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A metastable dominated convergence theorem

2012
*
Journal of Logic and Analysis
*

Tao has proved

doi:10.4115/jla.2012.4.3
fatcat:2slz7sdytrexdnkmddd3o62lde
*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*. ...##
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Model theory and metric convergence I: Metastability and dominated convergence
[article]

2019
*
arXiv
*
pre-print

As an instance of this phenomenon, we formulate an abstract version of Tao's

arXiv:1611.09971v6
fatcat:dej5ap6u6vcujaxawo4stjjl6y
*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. ...##
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Pseudocompactness and the Uniform Metastability Principle in Model Theory
[article]

2020
*
arXiv
*
pre-print

We prove that uniform

arXiv:2004.06206v1
fatcat:x544uxv3ozhplk2v2oq66xzzje
*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 ...##
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Metastable convergence and logical compactness
[article]

2019
*
arXiv
*
pre-print

The concept of

arXiv:1907.02398v2
fatcat:xq347ss23vex5iyw6l32lwk54m
*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. ...##
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On the Emergent Behavior of the 2-Choices Dynamics

2018
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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,

dblp:conf/ictcs/CrucianiNNS18
fatcat:m7omv3ujtbb6vo4xg72bnuhwya
*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. ...##
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Self-Switching Markov Chains: emerging dominance phenomena
[article]

2021
*
arXiv
*
pre-print

This article proposes

arXiv:2010.13279v2
fatcat:xlswds6ftzhoxjvd3ly6hl4lpq
*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. ...##
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Metastates in disordered mean-field models: Random field and hopfield models

1997
*
Journal of statistical physics
*

Moreover, it does not

doi:10.1007/bf02732434
fatcat:hcqoijrvvzge3ouef4hjgbvl5e
*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. ...##
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Metastable Equilibria

2006
*
Social Science Research Network
*

axiom:

doi:10.2139/ssrn.900612
fatcat:3k5knps73jgoti5opxhqj6kmvu
*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. ...##
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Short-Range Spin Glasses and Random Overlap Structures

2011
*
Journal of statistical physics
*

On the way,

doi:10.1007/s10955-011-0177-z
fatcat:ktd4noboifcoxo7cyhkukn5gt4
*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. ...##
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On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem
[article]

2021
*
arXiv
*
pre-print

Nonetheless, we obtain

arXiv:2012.09942v3
fatcat:ze67mh7agjgalaqq6g4jifrkse
*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. ...##
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Zero-Temperature Fluctuations in Short-Range Spin Glasses

2016
*
Journal of statistical physics
*

An essential aspect of our result is the use of the excitation

doi:10.1007/s10955-016-1516-x
fatcat:r7oncnyc45bbrpdd53eyret5hq
*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. ...##
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Extracting macroscopic stochastic dynamics: Model problems

2002
*
Communications on Pure and Applied Mathematics
*

equation exhibiting

doi:10.1002/cpa.10057
fatcat:hfkkvktv4bhujpgmuieco7dj3q
*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. ...##
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On the Borel-Cantelli Lemmas, the Erdős-Rényi Theorem, and the Kochen-Stone Theorem

2021
*
Journal of Logic and Analysis
*

Nonetheless, we obtain

doi:10.4115/jla.2021.13.6
fatcat:egivhblrmfarddzgck7ljyosm4
*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. ...##
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Limit theorems for sums of dependent random variables occurring in statistical mechanics

1980
*
Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
*

The relation of these results to multiple phases,

doi:10.1007/bf00536186
fatcat:45kw4thk5vdtvotwpylm25faqi
*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. ...##
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Eigenvalue bounds on restrictions of reversible nearly uncoupled Markov chains

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

doi:10.1016/j.laa.2004.10.018
fatcat:dl3ugrp3pffczox2f2t7wllcye
*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 ...
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