THEORIE DES CHAMPS NON STANDARDS

par Gauthier Mukerjee

Projet de thèse en Physique

Sous la direction de Kay Wiese.

Thèses en préparation à l'Université Paris sciences et lettres , dans le cadre de Physique en Île de France , en partenariat avec Laboratoire de Physique de l'École normale supérieure (laboratoire) et de Ecole normale supérieure (établissement opérateur d'inscription) depuis le 01-09-2019 .


  • Résumé

    Les marches aléatoires auto-évitantes et les marches aléatoires à boucles effacées sont deux types de chemins aléatoires ayant de nombreuses applications en mathématiques, physique statistique et théorie quantique des champs. Une marche aléatoire à boucles effacées consiste simplement en l'objet résultant de l'effaçage chronologique des différentes boucles d'une marche aléatoire usuelle. Si les marches auto-évitantes peuvent être décrites associées à une théorie des champs correspondant au modèle O(n) dans la limite n tend vers 0, il n'y a pas de description évidente pour les marches aléatoires à boucles effacées. Le but de cette thèse est de trouver une théorie des champs pour ces marches et d'explorer ses liens avec des modèles de croissance comme le Diffusion limited aggregation ou le Dielectric breakdown, et d'exploiter ces liens pour obtenir des prédictions quantitatives sur les exposants critiques de la théorie des champs sous-jacente.

  • Titre traduit

    Non Standard Field Theories


  • Résumé

    Self-avoiding walks or self-avoiding polymers (SAWs) and loop-erased random walks (LERWs) are two ensembles of random paths with numerous applications in mathematics, statistical physics and quantum field theory. Self-avoiding polymers exist in nature, and are well under- stood. Loop-erased random walks are constructed as follows (see figure): Follow the trajectory of a random walk, and whenever it comes back to its own trajectory, i.e. forms a loop, color this loop in red. The result is a LERW, i.e. the blue curve on the figure. While SAWs are described by the n → 0 limit of φ4-theory with O(n)-symmetry, LERWs have no obvious field-theoretic description. Two candidates for a field theory of LERWs have been proposed . The first such candidate is the O(n)-symmetric φ4 theory at n = −2 whose link to LERWs was known in two dimensions due to conformal field theory. The second candidate is a field theory for charge- density waves pinned by quenched disorder, whose relation to LERWs had been conjectured earlier using analogies with Abelian sandpiles. Both theories yield identical results to 4-loop order. As an example, for the fractal dimension of LERWs in d = 3, these theories give at 6-loop order z = 1.6243 ± 0.001, in agreement with the estimate z = 1.62400 ± 0.00005 of numerical simulations . The equivalence between the two theories can be proven perturbabtively, and using supersymmetry techniques Many open questions remain, or are motivated by the discovered equivalence. Some or them are relatively straightforward, others more exploratory. In order of increasing difficulty: • Charge-density waves have a notion of time, and renormalization thereof. What does this mean in φ4 theory? • The mapping in [1] from LERWs to φ4-theory was constructed perturbatively. One would like to find a non-perturbative proof. One approach is to use the equivalence of loop-erased random walks to Laplacian growth [3], where a random-walk explorer is sent off from the tip. If it reaches infinity, then we continue the Laplacian walk a step in that direction, otherwise we try again. The explorers can be interpreted as the cut-off loops. The goal is to provide this way an alternative, algebraic derivation of the above equivalence. • Allowing growth in the Laplacian-growth model to proceed from any point (instead of its tip) yields an isotropic growth process known as the dielectric breakdown model [4, 5, 6] (with one parameter set to 1), also equivalent to diffusion-limited aggregation [7]. It is proposed to explore these equivalences, and to tightly knit them into the theoretical frameworks established for LERWs, namely φ4 theory at N = −2, and charge-density waves at depinning. One obstacle to overcome is that the “established” field theory models DLA as a branching process, ignoring that a branch can come back onto an already occupied site. • On a more fundamental level, the equivalence between the functional field theory of charge-density waves and φ4 theory at n = −2 is quite striking, since the latter does not seem to know about disorder and as a consequence avalanches, while the former does. Are there other examples to be constructed? If yes, this would be a tremendous gain in computa- tional power. • Can we add to the zoo of φ4 theories with n = 1, 2, 3, ... for the physically intuitive ones, and n = 0 and n = −2 mentioned above other values, as e.g. n = −1?.