As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It relates the correlators involving n consecutive lattice sites to those with n-1 and n-2 sites. .

LPS, ENS

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Nature is ripe with dynamical aggregation phenomena, in which an initially homogeneous collection of weakly interacting particles fragment, disperse, and coalesce. Condensation and droplet formation is, of course, a well-known example in physics, galaxy formation and clustering another. The formation of swarms, schools, herds, or even the flocking of birds provide compelling zoological illustrations. Rich stochastic behavior, as well as phase transition phenomena, are evident in different evolutionary minority (e.g., El Farol Bar), public goods, and other societal selection games, such as the Seceder Model, which introduces a novel dynamical frustration via the competing tendencies to be distinct, yet part of the group. The Seceder Model reveals that an iterative microscopic mechanism favoring dissent, yet permitting conformity, cannot only lead to the genesis of distinct groups, but also yields an abundant diversity of cluster-forming dynamics. In this talk, we will discuss population fragmentation, ideological symmetry-breaking and nonlinear group dynamics characteristic of this intriguing model. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It relates the correlators involving n consecutive lattice sites to those with n-1 and n-2 sites. .

LPS, ENS

Après avoir rappelé comment on peut construire des théories comme l’électrodynamique avec un terme de Chern-Simons sur un réseau bidimensionnel, je discuterai plus particulièrement le cas dúne symétrie de jauge locale de type Ising. Dans ce cas, le modèle de Chern-Simons est équivalent à un système de spins 1/2 quantiques avec des interactions selon plusieurs directions dans léspace des spins. Quelques résultats concernant la dégénérescence des états fondamentaux et le gap déxcitation seront présentés. Enfin, je montrerai comment de tels modèles peuvent être simulés par certains réseaux de jonctions Josephson. The effect of quenched disorder is often relevant, as far as the long-time, long-distance behavior of many particle systems is considered. For relevant perturbations disorder and deterministic fluctuations are usually in the same order of magnitude and the random fixed point is conventional. There is, however, a class of problems for which the singular behavior is controlled by a strong disorder fixed point, in which disorder grows without limit during renormalization. This type of behavior is first studied in random quantum spin chains, but recently strong disorder fixed points have been observed in classical systems, such as in the random bond Potts model in the large-q limit and in nonequilibrium processes, such as in absorbing state phase transitions and in driven lattice gas models. In this talk we review these latter developments, in particular we show how the critical singularities of the different problems, which are related to each other, can be exactly calculated. We also mention results about the Griffiths phases and higher dimensional problems. I consider a particle (random walker) diffusing in the y direction, with a transverse drift velocity f(y) in the x-direction and an absorbing boundary at x=0. The survival probability of the particle decays as t^-q. For any odd function f(y), q = 1/4 exactly, a result conjectured by Redner and Krapivsky. For generic f(y), q is a nontrivial functional of f(y). Some new exact results will be presented for a certain class of functions f(y). De nombreux travaux récents étudient les limites continues dárbres aléatoires discrets. Ces arbres discrets peuvent être définis soit de manière combinatoire (arbre choisi au hasard parmi tous les arbres à n sommets dún certain type), soit de manière probabiliste (en donnant la loi du “nombre dénfants” de chaque sommet de lárbre). Un passage à la limite où le nombre de sommets de lárbre tend vers línfini et où simultanément la longueur de chaque arête tend vers 0, conduit à des arbres aléatoires continus, dont le prototype est le Continuum Random Tree (CRT) introduit par Aldous. Léxposé décrira la manière dont ces arbres sont codés et en quel sens ils sont limites des arbres discrets. On donnera aussi diverses propriétés géométriques et fractales de ces arbres continus. Si le temps le permet, on introduira le modèle appelé ISE (Integrated Super-Brownian Excursion) qui combine la structure de branchement du CRT avec un déplacement brownien (les ïndividus” dont le CRT décrit la généalogie se déplacent dans léspace de manière brownienne). Les travaux récents de Slade et de ses co-auteurs ont montré que lÍSE intervient dans lásymptotique de différents modèles de mécanique statistique (arbres sur réseaux, percolation, etc.). We discuss extremal problems related to planar conformal geometry. More specifically, we discuss the multifractal properties of the harmonic measure and boundary rotation, in relation to the boundary behavior of conformal maps.

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Nature is ripe with dynamical aggregation phenomena, in which an initially homogeneous collection of weakly interacting particles fragment, disperse, and coalesce. Condensation and droplet formation is, of course, a well-known example in physics, galaxy formation and clustering another. The formation of swarms, schools, herds, or even the flocking of birds provide compelling zoological illustrations. Rich stochastic behavior, as well as phase transition phenomena, are evident in different evolutionary minority (e.g., El Farol Bar), public goods, and other societal selection games, such as the Seceder Model, which introduces a novel dynamical frustration via the competing tendencies to be distinct, yet part of the group. The Seceder Model reveals that an iterative microscopic mechanism favoring dissent, yet permitting conformity, cannot only lead to the genesis of distinct groups, but also yields an abundant diversity of cluster-forming dynamics. In this talk, we will discuss population fragmentation, ideological symmetry-breaking and nonlinear group dynamics characteristic of this intriguing model. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It relates the correlators involving n consecutive lattice sites to those with n-1 and n-2 sites. .

LPS, ENS

Après avoir rappelé comment on peut construire des théories comme l’électrodynamique avec un terme de Chern-Simons sur un réseau bidimensionnel, je discuterai plus particulièrement le cas dúne symétrie de jauge locale de type Ising. Dans ce cas, le modèle de Chern-Simons est équivalent à un système de spins 1/2 quantiques avec des interactions selon plusieurs directions dans léspace des spins. Quelques résultats concernant la dégénérescence des états fondamentaux et le gap déxcitation seront présentés. Enfin, je montrerai comment de tels modèles peuvent être simulés par certains réseaux de jonctions Josephson. The effect of quenched disorder is often relevant, as far as the long-time, long-distance behavior of many particle systems is considered. For relevant perturbations disorder and deterministic fluctuations are usually in the same order of magnitude and the random fixed point is conventional. There is, however, a class of problems for which the singular behavior is controlled by a strong disorder fixed point, in which disorder grows without limit during renormalization. This type of behavior is first studied in random quantum spin chains, but recently strong disorder fixed points have been observed in classical systems, such as in the random bond Potts model in the large-q limit and in nonequilibrium processes, such as in absorbing state phase transitions and in driven lattice gas models. In this talk we review these latter developments, in particular we show how the critical singularities of the different problems, which are related to each other, can be exactly calculated. We also mention results about the Griffiths phases and higher dimensional problems. I consider a particle (random walker) diffusing in the y direction, with a transverse drift velocity f(y) in the x-direction and an absorbing boundary at x=0. The survival probability of the particle decays as t^-q. For any odd function f(y), q = 1/4 exactly, a result conjectured by Redner and Krapivsky. For generic f(y), q is a nontrivial functional of f(y). Some new exact results will be presented for a certain class of functions f(y). De nombreux travaux récents étudient les limites continues dárbres aléatoires discrets. Ces arbres discrets peuvent être définis soit de manière combinatoire (arbre choisi au hasard parmi tous les arbres à n sommets dún certain type), soit de manière probabiliste (en donnant la loi du “nombre dénfants” de chaque sommet de lárbre). Un passage à la limite où le nombre de sommets de lárbre tend vers línfini et où simultanément la longueur de chaque arête tend vers 0, conduit à des arbres aléatoires continus, dont le prototype est le Continuum Random Tree (CRT) introduit par Aldous. Léxposé décrira la manière dont ces arbres sont codés et en quel sens ils sont limites des arbres discrets. On donnera aussi diverses propriétés géométriques et fractales de ces arbres continus. Si le temps le permet, on introduira le modèle appelé ISE (Integrated Super-Brownian Excursion) qui combine la structure de branchement du CRT avec un déplacement brownien (les ïndividus” dont le CRT décrit la généalogie se déplacent dans léspace de manière brownienne). Les travaux récents de Slade et de ses co-auteurs ont montré que lÍSE intervient dans lásymptotique de différents modèles de mécanique statistique (arbres sur réseaux, percolation, etc.). We discuss extremal problems related to planar conformal geometry. More specifically, we discuss the multifractal properties of the harmonic measure and boundary rotation, in relation to the boundary behavior of conformal maps.

There are well-known conjectures about the precise value of the extreme, or universal, multifractal spectra, and about the extreme set where the “worst” multifractal behavior of harmonic measure and rotation is expected.

We show that this extreme behavior älmost” occurs for sets invariant under hyperbolic polynomial dynamics. It allows us to conclude that the extreme multifractal spectra are the same for connected and disconnected sets. Nature is ripe with dynamical aggregation phenomena, in which an initially homogeneous collection of weakly interacting particles fragment, disperse, and coalesce. Condensation and droplet formation is, of course, a well-known example in physics, galaxy formation and clustering another. The formation of swarms, schools, herds, or even the flocking of birds provide compelling zoological illustrations. Rich stochastic behavior, as well as phase transition phenomena, are evident in different evolutionary minority (e.g., El Farol Bar), public goods, and other societal selection games, such as the Seceder Model, which introduces a novel dynamical frustration via the competing tendencies to be distinct, yet part of the group. The Seceder Model reveals that an iterative microscopic mechanism favoring dissent, yet permitting conformity, cannot only lead to the genesis of distinct groups, but also yields an abundant diversity of cluster-forming dynamics. In this talk, we will discuss population fragmentation, ideological symmetry-breaking and nonlinear group dynamics characteristic of this intriguing model. Recent progress in studying supersymmetric Yang-Mills quantum mechanics in various space-time dimensions, D, will be reported.

As an introduction we present exact construction of the ground state of the D=2 model and its behaviour under the supersymmetry transformations. Then the new high precision results on the spectrum of low lying supermultiplets of the D=4 theory will be shown.

Finally, we will discuss in some details the unusual properties of the D=10 model, namely non-conservation of the fermion number and the resolution of this apparent paradox in terms of the content of SO(9) representations of the system. A new recursion formula is presented for the correlation functions of the integrable spin 1/2 XXX chain with inhomogeneity. It relates the correlators involving n consecutive lattice sites to those with n-1 and n-2 sites. .

LPS, ENS