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A New Look at the Interfaces in the Percolation and Ising Models Wei Zhou

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A New Look at the Interfaces in the Percolation and Ising Models Wei Zhou A new look at the interfaces in the percolation and Ising models Wei Zhou To cite this version: Wei Zhou. A new look at the interfaces in the percolation and Ising models. Probability [math.PR]. Université Paris-Saclay, 2019. English. NNT : 2019SACLS173. tel-02191676 HAL Id: tel-02191676 https://tel.archives-ouvertes.fr/tel-02191676 Submitted on 23 Jul 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. NNT : 2019SACLS173 THÈSE DE DOCTORAT de l’Université Paris-Saclay École doctorale de mathématiques Hadamard (EDMH, ED 574) Établissement d’inscription : Université Paris-Sud Établissement d’accueil : Ecole Normale Supérieure Laboratoire d’accueil : Département de mathématiques et applications, UMR 8553 CNRS Spécialité de doctorat : Mathématiques fondamentales Wei ZHOU Un nouveau regard sur les interfaces dans les modèles de percolation et d’Ising Date de soutenance : 25 Juin 2019 Jean-Baptiste GOUERE (Université de Tours) Après avis des rapporteurs : Yvan VELENIK (Université de Genève) Raphaël CERF (Université Paris-Saclay) Directeur de thèse Emilio CIRILLO (Université de Rome) Examinateur Jury de soutenance : Nathanaël ENRIQUEZ (Université Paris-Saclay) Président du jury Jean-Baptiste GOUERE (Université de Tours) Rapporteur Yvan VELENIK (Université de Genève) Rapporteur ii R´esum´e Titre : Un nouveau regard sur les interfaces dans les mod`elesde percolation et d'Ising Mots Clefs : Interface, localisation, percolation, FK-percolation, Ising. R´esum´e: Les interfaces dans les mod`elesde percolation et d'Ising jouent un r^olecrucial dans la compr´ehensionde ces mod`eleset sont au cœur de plusieurs probl´ematiques: la construction de Wulff, le mouvement par cour- bure moyenne, la th´eoriedu SLE. Dans son c´el`ebrearticle de 1972, Roland Dobrushin a montr´eque le mod`eled'Ising en dimension d > 3 admet une mesure de Gibbs qui n'est pas invariante par translation `al'aide d'une ´etude sur l'interface entre le haut et le bas d'une bo^ıtedroite de taille finie. Le cas d'une bo^ıte pench´eeest tr`esdiff´erent et plus difficile `aanalyser. Nous propo- sons dans cette th`eseune nouvelle d´efinitionde l'interface. Cette d´efinition est construite dans le mod`elede percolation Bernoulli `al'aide d'un couplage dynamique de deux configurations. Nous montrons que cette interface est localis´eeautour des ar^etespivot `aune distance d'ordre de ln2 n dans une bo^ıtede taille n. Notre m´ethode de preuve utilise les chemins espace-temps, qui permettent de contr^olerla vitesse de d´eplacement de l'interface. Nous montrons aussi que la vitesse des ar^etespivot est au plus de l'ordre de ln n. Nous ´etendons ces r´esultatsau mod`ele de FK-percolation, nous montrons la localisation de l'interface `adistance d'ordre ln2 n autour des ar^etespivot. En utilisant une modification du couplage classique d'Edwards-Sokal, nous obtenons des r´esultatsanalogues sur la localisation de l'interface dans le mod`eled'Ising. iii Abstract Title : A new look at the interfaces in the percolation and Ising models Keywords : Interface, localisation, percolation, FK-percolation, Ising mo- del Abstract : The interfaces in the percolation and Ising models play an important role in the understanding of these models and are at the heart of several problematics : the Wulff construction, the mean curvature motion and the SLE theory. In his famous 1972 paper, Roland Dobrushin showed that the Ising model in dimensions d > 3 has a Gibbs measure which is not invariant by translation by studying the interface between the top and the bottom of a straight finite box. The case of a tilted box is very different and more difficult to analyse. In this thesis, we propose a new definition of the interface. This definition is constructed in the Bernoulli percolation model with the help of a dynamical coupling between two configurations. We show that this interface is localised around the pivotal edges within a distance of order ln2 n inside a box of size n. The proof relies on space-time paths which allow us to control the speed of the interface. We also show that the speed of the pivotal edges is at most of order ln n. We extend these results to the FK-percolation model, we show the localisation of the interface at distance of order ln2 n around the pivotal edges. Using a modification of the classical Edwards-Sokal coupling, we obtain analogous results on the localisation of the interface in the Ising model. iv Harry Kesten, Rudolf Peierls et Roland Dobrushin, `aOxford, 1993 Remerciements Je tiens tout d'abord `aadresser mes plus sinc`eresremerciements `amon directeur de th`ese,Rapha¨elCerf. Il a su me guider durant ces ann´eesavec ses encouragements, sa bienveillance, tout en faisant preuve `ala fois d'une grande disponibilit´eet d'une grande g´en´erosit´edans le partage de ses id´ees. Sa cr´eativit´e,sa connaissance et sa compr´ehensiondes math´ematiquesainsi que sa rigueur resteront pour moi une importante source d'admiration et d'inspiration. Je suis tr`eshonor´ed'avoir eu la chance d'^etreson ´etudiant et travailler avec lui fut pour moi un grand plaisir. Je remercie vivement Jean-Baptiste Gou´er´eet Yvan Velenik d'avoir accept´e de rapporter cette th`ese.Leurs remarques et leurs commentaires m'ont ´et´e pr´ecieuxet je leur t´emoigneici mon respect et mon admiration math´ematique. Je suis par ailleurs tr`esreconnaissant `aNathana¨elEnriquez et Emilio Cirllo de me faire l'honneur de faire partie du jury. Ces ann´eesde th`eseont ´et´etr`esriches en discussions math´ematiqueset je souhaite remercier toutes les personnes avec qui j'ai eu le plaisir d'interagir. Je tiens tout particuli`erement `aremercier Barbara Dembin avec qui j'ai eu et ai encore la chance de travailler. Merci `atous les membres de l'ANR PPPP pour m'avoir invit´e`aleurs rencontres o`uj'ai pu avoir des discussions tr`esenrichissantes et d´ecouvrirune partie de la communaut´eprobabiliste fran¸caise. J'ai eu la chance de r´ealisercette th`esedans le DMA, dans lequel j'ai b´en´efici´ed'excellentes conditions de travail. Je remercie les ´equipes du la- boratoire qui ont su m'accueillir chaleureusement. En particulier, j'adresse mes remerciements `aB´en´edicteAuffray, Am´elieCastelain, Za¨ınaElmir et Albane Tr´emeaupour leur sympathie et leur efficacit´e. L'ambiance au sein des doctorants et doctorantes fut particuli`erement sym- pathique au cours de ces trois ann´eesde th`ese,et je tiens `asaluer mes anciens co-bureaux Guillaume, J´er´emy, Jessica, Maxence, Maxime, Nicolas, Tunan, Yichao pour de nombreuses discussions passionnantes. Merci aussi `atous les autres jeunes et/ou doctorants ou doctorantes avec qui j'ai eu v vi la chance de discuter autour d'un repas ou d'un caf´e,Aymeric, Ephr`eme, Louise, Micka¨el,Michel, Paul, Th´eophile,R´emy, Shariar, Thomas, Tobias, Yusuke. Ces derni`eresann´eesont ´et´eriches en rencontres et je remercie tr`es cha- leureusement Alejandro, Alexandre, Adrien, Antoine, Benjamin, Camille, Charles, Christophe, Dexiong, Fr´ed´eric,Geoffrey, Guillaume, L´eo, Louise, Mohammed, Jean, Luc, Luc (alias. Totoro), Philippe, Quentin, Rapha¨el, Rapha¨el(alias. Mr.pink), R´emi,Romain (alias. Trad), Salim, Thibaut, Tho- mas pour leur compagnie. Depuis mon arriv´eeen France il y a presque dix ans, j'ai eu la chance de rencontrer de nombreuses familles fran¸caisesqui m'ont accueilli pendant les weekends et les vacances scolaires. Je tiens `aremercier en particulier Claire, Chantal, Gabriel, Guy, Marianne et Samuel. Mes remerciements vont ´egalement `aleur famille pour leur aide et leur bienveillance. Un grand merci va tout particuli`erement `aPierre et toute la famille Man- ceron pour leur aide inestimable et sans qui rien de tout cela n'aurait ´et´e possible. 最后，我要感"我的妈妈和我的77FF。感"他们把我抚{长'，Y我 Zº。¡有`们无P的付ú，1¡有我Ê)的成1。 Table des mati`eres I Pr´esentation des r´esultats 1 1 Introduction g´en´erale 3 1.1 Les mod`eles de physique statistique . .3 1.1.1 Les objets g´eom´etriques . .3 1.1.2 Le mod`elede percolation Bernoulli . .4 1.1.3 Le mod`elede FK-percolation . .5 1.1.4 Le mod`eled'Ising . .7 1.2 Les dynamiques dans les mod`eles . 10 1.2.1 La percolation dynamique . 10 1.2.2 Les dynamiques de FK-percolation . 11 1.2.3 Les dynamiques du mod`eled'Ising . 11 1.3 L'interface classique . 12 1.3.1 La d´efinitionde l'interface . 12 1.3.2 La localisation des interfaces en dimensions d > 3 . 14 1.3.3 Les interfaces en dimension deux . 17 1.4 L'interface dynamique en percolation . 18 1.4.1 La d´efinitionde l'interface . 19 1.4.2 La localisation de l'interface . 20 1.4.3 La loi de la configuration conditionn´ee. 21 1.4.4 Une tentative d'am´elioration sur la localisation . 21 1.5 L'interface FK-Ising . 22 1.5.1 L'interface en FK-percolation . 22 1.5.2 L'interface dans le mod`eled'Ising . 23 1.6 Les chemins espace-temps . 25 1.7 Perspectives . 25 1.8 L'organisation de la th`ese .
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