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Theses

Directed polymers and the KPZ equation

Abstract : This thesis is dedicated to the study of the links between directed polymers in random environment, the stochastic heat equation with multiplicative noise (SHE) and the Kardar-Parisi-Zhang equation (KPZ), under different space dimensions. In dimension d= 1, the KPZ equation and the SHE equation belong to a particular class of models which feature non-standard scaling coefficients and non-standard scaling limits. This class is called the KPZ universality class. It is possible to prove that some specific polymer models, which are called exactly solvable models, belong to this class, but one of the open problems in this field is to show that this result should be universal, that is that polymer models should belong to the KPZ universality class for very general types of environment. Nevertheless, one can prove that under a scaling limit, the point-to-point partition function of general polymer models converges towards the solution of the SHE equation, which can be seen as a weak universality result for the polymer models. In higher space dimension, it is not clear whether the KPZ and SHE equations should be well-posed. In order to study these equations in higher dimension, we will consider them with a noise that is be regularized in space (in this case, the solutions of the equations are well-defined) and try to look at the limiting behaviour of the solutions when the regularization is removed. It turns out that for a certain choice of parameters, the solutions of the regularized equations are linked to the partition functions of a directed polymer model, and one can use standard polymer techniques to study the asymptotic behaviour of the solutions
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Clément Cosco. Directed polymers and the KPZ equation. General Mathematics [math.GM]. Université Sorbonne Paris Cité, 2019. English. ⟨NNT : 2019USPCC037⟩. ⟨tel-02886376⟩

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