| 1. |
- Angel, O., et al.
(författare)
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THE PHASE TRANSITION FOR DYADIC TILINGS
- 2014
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Ingår i: Transactions of the American Mathematical Society. - 0002-9947 .- 1088-6850. ; 366:2, s. 1029-1046
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Tidskriftsartikel (refereegranskat)abstract
- A dyadic tile of order n is any rectangle obtained from the unit square by n successive bisections by horizontal or vertical cuts. Let each dyadic tile of order n be available with probability p, independent of the others. We prove that for p sufficiently close to 1, there exists a set of pairwise disjoint available tiles whose union is the unit square, with probability tending to 1 as n -> infinity, as conjectured by Joel Spencer in 1999. In particular, we prove that if p = 7/8, such a tiling exists with probability at least 1 - (3/4)(n). The proof involves a surprisingly delicate counting argument for sets of unavailable tiles that prevent tiling.
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| 2. |
- Deijfen, Maria, et al.
(författare)
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Percolation in invariant Poisson graphs with i.i.d. degrees
- 2012
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Ingår i: Arkiv for Matematik. - : International Press of Boston. - 0004-2080 .- 1871-2487. ; 50:1, s. 41-58
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Tidskriftsartikel (refereegranskat)abstract
- Let each point of a homogeneous Poisson process in R-d independently be equipped with a random number of stubs (half-edges) according to a given probability distribution mu on the positive integers. We consider translation-invariant schemes for perfectly matching the stubs to obtain a simple graph with degree distribution mu. Leaving aside degenerate cases, we prove that for any mu there exist schemes that give only finite components as well as schemes that give infinite components. For a particular matching scheme which is a natural extension of Gale-Shapley stable marriage, we give sufficient conditions on mu for the absence and presence of infinite components.
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| 3. |
- Holroyd, A. E., et al.
(författare)
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Wald for non-stopping times: The rewards of impatient prophets
- 2014
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 19, s. 1-9
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Tidskriftsartikel (refereegranskat)abstract
- Let X-1 , X-2 , ... be independent identically distributed nonnegative random variables. Wald's identity states that the random sum S-T := X-1 + ... + X-T has expectation ET . EX1 provided T is a stopping time. We prove here that for any 1 < alpha <= 2, if T is an arbitrary nonnegative random variable, then S-T has finite expectation provided that X-1 has finite alpha-moment and T has finite 1/(alpha - 1)-moment. We also prove a variant in which T is assumed to have a finite exponential moment. These moment conditions are sharp in the sense that for any i.i.d. sequence X-i violating them, there is a T satisfying the given condition for which S-T (and, in fact, X-T) has infinite expectation. An interpretation is given in terms of a prophet being more rewarded than a gambler when a certain impatience restriction is imposed.
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