clusters, and also show uniform convergence of the survival probability from the $n$-th level along subsequences. These results also establish, as a corollary, the supercritical continuity of the percolation function. Some of our results are generalisations of results by Lyons (1992).

math.PR↗

arXiv2011

Stochastic domination and weak convergence of conditioned Bernoulli random vectors

For n>=1 let X_n be a vector of n independent Bernoulli random variables. We assume that X_n consists of M "blocks" such that the Bernoulli random variables in block i have success probability p_i. Here M does not depend on n and the size of each block is essentially linear in n. Let X'_n be a random vector having the conditional distribution of X_n, conditioned on the total number of successes being at least k_n, where k_n is also essentially linear in n. Define Y'_n similarly, but with success probabilities q_i>=p_i. We prove that the law of X'_n converges weakly to a distribution that we can describe precisely. We then prove that sup Pr(X'_n <= Y'_n) converges to a constant, where the supremum is taken over all possible couplings of X'_n and Y'_n. This constant is expressed explicitly in terms of the parameters of the system.

math.PR↗

arXiv2010

Long-range percolation on the hierarchical lattice

We study long-range percolation on the hierarchical lattice of order $N$, where any edge of length $k$ is present with probability $p_k=1-\exp(-β^{-k} α)$, independently of all other edges. For fixed $β$, we show that the critical value $α_c(β)$ is non-trivial if and only if $N < β< N^2$. Furthermore, we show uniqueness of the infinite component and continuity of the percolation probability and of $α_c(β)$ as a function of $β$. This means that the phase diagram of this model is well understood.

math.PR↗

arXiv2010

Bounding basic characteristics of spatial epidemics with a new percolation model

We introduce a new percolation model to describe and analyze the spread of an epidemic on a general directed and locally finite graph. We assign a two-dimensional random weight vector to each vertex of the graph in such a way that the weights of different vertices are i.i.d., but the two entries of the vector assigned to a vertex need not be independent. The probability for an edge to be open depends on the weights of its end vertices, but conditionally on the weights, the states of the edges are independent of each other. In an epidemiological setting, the vertices of a graph represent the individuals in a (social) network and the edges represent the connections in the network. The weights assigned to an individual denote its (random) infectivity and susceptibility, respectively. We show that one can bound the percolation probability and the expected size of the cluster of vertices that can be reached by an open path starting at a given vertex from above and below by the corresponding quantities for respectively independent bond and site percolation with certain densities; this generalizes a result of Kuulasmaa. Many models in the literature are special cases of our general model.

math.PR↗

arXiv2010

Trivial, Critical and Near-critical Scaling Limits of Two-dimensional Percolation

It is natural to expect that there are only three possible types of scaling limits for the collection of all percolation interfaces in the plane: (1) a trivial one, consisting of no curves at all, (2) a critical one, in which all points of the plane are surrounded by arbitrarily large loops and every deterministic point is almost surely surrounded by a countably infinite family of nested loops with radii going to zero, and (3) an intermediate one, in which every deterministic point of the plane is almost surely surrounded by a largest loop and by a countably infinite family of nested loops with radii going to zero. We show how one can prove this using elementary arguments, with the help of known scaling relations for percolation. The trivial limit corresponds to subcritical and supercritical percolation, as well as to the case when the density p approaches the critical probability, p_c, sufficiently slowly as the lattice spacing is sent to zero. The second type corresponds to critical percolation and to a faster approach of p to p_c. The third, or near-critical, type of limit corresponds to an intermediate speed of approach of p to p_c. The fact that in the near-critical case a deterministic point is a.s. surrounded by a largest loop demonstrates the persistence of a macroscopic correlation length in the scaling limit and the absence of scale invariance.

math.PR↗

arXiv2009

Existence and uniqueness of the stationary measure in the continuous Abelian sandpile

Let Λbe a finite subset of Z^d. We study the following sandpile model on Λ. The height at any given vertex x of Λis a positive real number, and additions are uniformly distributed on some interval [a,b], which is a subset of [0,1]. The threshold value is 1; when the height at a given vertex exceeds 1, it topples, that is, its height is reduced by 1, and the heights of all its neighbours in Λincrease by 1/2d. We first establish that the uniform measure μon the so called "allowed configurations" is invariant under the dynamics. When a < b, we show with coupling ideas that starting from any initial configuration of heights, the process converges in distribution to μ, which therefore is the unique invariant measure for the process. When a = b, that is, when the addition amount is non-random, and a is rational, it is still the case that μis the unique invariant probability measure, but in this case we use random ergodic theory to prove this; this proof proceeds in a very different way. Indeed, the coupling approach cannot work in this case since we also show the somewhat surprising fact that when a = b is rational, the process does not converge in distribution at all starting from any initial configuration.

math.PR↗

arXiv2009

Stabilizability and percolation in the infinite volume sandpile model

We study the sandpile model in infinite volume on $\mathbb{Z}^d$. In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure $μ$, are $μ$-almost surely stabilizable. We prove that stabilizability does not depend on the particular procedure of stabilization we adopt. In $d=1$ and $μ$ a product measure with density $ρ=1$ (the known critical value for stabilizability in $d=1$) with a positive density of empty sites, we prove that $μ$ is not stabilizable. Furthermore, we study, for values of $ρ$ such that $μ$ is stabilizable, percolation of toppled sites. We find that for $ρ>0$ small enough, there is a subcritical regime where the distribution of a cluster of toppled sites has an exponential tail, as is the case in the subcritical regime for ordinary percolation.

math.PR↗

arXiv2008

Uniqueness of the stationary distribution and stabilizability in Zhang's sandpile model

We show that Zhang's sandpile model (N,[a,b]) on N sites and with uniform additions on [a,b] has a unique stationary measure for all 0 <= a < b <= 1. This generalizes earlier results where this was shown in some special cases. We define the infinite volume Zhang's sandpile model in dimension d >= 1, in which topplings occur according to a Markov toppling process, and we study the stabilizability of initial configurations chosen according to some measure μ. We show that for a stationary ergodic measure μwith density ρ, for all ρ< 1/2, μis stabilizable; for all ρ>= 1, μis not stabilizable; for 1/2 <= ρ< 1, when ρis near to 1/2 or 1, both possibilities can occur.

math-ph↗

arXiv2008

The high temperature Ising model on the triangular lattice is a critical percolation model

The Ising model at inverse temperature $β$ and zero external field can be obtained via the Fortuin-Kasteleyn (FK) random-cluster model with $q=2$ and density of open edges $p=1-e^{-β}$ by assigning spin +1 or -1 to each vertex in such a way that (1) all the vertices in the same FK cluster get the same spin and (2) +1 and -1 have equal probability. We generalize the above procedure by assigning spin +1 with probability $r$ and -1 with probability $1-r$, with $r \in [0,1]$, while keeping condition (1). For fixed $β$, this generates a dependent (spin) percolation model with parameter $r$. We show that, on the triangular lattice and for $β<β_c$, this model has a percolation phase transition at $r=1/2$, corresponding to the Ising model. This sheds some light on the conjecture that the high temperature Ising model on the triangular lattice is in the percolation universality class and that its scaling limit can be described in terms of SLE$_6$. We also prove uniqueness of the infinite +1 cluster for $r>1/2$, sharpness of the percolation phase transition (by showing exponential decay of the cluster size distribution for $r<1/2$), and continuity of the percolation function for all $r \in [0,1]$.

math.PR↗

arXiv2007

Sharp phase transition and critical behaviour in 2D divide and colour models

Consider subcritical Bernoulli bond percolation with fixed parameter p<p_c. We define a dependent site percolation model by the following procedure: for each bond cluster, we colour all vertices in the cluster black with probability r and white with probability 1-r, independently of each other. On the square lattice, defining the critical probabilities for the site model and its dual, r_c(p) and r_c^*(p) respectively, as usual, we prove that r_c(p)+r_c^*(p)=1 for all subcritical p. On the triangular lattice, where our method also works, this leads to r_c(p)=1/2, for all subcritical p. On both lattices, we obtain exponential decay of cluster sizes below r_c(p), divergence of the mean cluster size at r_c(p), and continuity of the percolation function in r on [0,1]. We also discuss possible extensions of our results, and formulate some natural conjectures. Our methods rely on duality considerations and on recent extensions of the classical RSW theorem.

math.PR↗

arXiv2007

A probabilistic approach to Zhang's sandpile model

The current literature on sandpile models mainly deals with the abelian sandpile model (ASM) and its variants. We treat a less known - but equally interesting - model, namely Zhang's sandpile. This model differs in two aspects from the ASM. First, additions are not discrete, but random amounts with a uniform distribution on an interval $[a,b]$. Second, if a site topples - which happens if the amount at that site is larger than a threshold value $E_c$ (which is a model parameter), then it divides its entire content in equal amounts among its neighbors. Zhang conjectured that in the infinite volume limit, this model tends to behave like the ASM in the sense that the stationary measure for the system in large volumes tends to be peaked narrowly around a finite set. This belief is supported by simulations, but so far not by analytical investigations. We study the stationary distribution of this model in one dimension, for several values of $a$ and $b$. When there is only one site, exact computations are possible. Our main result concerns the limit as the number of sites tends to infinity, in the one-dimensional case. We find that the stationary distribution, in the case $a \geq E_c/2$, indeed tends to that of the ASM (up to a scaling factor), in agreement with Zhang's conjecture. For the case $a=0$, $b=1$ we provide strong evidence that the stationary expectation tends to $\sqrt{1/2}$.

math-ph↗

arXiv2006

A note on percolation in cocycle measures

We describe infinite clusters which arise in nearest-neighbour percolation for so-called cocycle measures on the square lattice. These measures arise naturally in the study of random transformations. We show that infinite clusters have a very specific form and direction. In concrete situations, this leads to a quick decision whether or not a certain cocycle measure percolates. We illustrate this with two examples which are interesting in their own right.

math.PR↗

arXiv2006

The size of components in continuum nearest-neighbor graphs

We study the size of connected components of random nearest-neighbor graphs with vertex set the points of a homogeneous Poisson point process in ${\mathbb{R}}^d$. The connectivity function is shown to decay superexponentially, and we identify the exact exponent. From this we also obtain the decay rate of the maximal number of points of a path through the origin. We define the generation number of a point in a component and establish its asymptotic distribution as the dimension $d$ tends to infinity.

math.PR↗

arXiv2006

Bounds for avalanche critical values of the Bak-Sneppen model

We study the Bak-Sneppen model on locally finite transitive graphs $G$, in particular on Z^d and on T_Delta, the regular tree with common degree Delta. We show that the avalanches of the Bak-Sneppen model dominate independent site percolation, in a sense to be made precise. Since avalanches of the Bak-Sneppen model are dominated by a simple branching process, this yields upper and lower bounds for the so-called avalanche critical value $p_c^{BS}(G)$. Our main results imply that 1/(Delta+1) <= \leq p_c^{BS}(T_Delta) \leq 1/(Delta -1)$, and that $1/(2d+1)\leq p_c^{BS}(Z^d)\leq 1/(2d)+ 1/(2d)^2+O(d^{-3}), as d\to\infty.

math.PR↗