Phénomènes d'accrochage et théorie des fluctuations.

Abstract : This work is made of two main parts: we first illustrate the notion of correlation length for homogeneous pinning models near from criticality by showing the convergence in law of the rescaled system towards a random closed subset of the interval [0,1] having an explicit density with respect to the law of the alpha regenerative set, where alpha is an initial parameter of the model. In a second part, we are interested in a wetting model in a stripe; the rewards/penalties are received in a stripe with fixed positive width, the free process being modeled by a centered square summable random walk S with continuous increments. We show that the system has a standard localization/delocalization phase transition, and we explicit the scaling limits of the system in each of them. To get these scaling limits, we show two independent results which are sharply linked to the theory of fluctuation for random walks; there results have their own interest and compose the last two parts of the thesis. The first one describes the asymptotic behavior of the distribution of the location of the first hitting point of the negative half-plane for S, where S starts from a (fixed) point which is located in the positive half plane. Note that this estimation is uniform over the negative half line. The second result is the convergence in law in the scaling limit of S conditioned to stay positive, to start close from the origin and to come back close to the origin. We show that this process converges in law towards the normalized brownian excursion.
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Contributor : Julien Sohier <>
Submitted on : Wednesday, November 10, 2010 - 1:22:28 PM
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  • HAL Id : tel-00534716, version 1


Julien Sohier. Phénomènes d'accrochage et théorie des fluctuations.. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2010. Français. ⟨tel-00534716⟩



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