Abstract. Let X1, X2, . . . denote i.i.d. random bits, each taking the values 1 and 0 with respective probabilities p and 1 − p. A well-known theorem of Erdős and Rényi (1970) describes the length of the longest contiguous stretch, or “run,” of ones in X1, . . . , Xn for large values of n. Benjamini, Häggström, Peres, and Steif (2003, Theorem 1.4) demonstrated the existence of unusual times, pr...

Let M be a compact d-dimensional Riemannian manifold without boundary. Given set E?M, we study the of distances from E to fixed point x?E. This is ??x(E)={?(x,y):y?E}, where ? metric on M. We prove that if Hausdorff dimension greater than d+12, then there exist many x?E such Lebesgue measure ??x(E) positive. result was previously established by Peres and Schlag in Euclidean setting. give simple...

It is shown that the " retrodiction paradox " recently introduced by Peres arises not because of the fallacy of the time-symmetric approach as he claimed, but due to an inappropriate usage of retrodiction.

We propose a general method by which the overlap parameter, first proposed by Peres, of a quantum mechanical chaotic system can be determined. We show explicitly how this could be carried out for the deltakicked harmonic oscillator, a system capable of displaying chaos classically. We propose an experimental configuration, within an ion trap, by which a quantum mechanical delta-kicked harmonic ...

Let X 1 , X 2 ,. .. denote i.i.d. random bits, each taking the values 1 and 0 with respective probabilities p and 1 − p. A well-known theorem of Erd˝ os and Rényi (1970) describes the length of the longest contiguous stretch, or " run, " of ones in X 1 ,. .. , X n for large values of n. Benjamini, Häggström, Peres, and Steif (2003, Theorem 1.4) demonstrated the existence of unusual times, provi...

We use a theorem by Ding, Lubetzky and Peres describing the structure of the giant component of random graphs in the strictly supercritical regime, in order to determine the typical size of MAXCUT of G ∼ G (

In this paper we discuss two different models of dependent percolation on the graph Z2. We show that they both exhibit phase transitions. This proves a conjecture of Jonasson, Mossel and Peres [6], who proved a similar result on Z3.