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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.
Universality of the REM for Dynamics of MeanField Spin Glasses
, 2008
"... Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the Ndimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β> 0, there exists a constant γβ,p> 0, such that for all exponential time scales, exp(γ N), with γ &l ..."
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Cited by 17 (2 self)
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Sherrington– Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the Ndimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β> 0, there exists a constant γβ,p> 0, such that for all exponential time scales, exp(γ N), with γ < γβ,p, the properly rescaled clock process (timechange process) converges to an αstable subordinator where α = γ /β2 < 1. Moreover, the dynamics exhibits aging at these time scales with a timetime correlation function converging to the arcsine law of this αstable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of pspin models ages in the same way as the REM, and by extension Bouchaud’s REMlike trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.
The GhirlandaGuerra identities
 J. Stat. Phys
, 2007
"... If the variance of a Gaussian spinglass Hamiltonian grows like the volume the model fulfills the GhirlandaGuerra identities in terms of the normalized Hamiltonian covariance. 1 1 ..."
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Cited by 14 (6 self)
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If the variance of a Gaussian spinglass Hamiltonian grows like the volume the model fulfills the GhirlandaGuerra identities in terms of the normalized Hamiltonian covariance. 1 1
Replica symmetry breaking in mean field spin glasses trough HamiltonJacobi technique
, 2010
"... During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field SherringtonKirkpatrick spin glass model have been firmly established. In particular, ..."
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Cited by 11 (10 self)
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During the last years, through the combined effort of the insight, coming from physical intuition and computer simulation, and the exploitation of rigorous mathematical methods, the main features of the mean field SherringtonKirkpatrick spin glass model have been firmly established. In particular, it has been possible to prove the existence and uniqueness of the infinite volume limit for the free energy, and its Parisi expression, in terms of a variational principle, involving a functional order parameter. Even the expected property of ultrametricity, for the infinite volume states, seems to be near to a complete proof. The main structural feature of this model, and related models, is the deep phenomenon of spontaneous replica symmetry breaking (RSB), discovered by Parisi many years ago. By expanding on our previous work, the aim of this paper is to investigate a general frame, where replica symmetry breaking is embedded in a kind of mechanical scheme of the HamiltonJacobi type. Here, the analog of the “time ” variable is a parameter characterizing the strength of the interaction, while the “space” variables rule out quantitatively the broken replica symmetry pattern. Starting from the simple cases, where annealing is assumed, or replica symmetry, we build up a progression of dynamical systems, with an increasing number of space variables, which allow to weaken the effect of the potential in the HamiltonJacobi equation, as the level of symmetry braking is increased. This new machinery allows to work out mechanically the general Kstep RSB solutions, in a different interpretation with respect to the replica trick, and lightens easily their properties as existence or uniqueness.
Sharp asymptotics for the partition function of some continuoustime directed polymers
, 2007
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Thermodynamical Limit for Correlated Gaussian Random Energy Models
"... Let fE (N)g 2N be a family of j N j = 2 random variables de ned by the covariance matrix CN of elements c N (; ) := Av (E (N)E (N)), and HN () = NE (N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N 1 + N 2 , and a ..."
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Cited by 9 (3 self)
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Let fE (N)g 2N be a family of j N j = 2 random variables de ned by the covariance matrix CN of elements c N (; ) := Av (E (N)E (N)), and HN () = NE (N) the corresponding random Hamiltonian. Then the quenched thermodynamical limit exists if, for every decomposition N = N 1 + N 2 , and all pairs (; ) 2 N N : c N (; ) c N1 ( 1 (); 1 ()) + c N2 ( 2 (); 2 ()) where k (); k = 1; 2 are the projections of 2 N into N k . The condition is explicitly verified for the SherringtonKirckpatrick, the even pspin, the Derrida REM and the DerridaGardner GREM models.
The AizenmanSimsStarr and Guerra’s schemes for the SK model with multidimensional spins
, 2008
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Evolution in random fitness landscapes: the infinite sites model
 J. Stat. Mech. Theory Exp
, 2008
"... Abstract. We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution g(w). This is the finite population versio ..."
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Abstract. We consider the evolution of an asexually reproducing population in an uncorrelated random fitness landscape in the limit of infinite genome size, which implies that each mutation generates a new fitness value drawn from a probability distribution g(w). This is the finite population version of Kingman’s house of cards model [J.F.C. Kingman, J. Appl. Probab. 15, 1 (1978)]. In contrast to Kingman’s work, the focus here is on unbounded distributions g(w) which lead to an indefinite growth of the population fitness. The model is solved analytically in the limit of infinite population size N → ∞ and simulated numerically for finite N. When the genomewide mutation probability U is small, the long time behavior of the model reduces to a point process of fixation events, which is referred to as a diluted record process (DRP). The DRP is similar to the standard record process except that a new record candidate (a number that exceeds all previous entries in the sequence) is accepted only with a certain probability that depends on the values of the current record and the candidate. We develop a systematic analytic approximation scheme for the DRP. At finite U the fitness frequency distribution of the population decomposes into a stationary part due to mutations and a traveling wave component due to selection, which is shown to imply a reduction of the mean fitness by a factor of 1 − U compared to the U → 0 limit. Evolution in random fitness landscapes: the infinite sites model 2 1.
On the Ising model with random boundary condition
 J. Stat. Phys
"... ABSTRACT. The infinitevolume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic sizedependence at low temperatures and we prove that the ‘+ ’ and ‘ ’ phases are the only almost sure limit Gibbs measures, assuming that the l ..."
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Cited by 6 (2 self)
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ABSTRACT. The infinitevolume limit behavior of the 2d Ising model under possibly strong random boundary conditions is studied. The model exhibits chaotic sizedependence at low temperatures and we prove that the ‘+ ’ and ‘ ’ phases are the only almost sure limit Gibbs measures, assuming that the limit is taken along a sparse enough sequence of squares. In particular, we give a multiscale perturbative argument to show that in a sufficiently large volume typical spin configuration under a typical boundary condition contains no interfaces. 1.