MINI COURSES
AND
LONG TALKS

Mini-courses:

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Title:  Localization transition for directed polymers in a random environment (in dimension larger than 3)

Abstract: The Directed Polymer in a Random Environment (DPRE) is obtained by reweighting the trajectories of finite length simple random walk using an i.i.d. random environment. It is one of the simplest disordered models in statistical mechanics, and one for which the disorder-induced phase transition has been intensively studied. When the intensity of the disorder increases, the systems behavior changes drastically: at high temperature (low disorder intensity) the trajectories of the polymer are diffusive with a behavior which is very similar to that of the simple random walk, at low temperature, the trajectories of the polymer are conjecture to concentrate on a narrow space-time corridor, and its end-point distribution is localized.

In this course, we will present the state of the art for the study of this transition in dimension larger than 3, and expose simple proof of some well established localization and delocalization results.

Long talks:

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Title: Some properties of 2D directed polymers and Stochastic Heat Flow

Abstract: I will review some recent results on the directed polymer model in dimension d=2, which can be seen as a discretised version of the stochastic heat equation. Recent results have shown that, in a suitable (and subtle) critical window of parameters, the model converges to a disordered limit, namely the Critical 2D Stochastic Heat Flow (SHF) introduced by Caravenna, Sun and Zygouras. This SHF is a non-Gaussian stochastic process of measures on R^2, and a lot of its properties have been studied over the last years. In this talk, I will first take some time to explain the (idea of the) construction of the SHF and then review some of the recent advances. In particular, I will comment on recent/ongoing works in collaboration with Caravenna, Turchi and Zygouras, which study how the mass assigned by the SHF to a ball behaves, either as time goes to infinity or as the radius of the ball goes to 0.

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Title: From 1d random Schrödinger operators to random Dirac operators

Abstract: We will consider random Schrödinger operators in 1d, both in the discrete and the continuum setting. It is known that for a large class of potentials, Anderson localisation holds: the spectrum is pure point and the eigenfunctions are exponentially localized. In the weak disorder limit, we will show that a random Dirac operator arises « generically ». This operator displays very nice properties. Joint work with Laure Dumaz (ENS). 

Title: Nonequilibrium fluctuations of interacting particle systems.

 
Abstract: We review some recent results on nonequilibrium fluctuations

of gradient and reaction-diffusion models.

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Title:  The periodic directed landscape  

Abstract: The Kardar--Parisi--Zhang universality class is a central topic in mathematical physics and probability theory. In this talk, we discuss the conjectural scaling limit of periodic models in the Kardar--Parisi--Zhang universality class: the periodic directed landscape. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape and give a variational characterization of the periodic KPZ fixed point, recently constructed by Baik, Liao, and Liu. We also establish the convergence of periodic asymmetric simple exclusion processes to periodic KPZ fixed points, coupled through the same periodic directed landscape. A key ingredient is a gluing technique for combining overlapping directed landscapes, which may be of independent interest. This is based on joint work with Amol Aggarwal and Ivan Corwin. 

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