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# Documents  82C22 | enregistrements trouvés : 27

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## Post-edited  Collective dynamics in life sciences - Lecture 1. Collective dynamics and self-organization in biological systems: challenges and some examples Degond, Pierre (Auteur de la Conférence) | CIRM (Editeur )

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.

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

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.

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

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 ...

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## Post-edited  Integrable probability - Lecture 1 Corwin, Ivan (Auteur de la Conférence) | CIRM (Editeur )

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 ...

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## Multi angle  Microscopic description of Coulomb gases Serfaty, Sylvia (Auteur de la Conférence) | CIRM (Editeur )

We are interested in the statistical mechanics of systems of N points with Coulomb interactions in general dimension for a broad temperature range. We discuss local laws characterizing the rigidity of the system at the microscopic level, as well as free energy expansion and Central Limit Theorems for fluctuations.

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## Multi angle  Cutoff phenomenon for the asymmetric simple exclusion process Labbé, Cyril (Auteur de la Conférence) | CIRM (Editeur )

I will consider the asymmetric simple exclusion process on a linear lattice of N sites, and I will present a result on the asymptotic (in N) behaviour of the distance to equilibrium of this process starting from the "worst" initial condition. This result shows a cutoff phenomenon: instead of decaying smoothly with time, the distance to equilibrium falls abruptly at some deterministic time. This is a joint work with Hubert Lacoin (IMPA).

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## Multi angle  Metastability of the contact process on evolving scale-free networks Mörters, Peter (Auteur de la Conférence) | CIRM (Editeur )

We study the contact process in the regime of small infection rates on scale-free networks evolving by stationary dynamics. A parameter allows us to interpolate between slow (static) and fast (mean-field) network dynamics. For two paradigmatic classes of networks we investigate transitions between phases of fast and slow extinction and in the latter case we analyse the density of infected vertices in the metastable state.
The talk is based on joint work with Emmanuel Jacob (ENS Lyon) and Amitai Linker (Universidad de Chile).
We study the contact process in the regime of small infection rates on scale-free networks evolving by stationary dynamics. A parameter allows us to interpolate between slow (static) and fast (mean-field) network dynamics. For two paradigmatic classes of networks we investigate transitions between phases of fast and slow extinction and in the latter case we analyse the density of infected vertices in the metastable state.
The talk is based on ...

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## Multi angle  Log-gases on a quadratic lattice via discrete loop equations Knizel, Alisa (Auteur de la Conférence) | CIRM (Editeur )

We study a general class of log-gas ensembles on a quadratic lattice. Using a variational principle we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance.
We apply our general results to analyze the asymptotic behavior of a q-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations of the height function on a fixed slice are described by a one-dimensional section of a pullback of the two-dimensional Gaussian free field.
Our approach is based on a q-analogue of the Schwinger-Dyson (or loop) equations, which originate in the work of Nekrasov and his collaborators, and extends the methods developed by Borodin, Gorin and Guionnet to a quadratic lattice.
Based on joint work with Evgeni Dimitrov
We study a general class of log-gas ensembles on a quadratic lattice. Using a variational principle we prove that the corresponding empirical measures satisfy a law of large numbers and that their global fluctuations are Gaussian with a universal covariance.
We apply our general results to analyze the asymptotic behavior of a q-boxed plane partition model introduced by Borodin, Gorin and Rains. In particular, we show that the global fluctuations ...

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## Multi angle  Blocks & gaps in the asymmetric simple exclusion process Tracy, Craig A. (Auteur de la Conférence) | CIRM (Editeur )

In earlier work (arXiv:1707.04927) the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time t a particle is at site x and is the beginning of a block of L consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\rightarrow\infty$) that a particle is the beginning of an L-block, given that it is at site x at time t. Using duality between occupied and unoccupied sites we obtain the analogous result for a gap of G unoccupied sites between the particle at x and the next one.
In earlier work (arXiv:1707.04927) the authors obtained formulas for the probability in the asymmetric simple exclusion process that at time t a particle is at site x and is the beginning of a block of L consecutive particles. Here we consider asymptotics. Specifically, for the KPZ regime with step initial condition, we determine the conditional probability (asymptotically as $t\rightarrow\infty$) that a particle is the beginning of an L-block, ...

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## Multi angle  Modelling of complex dynamical networks: from agent-based to continuum models Peurichard, Diane (Auteur de la Conférence) | CIRM (Editeur )

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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...

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## Multi angle  Steady states and long range correlations in driven systems - Lecture 1 Mukamel, David (Auteur de la Conférence) | CIRM (Editeur )

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 ...

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## Multi angle  Steady states and long range correlations in driven systems - Lecture 2 Mukamel, David (Auteur de la Conférence) | CIRM (Editeur )

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 ...

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## Multi angle  Collective dynamics in life sciences - Lecture 2. The Vicsek model as a paradigm for self-organization: from particles to fluid via kinetic descriptions Degond, Pierre (Auteur de la Conférence) | CIRM (Editeur )

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.

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## Multi angle  Collective dynamics in life sciences - Lecture 3. Phase transitions in the Vicsek model: mathematical analyses in the kinetic framework Degond, Pierre (Auteur de la Conférence) | CIRM (Editeur )

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.

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

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.

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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.

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## 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 ...

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## 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 ...

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## Multi angle  Macroscopic fluctuation theory. Lecture 2 Gabrielli, Davide (Auteur de la Conférence) | CIRM (Editeur )

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 ...

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