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Fall 2008 [Pdf] Le Bulletin du CRM • crm.math.ca • Automne/Fall 2008 | Volume 14 – No 2 | Le Centre de recherches mathématiques The Fall 2008 Aisenstadt Chairs Four Aisenstadt Chair lecturers will visit the CRM during the 2008-2009 thematic year “Probabilistic Methods in Mathemat- ical Physics.” We report here on the series of lectures of Wendelin Werner (Université Paris-Sud 11) and Andrei Okounkov (Princeton University), both of whom are Fields Medalists, who visited the CRM in August and September 2008 respectively. The other Aisenstadt Chairs will be held by Svante Janson (Uppsala University) and Craig Tracy (University of California at Davis). Wendelin Werner Andrei Okounkov de Yvan Saint-Aubin (Université de Montréal) by John Harnad (Concordia University) Wendelin Werner est un A metaphor from an ancient fragment by Archilochus “The Fox spécialiste de la théo- knows many things but the Hedgehog knows one big thing” rie des probabilités. Il a was used by Isaiah Berlin as title and theme of his essay on Tol- obtenu son doctorat en stoy’s view of history (“. by nature a fox, but believed in be- 1993 sous la direction de ing a hedgehog”) [1]. It is also very suitably applied to styles in Jean-François Le Gall. Il science. In his presentation of the work of Andrei Okounkov est professeur au labo- when he was awarded the Fields medal at the 2006 Interna- ratoire de mathématiques tional Congress of Mathematicians in Madrid, Giovanni Felder à l’Université Paris-Sud said: “Andrei Okounkov’s initial area of research was group XI à Orsay depuis 1997, representation theory, with particular emphasis on combinato- ainsi qu’à l’École nor- rial and asymptotic aspects. He used this subject as a start- male supérieure depuis 2005. Avec ses collaborateurs Greg Lawler ing point to obtain spectacular results in many different areas et Oded Schramm, il a montré que la probabilité de deux marches of mathematics and aléatoires planaires qui s’évitent décroît comme n−5/8 où n est la lon- mathematical physics, 4 gueur des marches et a déterminé la dimension de Hausdorff (= 3 ) from complex and real de la frontière extérieure du mouvement brownien planaire. Ses tra- algebraic geometry to vaux sur l’équation de Loewner stochastique et sur les ensembles de statistical mechanics, boucles conformes ont déjà eu un impact profond sur la description dynamical systems, mathématique des phénomènes critiques en deux dimensions. probability theory and topological string the- Werner a reçu le prix de la Société Européenne de Mathématiques ory. The research of en 2000, le prix Fermat en 2001, le prix Loève en 2005, le prix Pó- Okounkov has its roots lya l’année suivante. En 2006 il devenait le premier probabiliste à in very basic notions recevoir la médaille Fields. Il a détenu une des chaires Aisenstadt de such as partitions, which form a recurrent theme in his l’année thématique 2008-2009 sur les Méthodes probabilistes en phy- work. at its core (is) the idea that partitions and other no- sique mathématique. tions of representation theory should be considered as random objects with respect to natural probability measures. This idea Les phénomènes critiques was further developed by Okounkov, who showed that, to- L’étude des transitions de phase en physique a commencé il y gether with insights from geometry and ideas of high energy a plus d’un siècle, mais ses progrès importants ont été accom- physics, it can be applied to the most diverse areas of mathe- plis depuis les années quarante du XXe siècle. Son objet touche matics.” des phénomènes physiques variés, dont certains tombent sous This would appear to describe the work of a fox but, as with l’expérience de tous les jours, comme l’ébullition de l’eau ou Tolstoy, things are not necessarily as they seem. The fact la miscibilité de gas et de liquide, par exemple dans les bois- that Okounkov’s work intersects so many fields of mathe- sons gaseuses. Dans pratiquement tous ces phénomènes, une matics and mathematical physics may appear astonishing to quantité macroscopique possède un comportement singulier (continued on page 4) (suite à la page 2) • crm.math.ca • Wendelin Werner Une autre hypothèse importante est celle de l’invariance (suite de la page 1) conforme. Elle fut introduite dans le cadre des phénomènes cri- tiques au début des années 80 par Belavin, Polyakov et Za- en fonction des paramètres du système. Voici un exemple ma- molodchikov. Une forme de cette hypothèse, suggérée par Ai- thématique célèbre, celui de la percolation. zenman, est particulièrement simple à formuler. Changeons l’échelle de sorte que le côté de la région triangulaire conserve une longueur unité lorsque N varie et notons par ΓN l’interface ainsi obtenue. Fixons p = pc et considérons maintenant les in- terfaces commençant au sommet. Divisons la base de la région en deux segments, le gauche de longueur x ∈ [0, 1], l’autre de longueur 1 − x. Une seconde observable est la probabilité π(x) que, dans la limite N → ∞, l’interface ΓN issue du sommet touche le segment gauche avant le droit. Soit maintenant une transformation conforme φ sur un ouvert contenant la région triangulaire. L’image du triangle par cette transformation est (a) (b) maintenant tracée sur le réseau triangulaire qui, lui, demeure Fig. 1. Une interface γ12 est tracée en (a), une trajectoire du processus inchangé. L’hypothèse d’invariance conforme stipule que, dans SLE6 en (b). la limite de la maille du réseau allant vers zéro, la probabilité qu’une interface issue de l’image du sommet atteigne l’image D’un réseau triangulaire planaire de maille unité, considérons du segment gauche avant celle du droit demeure π(x). une région de la forme d’un triangle équilatéral dont les côtés ont longueur N, avec N pair. Le site médian de la base est l’ori- SLE gine. Les sites à la frontière de la région sont peints en blanc si leur coordonnée horizontale est positive, en noir autrement. Des énoncés rigoureux existent maintenant pour certains as- Pour les autres sites (intérieurs), leur couleur sera noir avec pects de ce domaine ; deux résultats particuliers en sont les pré- probabilité p ∈ [0, 1], blanc avec probabilité 1 − p. Une interface curseurs. Tout d’abord, Smirnov (1999) a montré qu’une cer- taine observable pour la percolation sur le réseau triangulaire γN, séparant les sites noirs des blancs, commence (presque) à l’origine, suit le réseau dual hexagonal et quitte la région tri- est effectivement conformément invariante lorsque la maille du angulaire (presque) au sommet. Soit d(p, N) la probabilité que réseau tend vers zéro. Et Schramm (2000) a lié la loi limite des cette interface, issue de l’origine, touche le côté droit avant le interfaces ΓN à celle du mouvement brownien unidimension- côté gauche. Il est possible de montrer que la limite de d(p, N) nel. Cette idée allait être développée par lui et ses collègues lorsque N → ∞ est 0 lorsque p < 1 et 1 lorsque p > 1 . Donc Lawler et Werner en un champ fertile en idées et outils nou- 2 2 veaux. Il est maintenant connu sous le nom d’équation de Loew- l’« observable » limN→∞ d(p, N) possède un comportement sin- 1 1 ner stochastique ou équation de Loewner-Schramm (SLE). gulier en p = 2 . (Pour cette raison pc = 2 est nommé la pro- babilité critique pour cet exemple.) Cette observable prend la 1 valeur 2 en p = pc. D’autres observables peuvent aussi être étudiées, par exemple le nombre moyen de sites dans la plage connexe de sites noirs contenant l’origine. Deux conjectures (a) La courbe c (b) L’image de la courbe c par g , t > 0 Cet exemple se prête à de nombreuses généralisations. Il est t Fig. 2. Une courbe c est progressivement retirée du demi-plan par la possible de changer le réseau triangulaire pour un autre ré- famille de transformations conformes g de Loewner. seau périodique ou d’étudier la percolation en d’autres dimen- t d sions, par exemple sur le réseau Z ou sur une région finie Le lien entre la loi limite de ΓN et le mouvement brow- de ce réseau. D’autres modèles peuvent être aussi proposés nien est faite à l’aide d’un résultat classique de la géo- pour lesquels la couleur d’un site (ou l’état de son spin) dé- métrie conforme. Soit une courbe c : [0, T] → H ∪ ∂H pend de celles de ses voisins, comme pour les modèles d’Ising dans le demi-plan supérieur H et l’axe réel dont le point et de Potts. Pour tous ces modèles, l’existence d’un point cri- initial est en l’origine. (Voir Figure 2(a).) Supposons que tique et les propriétés en ce point lorsqu’il existe dépendent de cette courbe ne touche ni l’axe réel (sauf l’origine) ni elle- la dimension du réseau. Mais les physiciens sont convaincus, même. (Ces hypothèse sont trop restrictives, mais elles sim- par des arguments théoriques et des simulations numériques, plifient la présentation.) Alors le complément C de cette que pour une dimension donnée les propriétés critiques sont courbe dans H est homéomorphe au demi-plan H. Par indépendantes du réseau choisi. Cette indépendance constitue le théorème de Riemann il existe donc une transformation l’hypothèse d’universalité. Cette conjecture demeure un des pro- conforme envoyant C sur H. Il est même possible de trouver blèmes majeurs du sujet. (suite à la page 6) BULLETIN CRM–2 • crm.math.ca • Prizes The 2008 CAP – CRM Prize in Theoretical and The 2008 CRM – SSC Prize in Statistics Mathematical Physics The CRM – SSC Prize is given conjointly by the Statisti- The CAP – CRM Prize is given conjointly by the Canadian cal Society of Canada (SSC) and the CRM every year since Association of Physicists (CAP) and the CRM every year 1999.
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