Levin, David A., Luczak, Malwina J., Peres, Yuval
We study the Glauber dynamics for the Ising model on the complete graph, also known as the Curie-Weiss Model. For beta < 1, we prove that the dynamics exhibits a cut-off: the distance to stationarity...
A Phase Transition in Random Coin Tossing (2007)
David A. Levin, Robin Pemantle, Yuval Peres, Are Singular, While If
this paper is organized as follows. In Section 2, we provide definitions and introduce notation. In Section 3, we prove a useful general zero-one law, to show that singularity and absolute continuity...
Cluster Growth in Hypergraph Processes (2007)
A Poisson random hypergraph, introduced in [DN01], contains for each k a Poisson number of k-hyperedges, distributed at random among all sets containing k vertices. In this paper, we introduce and...
A Dynamical Law of Large Numbers (2007)
Khoshnevisan, Davar, Levin, David A., Mendez-Hernandez, Pedro J.
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 Erdos and Renyi (1970) describes the length of the longest...
A Coupling, and the Darling-Erdos Conjectures (2005)
Khoshnevisan, Davar, Levin, David A.
We derive a new coupling of the running maximum of an Ornstein-Uhlenbeck process and the running maximum of an explicit i.i.d. sequence. We use this coupling to verify a conjecture of Darling and...
An Extreme-Value Analysis of the LIL for Brownian Motion (2005)
Khoshnevisan, Davar; University Of Utah, USa; Davar@math.utah.edu, Levin, David A.; University Of Oregon, USA; Dlevin@math.uoregon.edu, Shi, Zhan; Université Paris VI, France; Zhan@proba.jussieu.fr
We use excursion theory and the ergodic theorem to present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. A simplified version of our method also...
On dynamical Gaussian random walks (2005)
Khoshnevisan, Davar, Levin, David A., Méndez-Hernández, Pedro J.
Motivated by the recent work of Benjamini, Häggström, Peres and Steif [Ann. Probab. 34 (2003) 1–34] on dynamical random walks, we do the following: (i) Prove that, after a suitable normalization,...
An Extreme-Value Analysis of the LIL for Brownian Motion (2004)
Khoshnevisan, Davar, Levin, David A., Shi, Zhan
We present an extreme-value analysis of the classical law of the iterated logarithm (LIL) for Brownian motion. Our result can be viewed as a new improvement to the LIL.
Capacities in Wiener Space, Quasi-Sure Lower Functions, and Kolmogorov's Epsilon-Entropy (2004)
Khoshnevisan, Davar, Levin, David A., Mendez-Hernandez, Pedro J.
We propose a set-indexed family of capacities $\{\cap_G \}_{G \subseteq \R_+}$ on the classical Wiener space $C(\R_+)$. This family interpolates between the Wiener measure ($\cap_{\{0\}}$) on...
Exceptional Times and Invariance for Dynamical Random Walks (2004)
Khoshnevisan, Davar, Levin, David A., Mendez-Hernandez, Pedro J.
Consider a sequence {X(i,0) : i = 1, ..., n} of i.i.d. random variables. Associate to each X(i,0) an independent mean-one Poisson clock. Every time a clock rings replace that X-variable by an...
A phase transition in random coin tossing (2004)
Levin, David A., Pemantle, Robin, Peres, Yuval
Suppose that a coin with bias theta is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let mu_\theta be the distribution of the observed sequence of coin...
Abstract. Let V denote a set of N vertices. To construct a hypergraph process, create a new hyperedge at each event time of a Poisson process; the cardinality K of this hyperedge is random, with...
Identifying several biased coins encountered by a hidden random walk (2003)
Suppose that attached to each site z in Z is a coin with bias theta(z), and only finitely many of these coins have non-zero bias. Allow a simple random walker to generate observations by tossing, at...
A Phase Transition in Random coin Tossing (2001)
Levin, David A., Pemantle, Robin, Peres, Yuval
Suppose that a coin with bias $\theta$ is tossed at renewal times of a renewal process, and a fair coin is tossed at all other times. Let $\mu_\theta$ be the distribution ofthe observed sequence of...