Gibbs measures over permutations of point processes with low density

Autores
Armendáriz, María Inés; Ferrari, Pablo Augusto; Frevenza Maestrone, Nicolas Federico
Año de publicación
2019
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
We study a model of spatial random permutations over a discrete set of points. Formally, a permutation σ is sampled proportionally to the weight exp{−α∑_xV(σ(x)−x)}, where α>0 is the temperature and V is a non-negative and continuous potential. The most relevant case for physics is when V(x)=‖x‖^2, since it is related to Bose-Einstein condensation through a representation introduced by Feynman in 1953. In the context of statistical mechanics, the weights (1) define a probability when the set of points is finite, but the construction associated to an infinite set is not trivial and may fail without appropriate hypotheses. The first problem is to establish conditions for the existence of such a measure at infinite volume when the set of points is infinite. Once existence is derived, we are interested in establishing its uniqueness and the cycle structure of a typical permutation. We here consider the large temperature regime when the set of points is a Poisson point process in ℤ^d with intensity ρ∈(0,1/2), and the potential verifies some regularity conditions. In particular, we prove that if α is large enough, for almost every realization of the point process, there exists a unique Gibbs measure that concentrates on finite cycle permutations. We then extend these results to the continuous setting, when the set of points is given by a Poisson point process in ℝ^d with low enough intensity.
Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Frevenza Maestrone, Nicolas Federico. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
Materia
GIBBS MEASURES
PERMUTATIONS
FINITE CYCLES
POISSON POINT PROCESS
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/120189
Acceder
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network_acronym_str CONICETDig
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network_name_str CONICET Digital (CONICET)
spelling Gibbs measures over permutations of point processes with low densityArmendáriz, María InésFerrari, Pablo AugustoFrevenza Maestrone, Nicolas FedericoGIBBS MEASURESPERMUTATIONSFINITE CYCLESPOISSON POINT PROCESShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We study a model of spatial random permutations over a discrete set of points. Formally, a permutation σ is sampled proportionally to the weight exp{−α∑_xV(σ(x)−x)}, where α>0 is the temperature and V is a non-negative and continuous potential. The most relevant case for physics is when V(x)=‖x‖^2, since it is related to Bose-Einstein condensation through a representation introduced by Feynman in 1953. In the context of statistical mechanics, the weights (1) define a probability when the set of points is finite, but the construction associated to an infinite set is not trivial and may fail without appropriate hypotheses. The first problem is to establish conditions for the existence of such a measure at infinite volume when the set of points is infinite. Once existence is derived, we are interested in establishing its uniqueness and the cycle structure of a typical permutation. We here consider the large temperature regime when the set of points is a Poisson point process in ℤ^d with intensity ρ∈(0,1/2), and the potential verifies some regularity conditions. In particular, we prove that if α is large enough, for almost every realization of the point process, there exists a unique Gibbs measure that concentrates on finite cycle permutations. We then extend these results to the continuous setting, when the set of points is given by a Poisson point process in ℝ^d with low enough intensity.Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Frevenza Maestrone, Nicolas Federico. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaCornell University2019-04info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/120189Armendáriz, María Inés; Ferrari, Pablo Augusto; Frevenza Maestrone, Nicolas Federico; Gibbs measures over permutations of point processes with low density; Cornell University; arxiv.org; 4-2019; 1-252331-8422CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1904.03952info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:47:29Zoai:ri.conicet.gov.ar:11336/120189instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 10:47:29.967CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Gibbs measures over permutations of point processes with low density
title Gibbs measures over permutations of point processes with low density
spellingShingle Gibbs measures over permutations of point processes with low density
Armendáriz, María Inés
GIBBS MEASURES
PERMUTATIONS
FINITE CYCLES
POISSON POINT PROCESS
title_short Gibbs measures over permutations of point processes with low density
title_full Gibbs measures over permutations of point processes with low density
title_fullStr Gibbs measures over permutations of point processes with low density
title_full_unstemmed Gibbs measures over permutations of point processes with low density
title_sort Gibbs measures over permutations of point processes with low density
dc.creator.none.fl_str_mv Armendáriz, María Inés
Ferrari, Pablo Augusto
Frevenza Maestrone, Nicolas Federico
author Armendáriz, María Inés
author_facet Armendáriz, María Inés
Ferrari, Pablo Augusto
Frevenza Maestrone, Nicolas Federico
author_role author
author2 Ferrari, Pablo Augusto
Frevenza Maestrone, Nicolas Federico
author2_role author
author
dc.subject.none.fl_str_mv GIBBS MEASURES
PERMUTATIONS
FINITE CYCLES
POISSON POINT PROCESS
topic GIBBS MEASURES
PERMUTATIONS
FINITE CYCLES
POISSON POINT PROCESS
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv We study a model of spatial random permutations over a discrete set of points. Formally, a permutation σ is sampled proportionally to the weight exp{−α∑_xV(σ(x)−x)}, where α>0 is the temperature and V is a non-negative and continuous potential. The most relevant case for physics is when V(x)=‖x‖^2, since it is related to Bose-Einstein condensation through a representation introduced by Feynman in 1953. In the context of statistical mechanics, the weights (1) define a probability when the set of points is finite, but the construction associated to an infinite set is not trivial and may fail without appropriate hypotheses. The first problem is to establish conditions for the existence of such a measure at infinite volume when the set of points is infinite. Once existence is derived, we are interested in establishing its uniqueness and the cycle structure of a typical permutation. We here consider the large temperature regime when the set of points is a Poisson point process in ℤ^d with intensity ρ∈(0,1/2), and the potential verifies some regularity conditions. In particular, we prove that if α is large enough, for almost every realization of the point process, there exists a unique Gibbs measure that concentrates on finite cycle permutations. We then extend these results to the continuous setting, when the set of points is given by a Poisson point process in ℝ^d with low enough intensity.
Fil: Armendáriz, María Inés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Ferrari, Pablo Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Frevenza Maestrone, Nicolas Federico. Universidad de Buenos Aires; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina
description We study a model of spatial random permutations over a discrete set of points. Formally, a permutation σ is sampled proportionally to the weight exp{−α∑_xV(σ(x)−x)}, where α>0 is the temperature and V is a non-negative and continuous potential. The most relevant case for physics is when V(x)=‖x‖^2, since it is related to Bose-Einstein condensation through a representation introduced by Feynman in 1953. In the context of statistical mechanics, the weights (1) define a probability when the set of points is finite, but the construction associated to an infinite set is not trivial and may fail without appropriate hypotheses. The first problem is to establish conditions for the existence of such a measure at infinite volume when the set of points is infinite. Once existence is derived, we are interested in establishing its uniqueness and the cycle structure of a typical permutation. We here consider the large temperature regime when the set of points is a Poisson point process in ℤ^d with intensity ρ∈(0,1/2), and the potential verifies some regularity conditions. In particular, we prove that if α is large enough, for almost every realization of the point process, there exists a unique Gibbs measure that concentrates on finite cycle permutations. We then extend these results to the continuous setting, when the set of points is given by a Poisson point process in ℝ^d with low enough intensity.
publishDate 2019
dc.date.none.fl_str_mv 2019-04
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://hdl.handle.net/11336/120189
Armendáriz, María Inés; Ferrari, Pablo Augusto; Frevenza Maestrone, Nicolas Federico; Gibbs measures over permutations of point processes with low density; Cornell University; arxiv.org; 4-2019; 1-25
2331-8422
CONICET Digital
CONICET
url http://hdl.handle.net/11336/120189
identifier_str_mv Armendáriz, María Inés; Ferrari, Pablo Augusto; Frevenza Maestrone, Nicolas Federico; Gibbs measures over permutations of point processes with low density; Cornell University; arxiv.org; 4-2019; 1-25
2331-8422
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1904.03952
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
application/pdf
dc.publisher.none.fl_str_mv Cornell University
publisher.none.fl_str_mv Cornell University
dc.source.none.fl_str_mv reponame:CONICET Digital (CONICET)
instname:Consejo Nacional de Investigaciones Científicas y Técnicas
reponame_str CONICET Digital (CONICET)
collection CONICET Digital (CONICET)
instname_str Consejo Nacional de Investigaciones Científicas y Técnicas
repository.name.fl_str_mv CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas
repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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score 13.070432

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