arXiv subjects

Vlada Limic

Publications and source records attributed to Vlada Limic.

27 records · Page 2Linked to original sources

arXiv2011

Genealogies of regular exchangeable coalescents with applications to sampling

This article considers a model of genealogy corresponding to a regular exchangeable coalescent (also known as Xi-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 mutation-free lineages and the combined lineage lengths are derived using the same martingale-based technique. They are given in terms of convergence in probability, while extensions to convergence in moments and convergence almost surely are discussed. The above mentioned results have direct consequences on the sampling theory in the Xi-coalescent setting. In particular, the regular Xi-coalescents that come down from infinity (i.e., with locally finite genealogies), have an asymptotically equal number of families under the corresponding infinite alleles and infinite sites models. In special cases, quantitative asymptotic formulae for the number of families that contain a fixed number of individuals can be given.

math.PR↗

arXiv2010

VRRW on complete-like graphs: Almost sure behavior

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. In his thesis, Pemantle (1988) proved, via a dynamical system approach, that for a VRRW on any complete graph the asymptotic frequency of visits is uniform over vertices. These techniques do not easily extend even to the setting of complete-like graphs, that is, complete graphs ornamented with finitely many leaves at each vertex. In this work we combine martingale and large deviation techniques to prove that almost surely the VRRW on any such graph spends positive (and equal) proportions of time on each of its nonleaf vertices. This behavior was previously shown to occur only up to event of positive probability (cf. Volkov (2001)). We believe that our approach can be used as a building block in studying related questions on more general graphs. The same set of techniques is used to obtain explicit bounds on the speed of convergence of the empirical occupation measure.

math.PR↗

arXiv2010

Global divergence of spatial coalescents

We study several fundamental properties of a class of stochastic processes called spatial Lambda-coalescents. In these models, a number of particles perform independent random walks on some underlying graph G. In addition, particles on the same vertex merge randomly according to a given coalescing mechanism. A remarkable property of mean-field coalescent processes is that they may come down from infinity, meaning that, starting with an infinite number of particles, only a finite number remains after any positive amount of time, almost surely. We show here however that, in the spatial setting, on any infinite and bounded-degree graph, the total number of particles will always remain infinite at all times, almost surely. Moreover, if G=Z^d, and the coalescing mechanism is Kingman's coalescent, then starting with N particles at the origin, the total number of particles remaining is of order (log* N)^d at any fixed positive time (where log* is the inverse tower function). At sufficiently large times the total number of particles is of order (log* N)^{d-2}, when d>2. We provide parallel results in the recurrent case d=2. The spatial Beta-coalescents behave similarly, where log log N is replacing log* N.

math.PR↗

arXiv2009

Attraction time for strongly reinforced walks

We consider a class of strongly edge-reinforced random walks, where the corresponding reinforcement weight function is nondecreasing. It is known, from Limic and Tarrès [Ann. Probab. (2007), to appear], that the attracting edge emerges with probability 1 whenever the underlying graph is locally bounded. We study the asymptotic behavior of the tail distribution of the (random) time of attraction. In particular, we obtain exact (up to a multiplicative constant) asymptotics if the underlying graph has two edges. Next, we show some extensions in the setting of finite graphs, and infinite graphs with bounded degree. As a corollary, we obtain the fact that if the reinforcement weight has the form $w(k)=k^ρ$, $ρ>1$, then (universally over finite graphs) the expected time to attraction is infinite if and only if $ρ\leq1+\frac{1+\sqrt{5}}{2}$.

math.PR↗

arXiv2009

Coalescent processes arising in a study of diffusive clustering

This paper studies the spatial coalescent on $\Z^2$. In our setting, the partition elements are located at the sites of $\Z^2$ and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times. In addition, the partition elements perform independent random walks. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. These two situations are relevant if the coalescent is used to study the scaling limits for genealogies in Moran models respectively interacting Fisher-Wright diffusions (or Fleming-Viot processes), which is the key application of the present work. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $[-t^{α/2}, t^{α/2}]^2 \cap \Z^2$ and observed at time $t^β$ with $1 \geq β\geq α\ge 0$. We study both asymptotics, as $t\to\infty$, for a fixed value of $α$ as the parameter $β\in[α,1]$ varies, and for a fixed $β$, as the parameter $α\in [0,β]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. (... for more see the actual preprint)

math.PR↗

arXiv2007

Attracting edge and strongly edge reinforced walks

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 Report 94-26 (1994) Purdue Univ.]. An important corollary of this main result says that if $W$ is reciprocally summable and nondecreasing, the attracting edge exists on any graph of bounded degree, with probability 1. Another corollary is the main theorem of Limic [Ann. Probab. 31 (2003) 1615--1654], where the class of weights was restricted to reciprocally summable powers. The proof uses martingale and other techniques developed by the authors in separate studies of edge- and vertex-reinforced walks [Ann. Probab. 31 (2003) 1615--1654, Ann. Probab. 32 (2004) 2650--2701] and of nonconvergence properties of stochastic algorithms toward unstable equilibrium points of the associated deterministic dynamics [C. R. Acad. Sci. Sér. I Math. 330 (2000) 125--130].

math.PR↗

arXiv2005

The spatial $Λ$-coalescent

This paper extends the notion of the $\la$-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial $Λ$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $Λ$-coalescents that come down from infinity, in an analogous way to Schweinsberg (2000). Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $Λ$-coalescents on large tori in $d\ge 3$ dimensions. Our results generalize and strengthen those of Greven et al. (2005), who studied the spatial Kingman coalescent in this context.

math.PR↗

arXiv2005

More rigorous results on the Kauffman-Levin model of evolution

The purpose of this note is to provide proofs for some facts about the NK model of evolution proposed by 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 when K is unbounded. We also examine the role of the individual fitness distribution and find the model to be quite robust with respect to this.

math.PR↗

arXiv2004

The Beurling estimate for a class of random walks

An estimate of Beurling states that if K is a curve from 0 to the unit circle in the complex plane, then the probability that a Brownian motion starting at -eps reaches the unit circle without hitting the curve is bounded above by c eps^{1/2}. This estimate is very useful in analysis of boundary behavior of conformal maps, especially for connected but rough boundaries. The corresponding estimate for simple random walk was first proved by Kesten. In this note we extend this estimate to random walks with zero mean, and finite (3+delta) moment.

math.PR↗