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A metastable dominated convergence theorem

Avigad, Dean, Rute
2012 Journal of Logic and Analysis  
Tao has proved a quantitative version of this theorem: given a uniform bound on the rates of metastable convergence in the hypothesis, there is a bound on the rate of metastable convergence in the conclusion  ...  The dominated convergence theorem implies that if (f_n) is a sequence of functions on a probability space taking values in the interval [0,1], and (f_n) converges pointwise a.e., then the sequence of integrals  ...  A metastable dominated convergence theorem 13 Call the set just indicated A.  ... 
doi:10.4115/jla.2012.4.3 fatcat:2slz7sdytrexdnkmddd3o62lde

Model theory and metric convergence I: Metastability and dominated convergence [article]

Eduardo Dueñez, José N. Iovino
2019 arXiv   pre-print
As an instance of this phenomenon, we formulate an abstract version of Tao's Metastable Dominated Convergence Theorem as a statement about axiomatizable classes of metric structures, and show that it is  ...  Philosophically, this principle amounts to the following meta-theorem: "If a classical statement about convergence in metric structures is refined to a statement about metastable convergence with some  ...  Corollary (Metastable Dominated Convergence Theorem). Let (D, ≤, j 0 ) be a directed set with D countable.  ... 
arXiv:1611.09971v6 fatcat:dej5ap6u6vcujaxawo4stjjl6y

Pseudocompactness and the Uniform Metastability Principle in Model Theory [article]

Clovis Hamel, Franklin D. Tall
2020 arXiv   pre-print
We prove that uniform metastability is equivalent to all closed subspaces being pseudocompact and use this to provide a topological proof of the metatheorem introduced by Caicedo, Duenez and Iovino on  ...  uniform metastability and countable compactness for logics.  ...  It is a natural to ask when results regarding pointwise convergence of functions can be improved to uniform metastability in a way similar to that of Tao's metastable dominated convergence theorem [Tao08  ... 
arXiv:2004.06206v1 fatcat:x544uxv3ozhplk2v2oq66xzzje

Metastable convergence and logical compactness [article]

Xavier Caicedo, Eduardo Duenez, Jose Iovino
2019 arXiv   pre-print
The concept of metastable convergence was identified by Tao;it allows converting theorems about convergence into stronger theorems about uniform convergence.  ...  The Uniform Metastability Principle (UMP) states that if T is a theorem about convergence, then the fact that T is valid implies automatically that its (stronger) uniform version is valid, provided that  ...  Thus, for instance, Tao's uniformly metastable dominated convergence theorem follows from the classical dominated convergence theorem as a particular application of this metatheorem.  ... 
arXiv:1907.02398v2 fatcat:xq347ss23vex5iyw6l32lwk54m

On the Emergent Behavior of the 2-Choices Dynamics

Emilio Cruciani, Emanuele Natale, André Nusser, Giacomo Scornavacca
2018 Italian Conference on Theoretical Computer Science  
This short communication presents two recent results [13, 14] that study the the emergent behavior of the 2-Choices dynamics, a simple, local, and non-linear stochastic process on networks.  ...  We show that the 2-Choices dynamics can be exploited as an efficient distributed algorithm for graph clustering and as a model to explain phenomena in the contexts of sociology, biology, and neuroscience  ...  ; 3 if the dominance is less than c , then a metastable phase takes place where most of the nodes retain their initial opinion for n ω(1) rounds, with high probability.  ... 
dblp:conf/ictcs/CrucianiNNS18 fatcat:m7omv3ujtbb6vo4xg72bnuhwya

Self-Switching Markov Chains: emerging dominance phenomena [article]

S. Gallo, G. Iacobelli, G. Ost, D. Y. Takahashi
2021 arXiv   pre-print
This article proposes a Markov chain model, called Self-Switching Markov Chain (SSMC), in which the emergence of dominant dynamics can be rigorously addressed.  ...  Furthermore, we show that the switching between dynamics exhibits metastability-like property.  ...  GI and GO thank PIPGE-UFSCar (CAPES) for supporting a short visit to UFSCar where substantial part of this paper were done.  ... 
arXiv:2010.13279v2 fatcat:xlswds6ftzhoxjvd3ly6hl4lpq

Metastates in disordered mean-field models: Random field and hopfield models

Christof Külske
1997 Journal of statistical physics  
Moreover, it does not converge for a.e. r/, but rather in distribution, for whose limits we give explicit expressions. We also discuss another notion of metastates, due to Aizenman and Wehr.  ...  In both examples in the phase transition regime the empirical metastate is dispersed for large N.  ...  We see explicitly that, in both cases, the empirical metastate Ks(q) does not converge (see Theorems 1', 2') for a fixed realization.  ... 
doi:10.1007/bf02732434 fatcat:hcqoijrvvzge3ouef4hjgbvl5e

Metastable Equilibria

Srihari Govindan, Robert B. Wilson
2006 Social Science Research Network  
axiom: a metastable set of a global game projects to a metastable set of a local game.  ...  But the converse is slightly weaker than Mertens' decomposition property: a metastable set of a local game contains a metastable set that is the projection of a metastable set of a global game.  ...  A subset of a metastable set survives iterative elimination of weakly dominated strategies and strategies that are inferior replies at every equilibrium in the set.  ... 
doi:10.2139/ssrn.900612 fatcat:3k5knps73jgoti5opxhqj6kmvu

Short-Range Spin Glasses and Random Overlap Structures

Louis-Pierre Arguin, Michael Damron
2011 Journal of statistical physics  
On the way, a theorem of Newman and Stein about the pure state decomposition of the EA model is recovered and extended.  ...  In this spirit, it is shown, using translation invariance, that the ROSt of the EA model possesses a local stability that is stronger than stochastic stability, a property known to hold at almost all temperatures  ...  Proposition 5. 1 . 1 Let (N k ) be a subsequence for which (G N k ,β,J ) k converges to the metastate κ J in the sense of Theorem 2.1.  ... 
doi:10.1007/s10955-011-0177-z fatcat:ktd4noboifcoxo7cyhkukn5gt4

On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem [article]

Rob Arthan, Paulo Oliva
2021 arXiv   pre-print
Nonetheless, we obtain a quantitative formulation of the Kochen-Stone Theorem using Tao's notion of metastability.  ...  In this paper we present a quantitative analysis of the first and second Borel-Cantelli Lemmas and of two of their generalisations: the Erdős-Rényi Theorem, and the Kochen-Stone Theorem.  ...  Acknowledgements We would like to thank Iosif Pinelis for the proof of Theorem 4.7, and the anonymous referees for their constructive feedback, corrections and suggestions.  ... 
arXiv:2012.09942v3 fatcat:ze67mh7agjgalaqq6g4jifrkse

Zero-Temperature Fluctuations in Short-Range Spin Glasses

L.-P. Arguin, C. M. Newman, D. L. Stein, J. Wehr
2016 Journal of statistical physics  
An essential aspect of our result is the use of the excitation metastate.  ...  We consider the energy difference restricted to a finite volume for certain pairs of incongruent ground states (if they exist) in the d-dimensional Edwards-Anderson (EA) Ising spin glass at zero temperature  ...  such that the distributions converge weakly.  ... 
doi:10.1007/s10955-016-1516-x fatcat:r7oncnyc45bbrpdd53eyret5hq

Extracting macroscopic stochastic dynamics: Model problems

Wilhelm Huisinga, Christof Schütte, Andrew M. Stuart
2002 Communications on Pure and Applied Mathematics  
equation exhibiting a finite-state-space Markov chain structure.  ...  The model problems are ordinary differential equations constructed to have the property that, when projected onto a low-dimensional subspace, the dynamics is approximately that of a stochastic differential  ...  We are grateful to A. J. Majda for encouraging the collaboration that led to this work. We are grateful to A. J. Majda, E. Vanden Eijnden, and P. F.  ... 
doi:10.1002/cpa.10057 fatcat:hfkkvktv4bhujpgmuieco7dj3q

On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem

Rob Arthan, Paulo Oliva
2021 Journal of Logic and Analysis  
Nonetheless, we obtain a quantitative formulation of the Kochen-Stone Theorem using Tao's notion of metastability.  ...  In this paper we present a quantitative analysis of the first and second Borel-Cantelli Lemmas and of two of their generalisations: the Erdős–Rényi Theorem, and the Kochen-Stone Theorem.  ...  Acknowledgements We would like to thank Iosif Pinelis for the proof of Theorem 4.7, and the anonymous referees for their constructive feedback, corrections and suggestions.  ... 
doi:10.4115/jla.2021.13.6 fatcat:egivhblrmfarddzgck7ljyosm4

Limit theorems for sums of dependent random variables occurring in statistical mechanics

Richard S. Ellis, Charles M. Newman, Jay S. Rosen
1980 Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete  
The relation of these results to multiple phases, metastable states, and other physical phenomena is explained.  ...  , we obtain (3.5) by the dominated convergence theorem.Proof of Transfer Principle.  ...  Dominated convergence does the rest. To ease the notation, we set ~:= 1/2k(m) and fl= 1, General fl can be handled analogously.  ... 
doi:10.1007/bf00536186 fatcat:45kw4thk5vdtvotwpylm25faqi

Eigenvalue bounds on restrictions of reversible nearly uncoupled Markov chains

Eike Meerbach, Christof Schütte, Alexander Fischer
2005 Linear Algebra and its Applications  
We state upper bounds on the 2nd eigenvalue for restriction and stochastic complementation chains of reversible Markov chains, as well as a relation between them.  ...  We illustrate the obtained bounds analytically for bunkbed graphs, and furthermore apply them to restricted Markov chains that arise when analyzing conformation dynamics of a small biomolecule.  ...  For this non-trivial algorithmic task, spectral approaches that first identify a spectral gap (which is related to the number of metastable sets) and then exploit the structure of dominant eigenvectors  ... 
doi:10.1016/j.laa.2004.10.018 fatcat:dl3ugrp3pffczox2f2t7wllcye
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