7 research outputs found

    On concavity of TAP free energy in the SK model

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    We analyse the Hessian of the Thouless-Anderson-Palmer (TAP) free energy for the Sherrington-Kirkpatrick model, below the de Almeida-Thouless line, evaluated in Bolthausen's approximate solutions of the TAP equations. We show that while its empirical spectrum weakly converges to a measure with negative support, positive outlier eigenvalues occur for some (β,h)(\beta,h) below the AT line. In this sense, TAP free energy may lose concavity in the order parameter of the theory, i.e. the random spin-magnetisations, even below the AT line. Possible interpretations of these findings within Plefka's expansion of the Gibbs potential are not definitive and include the following: i) either higher order terms shall not be neglected even if Plefka's first convergence criterion (yielding, in infinite volume, the AT line) is satisfied, ii) Plefka's first convergence criterion (hence the AT line) is necessary yet hardly sufficient, or iii) Bolthausen's magnetizations do not approximate the TAP solutions sufficiently well up to the AT line.Comment: 29 pages, 1 figur

    AMP algorithms and Stein's method: Understanding TAP equations with a new method

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    We propose a new iterative construction of solutions of the classical TAP equations for the Sherrington-Kirkpatrick model, i.e. with finite-size Onsager correction. The algorithm can be started in an arbitrary point, and converges up to the AT line. The analysis relies on a novel treatment of mean field algorithms through Stein's method. As such, the approach also yields weak convergence of the effective fields at all temperatures towards Gaussians, and can be applied, upon proper alterations, to all models where TAP-like equations and a Stein-operator are available.Comment: 38 page

    Evolving genealogies for branching populations under selection and competition

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    For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by Donnelly and Kurtz (1999), namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach, we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of "potential" selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the "active" selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz's Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation

    Tree-valued Fleming-Viot processes : a generalization, pathwise constructions, and invariance principles

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    We study exchangeable coalescent trees and the evolving genealogical trees in models for neutral haploid populations. We show that every exchangeable infinite coalescent tree can be obtained as the genealogical tree of iid samples from a random marked metric measure space when the marks are added to the metric distances. We apply this representation to generalize the tree-valued Fleming-Viot process to include the case with dust in which the genealogical trees have isolated leaves. Using the Donnelly-Kurtz lookdown approach, we describe all individuals ever alive in the population model by a random complete and separable metric space, the lookdown space, which we endow with a family of sampling measures. This yields a pathwise construction of tree-valued Fleming-Viot processes. In the case of coming down from infinity, we also read off a process whose state space is endowed with the Gromov-Hausdorff-Prohorov topology. This process has additional jumps at the extinction times of parts of the population. In the case with only binary reproduction events, we construct the lookdown space also from the Aldous continuum random tree by removing the root and the highest leaf, and by deforming the metric in a way that corresponds to the time change that relates the Fleming-Viot process with a Dawson-Watanabe process. The sampling measures on the lookdown space are then image measures of the normalized local time measures. We also show invariance principles for Markov chains that describe the evolving genealogy in Cannings models. For such Markov chains with values in the space of distance matrix distributions, we show convergence to tree-valued Fleming-Viot processes under the conditions of Möhle and Sagitov for the convergence of the genealogy at a fixed time to a coalescent with simultaneous multiple mergers. For the convergence of Markov chains with values in the space of marked metric measure spaces, an additional assumption is needed in the case with dust

    Evolving genealogies for branching populations under selection and competition

    No full text
    For a continuous state branching process with two types of individuals which are subject to selection and density dependent competition, we characterize the joint evolution of population size, type configurations and genealogies as the unique strong solution of a system of SDE's. Our construction is achieved in the lookdown framework and provides a synthesis as well as a generalization of cases considered separately in two seminal papers by Donnelly and Kurtz (1999), namely fluctuating population sizes under neutrality, and selection with constant population size. As a conceptual core in our approach, we introduce the selective lookdown space which is obtained from its neutral counterpart through a state-dependent thinning of "potential" selection/competition events whose rates interact with the evolution of the type densities. The updates of the genealogical distance matrix at the "active" selection/competition events are obtained through an appropriate sampling from the selective lookdown space. The solution of the above mentioned system of SDE's is then mapped into the joint evolution of population size and symmetrized type configurations and genealogies, i.e. marked distance matrix distributions. By means of Kurtz's Markov mapping theorem, we characterize the latter process as the unique solution of a martingale problem. For the sake of transparency we restrict the main part of our presentation to a prototypical example with two types, which contains the essential features. In the final section we outline an extension to processes with multiple types including mutation
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