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2 y

Research School

Lecture 1. Collective dynamics and self-organization in biological systems : challenges and some examples.

Lecture 2. The Vicsek model as a paradigm for self-organization : from particles to fluid via kinetic descriptions

Lecture 3. Phase transitions in the Vicsek model : mathematical analyses in the kinetic framework.

35L60 ; 82C22 ; 82B26 ; 82C26 ; 92D50

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2 y

Research School

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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2 y

Research talks

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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2 y

Research talks

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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V

- x; 343 p.
ISBN 978-3-540-92795-2

Lecture notes in mathematics , 1970

Localisation : Collection 1er étage

modèle de treillis # mécanique statistique # méta-stabilité # transition de phase # géométrie stochastique

82B20 ; 82B26 ; 82C20 ; 60K35 ; 82C26 ; 82C35 ; 82B31 ; 82B41 ; 82B44 ; 82B10 ; 82C22 ; 60K05 ; 82B05 ; 82C05 ; 82-06 ; 00B25

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V

- 105 p.
ISBN 978-3-540-73704-9

Lecture notes in mathematics , 1916

Localisation : Collection 1er étage

mécanique des fluides # structure anatomique # structure moléculaire # théorie cinétique du gaz # théorie de l'information # mécanique statistique # processus aléatoire # système de particule en intéraction aléatoire

76-02 ; 76P05 ; 82B40 ; 94A15 ; 60K35 ; 82C22

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V

- 141 p.
ISBN 978-0-8218-1993-7

Fields institute communications , 0027

Localisation : Collection 1er étage

probabilité # mécanique statistique # physique statistique # limite hydrodynamique # diffusion # problème à frontière libre # entropie # entropie relative # grande déviation

60K35 ; 82C22

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y

Research talks

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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y

Research School

Lecture 1. Collective dynamics and self-organization in biological systems : challenges and some examples.

Lecture 2. The Vicsek model as a paradigm for self-organization : from particles to fluid via kinetic descriptions

Lecture 3. Phase transitions in the Vicsek model : mathematical analyses in the kinetic framework.

35L60 ; 82C22 ; 82B26 ; 82C26 ; 92D50

... Lire [+]

y

Research School

Lecture 1. Collective dynamics and self-organization in biological systems : challenges and some examples.

Lecture 2. The Vicsek model as a paradigm for self-organization : from particles to fluid via kinetic descriptions

Lecture 3. Phase transitions in the Vicsek model : mathematical analyses in the kinetic framework.

35L60 ; 82C22 ; 82B26 ; 82C26 ; 92D50

... Lire [+]

y

Research School

In these three lectures steady states and dynamical properties of nonequilibrium systems will be discussed.
Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. This is very different from systems in thermal equilibrium where long-range correlations develop only at phase transition points. In some cases these correlations even lead to long-range order in d=1 dimension, of the type occurring in traffic jams. Simple examples of such correlations induced in the steady state of driven systems will be presented and discussed. Close correspondence of these nonequilibrium steady states to electrostatic potentials induces by charge distribution will be pointed out.
Another class which will be discussed is that of systems with boundary drive, such as in heat conduction problems, where anomalous heat conduction takes place in low dimensions. In addition some similarities between driven systems and equilibrium systems with long-range interactions will be elucidated.
In these three lectures steady states and dynamical properties of nonequilibrium systems will be discussed.
Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. This is very different from systems in thermal equilibrium where long-range correlations develop only at phase transition points. In some cases these correlations even lead to long-range order in d=1 ...

82C26 ; 82C22

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y

Research School

In these three lectures steady states and dynamical properties of nonequilibrium systems will be discussed.
Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. This is very different from systems in thermal equilibrium where long-range correlations develop only at phase transition points. In some cases these correlations even lead to long-range order in d=1 dimension, of the type occurring in traffic jams. Simple examples of such correlations induced in the steady state of driven systems will be presented and discussed. Close correspondence of these nonequilibrium steady states to electrostatic potentials induces by charge distribution will be pointed out.
Another class which will be discussed is that of systems with boundary drive, such as in heat conduction problems, where anomalous heat conduction takes place in low dimensions. In addition some similarities between driven systems and equilibrium systems with long-range interactions will be elucidated.
In these three lectures steady states and dynamical properties of nonequilibrium systems will be discussed.
Systems driven out of thermal equilibrium often reach a steady state which under generic conditions exhibits long-range correlations. This is very different from systems in thermal equilibrium where long-range correlations develop only at phase transition points. In some cases these correlations even lead to long-range order in d=1 ...

82C26 ; 82C22

... Lire [+]

y

Research School

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

... Lire [+]

y

Research School

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

... Lire [+]

y

Research School

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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y

Research School

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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y

Research talks

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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y

Research talks

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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y

Research talks

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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y

Research talks

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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