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Documents  82C43 | enregistrements trouvés : 6

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Variational formulas for limit shapes of directed last-passage percolation models. Connections of minimizing cocycles of the variational formulas to geodesics, Busemann functions, and stationary percolation.

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

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

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

Multi angle  The KPZ fixed point - Lecture 1
Remenik, Daniel (Auteur de la Conférence) | CIRM (Editeur )

In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion process (TASEP). Then I will present a Fredholm determinant formula for its distribution (at a fixed time) and show how all the main properties of the fixed point (including the Markov property, space and time regularity, symmetries and scaling invariance, and variational formulas) can be derived from the formula and the construction, and also how the formula reproduces known self-similar solutions such as the $Airy_1andAiry_2$ processes.
The second part of the course will be devoted to explaining how the KPZ fixed point can be computed starting from TASEP. The method is based on solving, for any initial condition, the biorthogonal ensemble representation for TASEP found by Sasamoto '05 and Borodin-Ferrari-Prähofer-Sasamoto '07. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.
Based on joint work with K. Matetski and J. Quastel.
In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion ...

82C31 ; 82C23 ; 82D60 ; 82C22 ; 82C43

Multi angle  The KPZ fixed point - Lecture 2
Remenik, Daniel (Auteur de la Conférence) | CIRM (Editeur )

In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion process (TASEP). Then I will present a Fredholm determinant formula for its distribution (at a fixed time) and show how all the main properties of the fixed point (including the Markov property, space and time regularity, symmetries and scaling invariance, and variational formulas) can be derived from the formula and the construction, and also how the formula reproduces known self-similar solutions such as the $Airy_1andAiry_2$ processes.
The second part of the course will be devoted to explaining how the KPZ fixed point can be computed starting from TASEP. The method is based on solving, for any initial condition, the biorthogonal ensemble representation for TASEP found by Sasamoto '05 and Borodin-Ferrari-Prähofer-Sasamoto '07. The resulting kernel involves transition probabilities of a random walk forced to hit a curve defined by the initial data, and in the KPZ 1:2:3 scaling limit the formula leads in a transparent way to a Fredholm determinant formula given in terms of analogous kernels based on Brownian motion.
Based on joint work with K. Matetski and J. Quastel.
In these lectures I will present the recent construction of the KPZ fixed point, which is the scaling invariant Markov process conjectured to arise as the universal scaling limit of all models in the KPZ universality class, and which contains all the fluctuation behavior seen in the class.
In the first part of the minicourse I will describe this process and how it arises from a particular microscopic model, the totally asymmetric exclusion ...

82C31 ; 82C23 ; 82D60 ; 82C22 ; 82C43

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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