Browsing by Author "den Hollander, Frank"
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- ItemNetwork Models: Structure and Function(Zürich : EMS Publ. House, 2017) Bhamidi, Shankar; van der Hofstad, Remco; den Hollander, FrankThe focus of the meeting was on the mathematical analysis of complex networks, both on how networks emerge through microscopic interaction rules as well as on dynamic processes and optimization problems on networks, including random walks, interacting particle systems and search algorithms. Topics that were addressed included: percolation on graphs and critical regimes for the emergence of a giant component; graph limits and graphons; epidemics, propagation and competition; trees and forests; dynamic random graphs; local versus global algorithms; statistical learning on networks.
- ItemThe parabolic Anderson model on a Galton--Watson tree(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2020) den Hollander, Frank; König, Wolfgang; Soares dos Santos, RenatoWe study the long-time asymptotics of the total mass of the solution to the parabolic Anderson model ( PAM) on a supercritical Galton-Watson random tree with bounded degrees. We identify the second-order contribution to this asymptotics in terms of a variational formula that gives information about the local structure of the region where the solution is concentrated. The analysis behind this formula suggests that, under mild conditions on the model parameters, concentration takes place on a tree with minimal degree. Our approach can be applied to finite locally tree-like random graphs, in a coupled limit where both time and graph size tend to infinity. As an example, we consider the configuration model or, more precisely, the uniform simple random graph with a prescribed degree sequence.
- ItemQuenched LDP for words in a letter sequence(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2008) Birkner, Matthias; Greven, Andreas; den Hollander, FrankWhen we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential tail, in which case the rate function turns out to be the first term on the set where the second term vanishes and to be infinite elsewhere ...
- ItemRandom Trees(Zürich : EMS Publ. House, 2009) Dawson, Donald A.; Greven, Andreas; den Hollander, FrankThe meeting was devoted to random trees, a central concept in mathematics that provides a key way of thinking about relationships between objects such as particles in a fluid, individuals in a population, or labels in a search algorithm. Particular emphasis was put on the role of random trees in physics, biology, and computer science.
- ItemSelf-interacting Random Processes(Oberwolfach-Walke : Mathematisches Forschungsinstitut Oberwolfach, 2000) Brydges, David C.; den Hollander, Frank[no abstract available]