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Any of these conditions can be waived by permission of the Corresponding Author. ecp@iam.uni-bonn.de (Anton Bovier (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 09 Jan 2014 00:05:12 -0800 OJS http://blogs.law.harvard.edu/tech/rss 60 The power of choice combined with preferential attachement http://ecp.ejpecp.org/article/view/3461 <div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p>We prove almost sure convergence of the maximum degree in an evolving tree model combining local choice and preferential attachment. At each step in the growth of the graph, a new vertex is introduced. A fixed, finite number of possible neighbors are sampled from the existing vertices with probability proportional to degree. Of these possibilities, the new vertex attaches to the vertex from the sample that has the highest degree. The maximal degree in this model has linear or near-linear behavior. This behavior contrasts sharply with the behavior in the same choice model with uniform attachment as well as the preferential attachment model without choice. The proof is based on showing the tree has a persistent hub by comparison with the standard preferential attachment model, as well as martingale and stochastic approximation arguments.</p></div></div></div> Yury Malyshkin, Elliot Paquette http://ecp.ejpecp.org/article/view/3461 Sat, 12 Jul 2014 07:15:23 -0700 Uniqueness of degenerate Fokker-Planck equations with weakly differentiable drift whose gradient is given by a singular integral http://ecp.ejpecp.org/article/view/3407 In this paper we prove the uniqueness of solutions to degenerate Fokker-Planck equations with bounded coefficients, under the additional assumptions that the diffusion coefficient has $W^{1,2}_{loc}$ regularity, while the gradient of the drift coefficient is merely given by a singular integral. Dejun Luo http://ecp.ejpecp.org/article/view/3407 Sat, 12 Jul 2014 06:54:14 -0700 Lower bounds for bootstrap percolation on Galton-Watson trees http://ecp.ejpecp.org/article/view/3315 Bootstrap percolation is a cellular automaton modelling the spread of an `infection' on a graph. In this note, we prove a family lower bounds on the critical probability for r-neighbour bootstrap percolation on Galton-Watson trees in terms of moments of the offspring distributions. With this result we confirm a conjecture of Bollobás, Gunderson, Holmgren, Janson and Przykucki. We also show that these bounds are best possible up to positive constants not depending on the offspring distribution. Karen Gunderson, Michal Przykucki http://ecp.ejpecp.org/article/view/3315 Sat, 12 Jul 2014 06:45:48 -0700 Large deviations for weighted sums of stretched exponential random variables http://ecp.ejpecp.org/article/view/3266 We consider the probability that a weighted sum of n i.i.d. random variables $X_j, j = 1,\ldots,n$, with stretched exponential tails is larger than its expectation and   determine the rate of its decay, under suitable conditions on the weights. We show that the decay is subexponential, and  identify the rate function in terms of the tails of $X_j$ and the weights. Our result generalizes the large deviation principle given by Kiesel and Stadtmüller as well as the tail asymptotics for sums of i.i.d. random variables provided by Nagaev. As an application of our result, motivated by random projections of high-dimensional vectors, we consider the case of random, self-normalized weights that are independent of the sequence $X_j$, identify the decay rate for both the quenched and annealed large deviations in this case, and show that they coincide. As another example we consider weights derived from kernel functions that arise in nonparametric regression. Nina Gantert, Kavita Ramanan, Franz Rembart http://ecp.ejpecp.org/article/view/3266 Sat, 12 Jul 2014 06:37:12 -0700 Bernoulli and self-destructive percolation on non-amenable graphs http://ecp.ejpecp.org/article/view/2611 <p>In this note we study some properties of infinite percolation clusters on non-amenable graphs. In particular, we study the percolative properties of the complement  of infinite percolation clusters. An approach based on mass-transport is adapted to show that for a large class of non-amenable graphs, the graph obtained by removing each site contained in an infinite percolation cluster has critical percolation threshold which can be arbitrarily close to the critical threshold for the original graph, almost surely, as <span style="font-size: 10px;">$p\searrow p_c$. Closely related is the self-destructive percolation process, introduced by J. van den Berg and R. Brouwer, for which we prove that an infinite cluster emerges for any small reinforcement.</span></p> Daniel Ahlberg, Vladas Sidoravicius, Johan Tikesson http://ecp.ejpecp.org/article/view/2611 Sat, 12 Jul 2014 06:21:42 -0700