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A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide ...[+]

82C22 ; 82B23 ; 60H15

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A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide background on this growing area of research and delve into a few of the recent developments.

Kardar-Parisi-Zhang - interacting particle systems - random growth processes - directed polymers - Markov duality - quantum integrable systems - Bethe ansatz - asymmetric simple exclusion process - stochastic partial differential equations[-]

A number of probabilistic systems which can be analyzed in great detail due to certain algebraic structures behind them. These systems include certain directed polymer models, random growth process, interacting particle systems and stochastic PDEs; their analysis yields information on certain universality classes, such as the Kardar-Parisi-Zhang; and these structures include Macdonald processes and quantum integrable systems. We will provide ...[+]

82C22 ; 82B23 ; 60H15

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In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the understanding of the structure of the stationary non equilibrium states. I will discuss these problems in specific examples and from two different perspectives. The stochastic microscopic and combinatorial point of view and the macroscopic variational approach where the microscopic details of the models are encoded just by the transport coefficients.[-]

In this first lecture I will introduce a class of stochastic microscopic models very useful as toy models in non equilibrium statistical mechanics. These are multi-component stochastic particle systems like the exclusion process, the zero range process and the KMP model. I will discuss their scaling limits and the corresponding large deviations principles. Problems of interest are the computation of the current flowing across a system and the ...[+]

82C05 ; 82C22 ; 60F10

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In this second lecture I will discuss the basic ideas of the macroscopic fluctuation theory as an effective theory in non equilibrium statistical mechanics. All the theory develops starting from a principal formula that describes the distribution at large deviations scale of the joint fluctuations of the density and the current for a diffusive system. The validity of such a formula can be proved for diffusive stochastic lattice gases. I will discuss an infinite dimensional Hamilton-Jacobi equation for the quasi-potential of stationary non equilibrium states, fluctuation-dissipation relationships, the underlying Hamiltonian structure, a relation with work and Clausius inequality, a large deviations functional for the current flowing through a system.[-]

In this second lecture I will discuss the basic ideas of the macroscopic fluctuation theory as an effective theory in non equilibrium statistical mechanics. All the theory develops starting from a principal formula that describes the distribution at large deviations scale of the joint fluctuations of the density and the current for a diffusive system. The validity of such a formula can be proved for diffusive stochastic lattice gases. I will ...[+]

60F10 ; 82C05 ; 82C22

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In the last lecture I will apply the macroscopic fluctuation theory to solve specific problems. I will show that several features and behaviors of non equilibrium systems can be deduced within the theory. In particular I will discuss the following issues: the presence of long range correlations in stationary non equilibrium states; the explicit computation of the large deviations rate functional for a few one dimensional stationary non equilibrium states; the existence of dynamical phase transitions in terms of the current flowing across the system, the existence of Lagrangian phase transitions.[-]

In the last lecture I will apply the macroscopic fluctuation theory to solve specific problems. I will show that several features and behaviors of non equilibrium systems can be deduced within the theory. In particular I will discuss the following issues: the presence of long range correlations in stationary non equilibrium states; the explicit computation of the large deviations rate functional for a few one dimensional stationary non ...[+]

60F10 ; 82C05 ; 82C22

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The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

35Q82 ; 60K35 ; 82C22 ; 82C24

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The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

35Q82 ; 60K35 ; 82C22 ; 82C24

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The aim of this series of lectures is to explain what the weak KPZ universality conjecture is, and to present a proof of it in the stationary case.
Lecture 1: The KPZ equation, the KPZ universality class and the weak and strong KPZ universality conjectures.
Lecture 2: The martingale approach and energy solutions of the KPZ equation.
Lecture 3: A proof of the weak KPZ universality conjecture in the stationary case.

35Q82 ; 60K35 ; 82C22 ; 82C24

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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

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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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Documents 82C22 33 résultats

Integrable probability - Lecture 1 - Corwin, Ivan (Auteur de la Conférence) | CIRM H Nouveau

Integrable probability - Lecture 2 - Corwin, Ivan (Auteur de la Conférence) | CIRM H Nouveau

Macroscopic fluctuation theory. Lecture 1: Particle systems, scaling limits and large deviations - Gabrielli, Davide (Auteur de la Conférence) | CIRM H Nouveau

Macroscopic fluctuation theory. Lecture 2 - Gabrielli, Davide (Auteur de la Conférence) | CIRM H Nouveau

Macroscopic fluctuation theory. Lecture 3: Examples and applications - Gabrielli, Davide (Auteur de la Conférence) | CIRM H Nouveau

The weak KPZ universality conjecture. Lecture 1 - Jara, Milton | CIRM H Nouveau

The weak KPZ universality conjecture. Lecture 2 - Jara, Milton (Auteur de la Conférence) | CIRM H Nouveau

The weak KPZ universality conjecture. Lecture 3 - Jara, Milton (Auteur de la Conférence) | CIRM H Nouveau

Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights - Lecture 1 - Seppäläinen, Timo (Auteur de la Conférence) | CIRM H Nouveau

Variational formulas, Busemann functions, and fluctuation exponents for the corner growth model with exponential weights - Lecture 2 - Seppäläinen, Timo (Auteur de la Conférence) | CIRM H Nouveau