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Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 48 (3 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
On the irrelevant disorder regime of pinning models
, 2007
"... Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that, albeit still partial, is now rigorous. In this wor ..."
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Cited by 14 (4 self)
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Recent results have lead to substantial progress in understanding the role of disorder in the (de)localization transition of polymer pinning models. Notably, there is an understanding of the crucial issue of disorder relevance and irrelevance that, albeit still partial, is now rigorous. In this work we exploit interpolation and replica coupling methods to get sharper results on the irrelevant disorder regime of pinning models. In particular, we compute in this regime the first order term in the expansion of the free energy close to criticality, which coincides with the first order of the formal expansion obtained by field theory methods. We also show that the quenched and the quenched averaged correlation length exponents coincide, while in general they are expected to be different. Interpolation and replica coupling methods in this class of models naturally lead to studying the behavior of the intersection of certain renewal sequences and one of the main tools in this work is precisely renewal theory and the study of these intersection renewals.
Almost Sure Invariance Principle for ContinuousSpace Random Walk in Dynamic Random Environment
"... Abstract. We consider a random walk on R d in a polynomially mixing random environment that is refreshed at each time step. We use a martingale approach to give a necessary and sufficient condition for the almostsure functional central limit theorem to hold. 1. Introduction and ..."
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Abstract. We consider a random walk on R d in a polynomially mixing random environment that is refreshed at each time step. We use a martingale approach to give a necessary and sufficient condition for the almostsure functional central limit theorem to hold. 1. Introduction and
ON HOMOGENIZATION OF A DIFFUSION PERTURBED BY A PERIODIC REFLECTION INVARIANT VECTOR FIELD
, 2006
"... Abstract. In this paper the author studies the problem of the homogenization of a diffusion perturbed by a periodic reflection invariant vector field. The vector field is assumed to have fixed direction but varying amplitude. The existence of a homogenized limit is proven and formulas for the effect ..."
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Abstract. In this paper the author studies the problem of the homogenization of a diffusion perturbed by a periodic reflection invariant vector field. The vector field is assumed to have fixed direction but varying amplitude. The existence of a homogenized limit is proven and formulas for the effective diffusion constant are given. In dimension d = 1 the effective diffusion constant is always less than the constant for the pure diffusion. In d> 1 this property no longer holds in general. 1.