| 1. |
- Ahlberg, Daniel, 1982, et al.
(författare)
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Bernoulli and self-destructive percolation on non-amenable graphs
- 2014
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 19, s. article 40-
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Tidskriftsartikel (refereegranskat)abstract
- 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 p approaches 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.
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| 2. |
- Ahlberg, Daniel, et al.
(författare)
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Existence of an unbounded vacant set for subcritical continuum percolation
- 2018
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 23
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Tidskriftsartikel (refereegranskat)abstract
- We consider the Poisson Boolean percolation model in R-2, where the radius of each ball is independently chosen according to some probability measure with finite second moment. For this model, we show that the two thresholds, for the existence of an unbounded occupied and an unbounded vacant component, coincide. This complements a recent study of the sharpness of the phase transition in Poisson Boolean percolation by the same authors. As a corollary it follows that for Poisson Boolean percolation in R-d, for any d >= 2, finite moment of order d is both necessary and sufficient for the existence of a nontrivial phase transition for the vacant set.
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| 3. |
- Albin, Patrik, 1960, et al.
(författare)
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A New Proof of an Old Result by Pickands
- 2010
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 15, s. 339-345
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Tidskriftsartikel (refereegranskat)abstract
- Let {xi(t)}(t is an element of[0,h]) be a stationary Gaussian process with covariance function r such that r(t) = 1 - C vertical bar t vertical bar(alpha) + o(vertical bar t vertical bar(alpha)) as t -> 0. We give a new and direct proof of a result originally obtained by Pickands, on the asymptotic behaviour as u -> infinity of the probability P{sup(t is an element of vertical bar 0,h vertical bar) xi(t) > u} that the process xi exceeds the level u. As a by-product, we obtain a new expression for Pickands constant H alpha
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| 4. |
- Aly, Sidi Mohamed
(författare)
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Parameter sensitivity of CIR process
- 2013
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 18
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Tidskriftsartikel (refereegranskat)abstract
- We study the differentiability of the CIR process with respect to its parameters. We give a stochastic representation for these derivatives in terms of the paths of V.
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| 5. |
- Andersson, Jenny, 1976, et al.
(författare)
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The Volume Fraction of a Non-overlapping Germ-grain Model
- 2006
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 11, s. 78-88
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Tidskriftsartikel (refereegranskat)abstract
- We discuss the volume fraction of a model of non--overlapping convex grains. It is obtained from thinning a Poisson process where each point has a weight and is the centre of a grain, by removing any grain that is overlapped by one of larger or equal weight. In the limit as the intensity of the Poisson process tends to infinity, the model can be identified with the intact grains in the dead leaves model if the weights are independent of the grain sizes. In this case we can show that the volume fraction is at most $1/2^d$ for $d=1$ or $2$ if the shape is fixed, but the size and the orientation are random. The upper bound is achieved for centrally symmetric sets of the same size and orientation. For general $d$ we can show the upper bound, $1/2^d$, for spherical grains with two--point radius distribution. If dependence between weight and size is allowed, it is possible to achieve a volume fraction arbitrarily close to one.
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| 6. |
- Bergqvist, Göran, et al.
(författare)
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Rank probabilities for real random NxNx2 tensors
- 2011
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Ingår i: Electronic Communications in Probability. - : Institute of Mathematical Statistics / Bernoulli society.. - 1083-589X. ; 16, s. 630-637
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Tidskriftsartikel (refereegranskat)abstract
- We prove that the probability P_N for a real random Gaussian NxNx2 tensor to be of real rank N is P_N=(Gamma((N+1)/2))^N/G(N+1), where Gamma(x) and G(x) denote the gamma and the Barnes G-functions respectively. This is a rational number for N odd and a rational number multiplied by pi^{N/2} for N even. The probability to be of rank N+1 is 1-P_N. The proof makes use of recent results on the probability of having k real generalized eigenvalues for real random Gaussian N x N matrices. We also prove that log P_N= (N^2/4)log (e/4)+(log N-1)/12-zeta'(-1)+O(1/N) for large N, where zeta is the Riemann zeta function.
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| 7. |
- Björklund, Johan, et al.
(författare)
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On percolation in one-dimensional stable Poisson graphs
- 2015
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 20:50, s. 1-6
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Tidskriftsartikel (refereegranskat)abstract
- Equip each point x of a homogeneous Poisson point process P on R with D-x edge stubs, where the D-x are i.i.d. positive integer-valued random variables with distribution given by mu. Following the stable multi-matching scheme introduced by Deijfen, Haggstrom and Holroyd [1], we pair off edge stubs in a series of rounds to form the edge set of a graph G on the vertex set P. In this note, we answer questions of Deijfen, Holroyd and Peres [2] and Deijfen, Haggstrom and Holroyd [1] on percolation (the existence of an infinite connected component) in G. We prove that percolation may occur a.s. even if mu has support over odd integers. Furthermore, we show that for any epsilon > 0, there exists a distribution mu such that mu ({1}) > 1 - epsilon, but percolation still occurs a.s..
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| 8. |
- Björnberg, Jakob, 1983, et al.
(författare)
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Characterising random partitions by random colouring
- 2020
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 25
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Tidskriftsartikel (refereegranskat)abstract
- Let (X-1, X-2, ...) be a random partition of the unit interval [0, 1], i.e. X-i >= 0 and Sigma(i >= 1) X-i = 1, and let (epsilon(1), epsilon(2), ...) be i.i.d. Bernoulli random variables of parameter p is an element of (0, 1). The Bernoulli convolution of the partition is the random variable Z = Sigma(i >= 1) epsilon X-i(i). The question addressed in this article is: Knowing the distribution of Z for some fixed p is an element of (0, 1), what can we infer about the random partition (X-1, X-2, ...)? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter p is not equal to 1/2.
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| 9. |
- Björnberg, Jakob E., 1983, et al.
(författare)
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Characterising random partitions by random colouring
- 2020
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Ingår i: Electronic Communications in Probability. - : Institute of Mathematical Statistics. - 1083-589X. ; 25
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Tidskriftsartikel (refereegranskat)abstract
- Let (X-1, X-2, ...) be a random partition of the unit interval [0, 1], i.e. X-i >= 0 and Sigma(i >= 1) X-i = 1, and let (epsilon(1), epsilon(2), ...) be i.i.d. Bernoulli random variables of parameter p is an element of (0, 1). The Bernoulli convolution of the partition is the random variable Z = Sigma(i >= 1) epsilon X-i(i). The question addressed in this article is: Knowing the distribution of Z for some fixed p is an element of (0, 1), what can we infer about the random partition (X-1, X-2, ...)? We consider random partitions formed by residual allocation and prove that their distributions are fully characterised by their Bernoulli convolution if and only if the parameter p is not equal to 1/2.
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| 10. |
- Björnberg, Jakob, 1983
(författare)
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Large cycles in random permutations related to the Heisenberg model
- 2015
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 20
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Tidskriftsartikel (refereegranskat)abstract
- We study the weighted version of the interchange process where a permutation receives weight theta(#cycles). For theta = 2 this is Toth's representation of the quantum Heisenberg ferromagnet on the complete graph. We prove, for theta > 1, that large cycles appear at 'low temperature'.
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