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Multi angle  Introduction to hyperbolic sigma models and Edge Reinforced Random Walk
Spencer, Tom (Auteur de la Conférence) | CIRM (Editeur )

This talk will introduce two statistical mechanics models on the lattice. The spins in these models have a hyperbolic symmetry. Correlations for these models can be expressed in terms of a random walk in a highly correlated random environment. In the SUSY hyperbolic case these walks are closely related to the vertex reinforced jump process and to the edge reinforced random walk. (Joint work with M. Disertori and M. Zirnbauer.)

60K37 ; 60G50 ; 60K35 ; 60J75 ; 81T25 ; 81T60

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Busemann functions for the two-dimensional corner growth model with exponential weights. Derivation of the stationary corner growth model and its use for calculating the limit shape and proving existence of Busemann functions.

60K35 ; 60K37 ; 82C22 ; 82C43 ; 82D60

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Multi angle  Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights - Lecture 3
Seppäläinen, Timo (Auteur de la Conférence) | CIRM (Editeur )

Kardar-Parisi-Zhang fluctuation exponent for the last-passage value of the two-dimensional corner growth model with exponential weights. We sketch the proof of the fluctuation exponent for the stationary corner growth process, and if time permits indicate how the exponent is derived for the percolation process with i.i.d. weights.

60K35 ; 60K37 ; 82C22 ; 82C43 ; 82D60

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Multi angle  Low temperature interfaces and level lines in the critical prewetting regime
Ioffe, Dmitry (Auteur de la Conférence) | CIRM (Editeur )

Complete wetting in the context of the low temperature two-dimensional Ising model is characterized by creation of a mesoscopic size layer of the "-" phase above an active substrate. Adding a small positive magnetic field h makes "-"-phase unstable, and the layer becomes only microscopically thick. Critical prewetting corresponds to a continuous divergence of this layer as h tends to zero. There is a conjectured 1/3 (diffusive) scaling leading to Ferrari-Spohn diffusions. Rigorous results were established for polymer models of random and self-avoiding walks under vanishing area tilts.
A similar 1/3-scaling is conjectured to hold for top level lines of low temperature SOS-type interfaces in three dimensions. In the latter case, the effective local structure is that of ordered walks, again under area tilts. The conjectured scaling limits (rigorously established in the random walk context) are ordered diffusions driven by Airy Slatter determinants.
Based on joint walks with Senya Shlosman, Yvan Velenik and Vitali Wachtel.
Complete wetting in the context of the low temperature two-dimensional Ising model is characterized by creation of a mesoscopic size layer of the "-" phase above an active substrate. Adding a small positive magnetic field h makes "-"-phase unstable, and the layer becomes only microscopically thick. Critical prewetting corresponds to a continuous divergence of this layer as h tends to zero. There is a conjectured 1/3 (diffusive) scaling leading ...

60K35 ; 82B41 ; 60G50 ; 60F17

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Multi angle  A 2d growth model in the anisotropic KPZ class
Toninelli, Fabio (Auteur de la Conférence) | CIRM (Editeur )

Dimer models provide natural models of (2+1)-dimensional random discrete interfaces and of stochastic interface dynamics. I will discuss two examples of such dynamics, a reversible one and a driven one (growth process). In both cases we can prove the convergence of the stochastic interface evolution to a deterministic PDE after suitable (diffusive or hyperbolic respectively in the two cases) space-time rescaling.
Based on joint work with B. Laslier and M. Legras.
Dimer models provide natural models of (2+1)-dimensional random discrete interfaces and of stochastic interface dynamics. I will discuss two examples of such dynamics, a reversible one and a driven one (growth process). In both cases we can prove the convergence of the stochastic interface evolution to a deterministic PDE after suitable (diffusive or hyperbolic respectively in the two cases) space-time rescaling.
Based on joint work with B. ...

60K35 ; 82C20

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Multi angle  Multi-time distribution of periodic TASEP
Baik, Jinho (Auteur de la Conférence) | CIRM (Editeur )

We consider periodic TASEP with periodic step initial condition, and evaluate the joint distribution of the locations of m particles. For arbitrary indices and times, we find a formula for the multi-time, multi-space joint distribution in terms of an integral of a Fredholm determinant. We then discuss the large time limit in the so-called relaxation scale. The one-point distributions for other initial conditions are also going to discussed.
Based on joint work with Zhipeng Liu (NYU).
We consider periodic TASEP with periodic step initial condition, and evaluate the joint distribution of the locations of m particles. For arbitrary indices and times, we find a formula for the multi-time, multi-space joint distribution in terms of an integral of a Fredholm determinant. We then discuss the large time limit in the so-called relaxation scale. The one-point distributions for other initial conditions are also going to discus...

82C22 ; 60K35 ; 82C43

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Multi angle  Random walks on dynamical percolation
Sousi, Perla (Auteur de la Conférence) | CIRM (Editeur )

We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate $\mu$, while at the same time a random walker moves on $G$ at rate 1, but only along edges which are open. On the d-dimensional torus with side length $n$, when the bond parameter is subcritical, the mixing times for both the full system and the random walker were determined by Peres, Stauffer and Steif. I will talk about the supercritical case, which was left open, but can be analysed using evolving sets.

Joint work with Y. Peres and J. Steif.
We study the behaviour of random walk on dynamical percolation. In this model, the edges of a graph are either open or closed and refresh their status at rate $\mu$, while at the same time a random walker moves on $G$ at rate 1, but only along edges which are open. On the d-dimensional torus with side length $n$, when the bond parameter is subcritical, the mixing times for both the full system and the random walker were determined by Peres, ...

60K35 ; 60J10 ; 60G50 ; 82B43