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- Rick Durrett, Vlada Limic
- 2002

Motivated by the work of Tilman (Ecology 75 (1994) 2) and May and Nowak (J. Theoret. Biol. 170 (1994) 95) we consider a process in which points are inserted randomly into the unit interval and a new point kills each point to its left independently and with probability a. Intuitively this dynamic will create a negative dependence between the number of points… (More)

Motivated by the problem of the evolution of DNA sequences, Kauffman and Levin introduced a model in which fitnesses were assigned to strings of 0’s and 1’s of length N based on the values observed in a sliding window of length K + 1. When K ≥ 1, the landscape is quite complicated with many local maxima. Its properties have been extensively investigated by… (More)

- Vlada Limic
- 2009

The Ξ-coalescent processes were initially studied by Möhle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics as the complete class of scaling limits for genealogies of Cannings’ models. The Ξ-coalescents generalize Λ-coalescents, where now simultaneous multiple collisions… (More)

- Vlada Limic, Pierre Tarres
- 2007

The goal is to show that an edge-reinforced random walk on a graph of bounded degree, with reinforcement weight function W taken from a general class of reciprocally summable reinforcement weight functions, traverses a random attracting edge at all large times. The statement of the main theorem is very close to settling a conjecture of Sellke [Technical… (More)

- Vlada Limic
- 1999

This paper studies heavy traffic behavior of a G/G/1 last-in-first-out (LIFO) preemptive resume queue, by extending the techniques developed in Limic (1999). The queue length process exhibits a perhaps unexpected heavy traffic behavior. The diffusion limit depends on the type of arrivals (and services) in a fairly intricate way, related to the Wiener-Hopf… (More)

The purpose of this note is to provide proofs for some facts about the NK model of evolution due to Kauffman and Levin. In the case of normally distributed fitness summands, some of these facts have been previously conjectured and heuristics given. In particular, we provide rigorous asymptotic estimates for the number of local fitness maxima in the case… (More)

- Rick Durrett, Vlada Limic
- 2000

Single nucleotide polymorphisms (SNPs) are useful markers for locating genes since they occur throughout the human genome and thousands can be scored at once using DNA microarrays. Here, we use branching processes and coalescent theory to show that if one uses Kruglyak’s (Nature Gen. 12 (1999) 139–144) model of the growth of the human population and one… (More)

We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a Λcoalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and… (More)

By a theorem of Volkov (2001) we know that on most graphs, with positive probability, the linearly vertex-reinforced random walk (VRRW) stays within a finite “trapping” subgraph at all large times. The question of whether this tail behavior occurs with probability one is open in general. R. Pemantle (1988) in his thesis proved, via a dynamical system… (More)

- Vlada Limic
- 2010

This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as Ξ-coalescent) started from a large finite configuration, and undergoing neutral mutations. Asymptotic expressions for the number of active lineages were obtained by the author in a previous work. Analogous results for the number of active… (More)