24,265 research outputs found

    Percolation and coarse conformal uniformization

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    We formulate conjectures regarding percolation on planar triangulations suggested by assuming (quasi) invariance under coarse conformal uniformization

    Exponential clogging time for a one dimensional DLA

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    When considering DLA on a cylinder it is natural to ask how many particles it takes to clog the cylinder, e.g. modeling clogging of arteries. In this note we formulate a very simple DLA clogging model and establish an exponential lower bound on the number of particles arriving before clogging appears

    Unimodular Random Trees

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    We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We also prove that uniform integrability of the degree distribution of a family of finite graphs implies tightness of that family for local convergence, also known as random weak convergence.Comment: 19 pages, 4 figure

    Graph diameter in long-range percolation

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    We study the asymptotic growth of the diameter of a graph obtained by adding sparse "long" edges to a square box in Zd\Z^d. We focus on the cases when an edge between xx and yy is added with probability decaying with the Euclidean distance as xys+o(1)|x-y|^{-s+o(1)} when xy|x-y|\to\infty. For s(d,2d)s\in(d,2d) we show that the graph diameter for the graph reduced to a box of side LL scales like (logL)Δ+o(1)(\log L)^{\Delta+o(1)} where Δ1:=log2(2d/s)\Delta^{-1}:=\log_2(2d/s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance LL. We also show that a ball of radius rr in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp{r1/Δ+o(1)}\exp\{r^{1/\Delta+o(1)}\} in the Euclidean metric.Comment: 17 pages, extends the results of arXiv:math.PR/0304418 to graph diameter, substantially revised and corrected, added a result on volume growth asymptoti

    The shuffle estimator for explainable variance in fMRI experiments

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    In computational neuroscience, it is important to estimate well the proportion of signal variance in the total variance of neural activity measurements. This explainable variance measure helps neuroscientists assess the adequacy of predictive models that describe how images are encoded in the brain. Complicating the estimation problem are strong noise correlations, which may confound the neural responses corresponding to the stimuli. If not properly taken into account, the correlations could inflate the explainable variance estimates and suggest false possible prediction accuracies. We propose a novel method to estimate the explainable variance in functional MRI (fMRI) brain activity measurements when there are strong correlations in the noise. Our shuffle estimator is nonparametric, unbiased, and built upon the random effect model reflecting the randomization in the fMRI data collection process. Leveraging symmetries in the measurements, our estimator is obtained by appropriately permuting the measurement vector in such a way that the noise covariance structure is intact but the explainable variance is changed after the permutation. This difference is then used to estimate the explainable variance. We validate the properties of the proposed method in simulation experiments. For the image-fMRI data, we show that the shuffle estimates can explain the variation in prediction accuracy for voxels within the primary visual cortex (V1) better than alternative parametric methods.Comment: Published in at http://dx.doi.org/10.1214/13-AOAS681 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org

    On the trace of branching random walks

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    We study branching random walks on Cayley graphs. A first result is that the trace of a transient branching random walk on a Cayley graph is a.s. transient for the simple random walk. In addition, it has a.s. critical percolation probability less than one and exponential volume growth. The proofs rely on the fact that the trace induces an invariant percolation on the family tree of the branching random walk. Furthermore, we prove that the trace is a.s. strongly recurrent for any (non-trivial) branching random walk. This follows from the observation that the trace, after appropriate biasing of the root, defines a unimodular measure. All results are stated in the more general context of branching random walks on unimodular random graphs.Comment: revised versio
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