Planar random-cluster model: scaling relations
This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in [1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $\beta $ , $\gamma $...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2022-01-01
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| Series: | Forum of Mathematics, Pi |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050508622000166/type/journal_article |
| _version_ | 1811156238954659840 |
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| author | Hugo Duminil-Copin Ioan Manolescu |
| author_facet | Hugo Duminil-Copin Ioan Manolescu |
| author_sort | Hugo Duminil-Copin |
| collection | DOAJ |
| description | This paper studies the critical and near-critical regimes of the planar random-cluster model on
$\mathbb Z^2$
with cluster-weight
$q\in [1,4]$
using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents
$\beta $
,
$\gamma $
,
$\delta $
,
$\eta $
,
$\nu $
,
$\zeta $
as well as
$\alpha $
(when
$\alpha \ge 0$
). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of the influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalisation of Kesten’s classical scaling relation for Bernoulli percolation involving the ‘mixing rate’ critical exponent
$\iota $
replacing the four-arm event exponent
$\xi _4$
. |
| first_indexed | 2024-04-10T04:48:17Z |
| format | Article |
| id | doaj.art-6c3c3af98b9941c9aa00ecedcf181799 |
| institution | Directory Open Access Journal |
| issn | 2050-5086 |
| language | English |
| last_indexed | 2024-04-10T04:48:17Z |
| publishDate | 2022-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Pi |
| spelling | doaj.art-6c3c3af98b9941c9aa00ecedcf1817992023-03-09T12:34:22ZengCambridge University PressForum of Mathematics, Pi2050-50862022-01-011010.1017/fmp.2022.16Planar random-cluster model: scaling relationsHugo Duminil-Copin0https://orcid.org/0000-0002-7609-2816Ioan Manolescu1https://orcid.org/0000-0001-9140-5430Université de Genève, 7-9 Rue du Conseil Général, 1205 Genève, Switzerland Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-Sur-Yvette, France; E-mail:Université de Fribourg, Département de mathématiques, 23 chemin du Musée, CH-1700 Fribourg, Switzerland; E-mail:This paper studies the critical and near-critical regimes of the planar random-cluster model on $\mathbb Z^2$ with cluster-weight $q\in [1,4]$ using novel coupling techniques. More precisely, we derive the scaling relations between the critical exponents $\beta $ , $\gamma $ , $\delta $ , $\eta $ , $\nu $ , $\zeta $ as well as $\alpha $ (when $\alpha \ge 0$ ). As a key input, we show the stability of crossing probabilities in the near-critical regime using new interpretations of the notion of the influence of an edge in terms of the rate of mixing. As a byproduct, we derive a generalisation of Kesten’s classical scaling relation for Bernoulli percolation involving the ‘mixing rate’ critical exponent $\iota $ replacing the four-arm event exponent $\xi _4$ .https://www.cambridge.org/core/product/identifier/S2050508622000166/type/journal_article60K35 |
| spellingShingle | Hugo Duminil-Copin Ioan Manolescu Planar random-cluster model: scaling relations Forum of Mathematics, Pi 60K35 |
| title | Planar random-cluster model: scaling relations |
| title_full | Planar random-cluster model: scaling relations |
| title_fullStr | Planar random-cluster model: scaling relations |
| title_full_unstemmed | Planar random-cluster model: scaling relations |
| title_short | Planar random-cluster model: scaling relations |
| title_sort | planar random cluster model scaling relations |
| topic | 60K35 |
| url | https://www.cambridge.org/core/product/identifier/S2050508622000166/type/journal_article |
| work_keys_str_mv | AT hugoduminilcopin planarrandomclustermodelscalingrelations AT ioanmanolescu planarrandomclustermodelscalingrelations |