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Evolutions de Schramm-Loewner et théories conformes;
Deux exemples de systèmes désordonnés de basse dimension

Abstract : In the first part of this thesis we study of two-dimensional critical interfaces by means of Schramm-Loewner evolutions (SLEs) and conformal field theories. In particular, we study SLE(2) which corresponds to the scaling limit of loop-erased random walks. The explicit solution of the winding problem on doubly connected domains is given. Moreover we establish a generalisation of Schramm's formula for SLE(2) to the doubly connected case, and extend the solution to mixed Dirichlet-Neumann boundary conditions. Conformal field theory methods allow for an identification of the corresponding boundary-condition changing operator.
Moreover, upon studying the discontinuity lines of the Gaussian free field on doubly connected domains we point out a relation between SLE(4) and Brownian bridges.

The topic of the second part is the study of two different low-dimensional disordered systems. First, we establish spectral and localisation properties of a one-dimensional random Hamiltonian which interpolates between the well-known cases of Halperin's model and the disordered supersymmetric model. A mapping to one-dimensional diffusion in a random potential allows to study the modification of Sinai's ultra-slow dynamics in the presence of absorbers. The topic of the second part is the freezing transition of RNA with random sequences by means of the Lässig-Wiese-David field theory. We establish the force-extension characteristics for heterogeneous sequences subject to external forces. To better understand the glassy phase, we consider a combinatorial hierarchical model whose exact scaling laws and exponents, as well as finite-size properties are calculated analytically.
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Contributor : Christian Hagendorf <>
Submitted on : Tuesday, October 6, 2009 - 4:04:29 PM
Last modification on : Thursday, December 10, 2020 - 12:36:42 PM
Long-term archiving on: : Tuesday, October 16, 2012 - 11:55:09 AM


  • HAL Id : tel-00422366, version 1


Christian Hagendorf. Evolutions de Schramm-Loewner et théories conformes;
Deux exemples de systèmes désordonnés de basse dimension. Physique mathématique [math-ph]. Université Pierre et Marie Curie - Paris VI, 2009. Français. ⟨tel-00422366⟩



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