Mathematics Meets Physics on Disordered Systems

25 avr. 2022, 08:00 → 6 mai 2022, 19:00 Europe/Paris

Systems of many degrees of freedom interacting in a random or heterogeneous way are paradigmatic examples where complex behavior emerges. Their study has grown enormously in the last 50 years starting from statistical physics and going across many different disciplines such as probability, applied mathematics, statistics, optimization theory as well as inference and machine learning. The strong interdisciplinarity of the field of disordered systems has recently boosted the research on this topic which is now seeing a flourishing of activities and results. The purpose of this workshop is to bring together mathematicians and physicists that work actively on disordered systems and their applications to present the main theoretical tools, concepts and the most recent advances to a target audience of students as well as postdocs and young researchers.

 

Organizers: E. Marinari (Università La Sapienza), G. Parisi (Università La Sapienza), F. Ricci-Tersenghi (Università La Sapienza), L. Sutera (Università La Sapienza), P. Urbani (Université Paris-Saclay)

Sponsors: We acknowledge funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No.694925, LoTGlasSy).

 

 

mar. 26 avril

1 Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Orateur: Prof. Slava Rychkov (IHES)

10:30 Break

2 Some intriguing maths in dynamical mean-field for glasses

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

Orateur: Prof. Jorge Kurchan (ENS Paris)

12:30 Lunch

3 The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models.
However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Orateur: Prof. Nicholas Read (Yale University)

15:00 Break

4 The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models.
However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Orateur: Prof. Nicholas Read (Yale University)

mer. 27 avril

5 Optimization and sampling from mean-field spin glass models

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures:

1- Optimization: Can we efficiently find ground state configurations whenever they exist?

I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case.

2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure?

I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature.

For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Orateur: Prof. Ahmed El Alaoui (Cornell University)

10:30 Break

6 Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Orateur: Prof. Slava Rychkov (IHES)

12:30 Lunch

7 Some intriguing maths in dynamical mean-field for glasses

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

Orateur: Prof. Jorge Kurchan (ENS Paris)

15:00 Break

8 The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models.
However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Orateur: Prof. Nicholas Read (Yale University)

jeu. 28 avril

9 Some intriguing maths in dynamical mean-field for glasses

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

Orateur: Prof. Jorge Kurchan (ENS Paris)

10:30 Break

10 Optimization and sampling from mean-field spin glass models

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures:

1- Optimization: Can we efficiently find ground state configurations whenever they exist?

I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case.

2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure?

I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature.

For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Orateur: Prof. Ahmed El Alaoui (Cornell University)

12:30 Lunch

11 Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Orateur: Prof. Slava Rychkov (IHES)

15:00 Break

12 The Metastate interpretation of replica symmetry breaking in short range Ising spin glasses

The mean field analysis of Ising spin glasses shows that at low temperature, these system display a non ergodic phase characterized by an exponential number of metastable states which is captured by the so called replica symmetry breaking (RSB) solution of the models.
However when this mean field theory picture is applied to finite dimensional systems, it runs into troubles. In these lectures a way out to this conundrum will be discussed through the introduction of the concept of metastate, a probability measure over infinite-size Gibbs states in a given disorder realization, in conjunction with a non-standard interpretation of the RSB results.

Orateur: Prof. Nicholas Read (Yale University)

ven. 29 avril

13 Optimization and sampling from mean-field spin glass models

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures:

1- Optimization: Can we efficiently find ground state configurations whenever they exist?

I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case.

2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure?

I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature.

For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Orateur: Prof. Ahmed El Alaoui (Cornell University)

10:30 Break

14 Random Field Ising Model and Parisi-Sourlas Supersymmetry - some recent developments

Numerical simulations of Random Field Ising Model suggest that Parisi-Sourlas supersymmetry and dimensional reduction are present for spatial dimension d>d_c and are lost for d<d_c, where the critical dimension d_c is somewhere between 4 and 5. I will lecture about a recent theoretical framework, based on perturbative renormalization group, which predicts a mechanism for the loss of supersymmetry and allows to compute the value of d_c. Alternative scenarios will also be discussed.

Orateur: Prof. Slava Rychkov (IHES)

12:30 Lunch

15 Optimization and sampling from mean-field spin glass models

These lectures will cover two main topics relating to some computational aspects of mean-field spin glass Gibbs measures:

1- Optimization: Can we efficiently find ground state configurations whenever they exist?

I will introduce the framework of Incremental Approximate Message Passing and the associated optimal control problem. This will be used to compute near optimal ground state configurations in models presenting full replica symmetry breaking, as well as to obtain good lower bounds on the (free) energy in the general case.

2- Sampling: Can we efficiently produce an approximate sample from the associated Gibbs measure?

I will introduce Eldan's Stochastic Localization process, which is a way of decomposing any probability measure into a low entropy mixture of increasingly concentrated components, and use it to design an algorithm for sampling from the Gibbs measure at high-temperature.

For both questions, we will highlight the limitations of these approaches, and show that whenever they fail, a host of other algorithms must fail as well. We will use the SK model as our main playground.

Orateur: Prof. Ahmed El Alaoui (Cornell University)

15:30 Break

16 Some intriguing maths in dynamical mean-field for glasses

The solution of the dynamics of glasses at the mean-field level - for example in the limit of high dimension -- has a few surprises. The equations possess an emergent reparametrization invariance. The supersymmetry associated with thermal equilibrium is broken, but a new one arises ‘from nowhere’. These two intriguing mathematical features dominate the glassy phenomenology.

Orateur: Prof. Jorge Kurchan (ENS Paris)

sam. 30 avril

17 Statistical physics analysis of artificial neural networks

The scope of these lectures will be to discuss the statistical physics approach to the phase diagram and landscape of machine learning problems focusing on the loss landscape of artificial neural networks. We will show how this analysis suggest new powerful training algorithms based on the role of wide flat minima of the corresponding cost functions.

Orateur: Prof. Riccardo Zecchina (Bocconi University)

10:30 Break

18 Statistical physics analysis of artificial neural networks

The scope of these lectures will be to discuss the statistical physics approach to the phase diagram and landscape of machine learning problems focusing on the loss landscape of artificial neural networks. We will show how this analysis suggest new powerful training algorithms based on the role of wide flat minima of the corresponding cost functions.

Orateur: Prof. Riccardo Zecchina (Bocconi University)

lun. 2 mai

19 On the solution of the SK model

Orateur: Prof. Giorgio Parisi (Università di Roma La Sapienza)

10:30 Break

20 Statistical physics of inference and learning: from algorithmic strategies to proofs.

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Orateur: Prof. Florent Krzakala (EPFL)

12:30 Lunch

21 Statistical physics of inference and learning: from algorithmic strategies to proofs.

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Orateur: Prof. Florent Krzakala (EPFL)

15:00 Break

22 Rigorous analysis of pure states of spin glass models on sparse random graphs

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs).
Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Orateur: Prof. Amin Coja-Oghlan (Goethe University Frankfurt)

talk_cortona1.pdf

mar. 3 mai

23 Rigorous analysis of the complexity of the landscape of spin glass models

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent).
The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

Orateur: Prof. Gerard Ben Arous (New York University)

10:30 Break

24 Statistical physics of inference and learning: from algorithmic strategies to proofs.

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Orateur: Prof. Florent Krzakala (EPFL)

12:30 Lunch

25 Statistical physics of inference and learning: from algorithmic strategies to proofs.

The scope of these lectures is to discuss the statistical physics approach to high dimensional inference and learning. In these problems one seeks for a particular configuration of some variables (the signal) which is hidden in a rough energy landscape of spurious non-informative minima. We will focus on two aspects of these problems: on the one hand we will show how statistical physics provides powerful algorithms to reconstruct the signal from some measurements. On the other hand we will show how the performances of such algorithms can be analyzed on a rigorous basis and compared to the information theoretic limits that can be proven using the techniques developed to analyze rigorously mean field spin glasses.

Orateur: Prof. Florent Krzakala (EPFL)

15:00 Break

26 The low temperature properties of the Gibbs measure in mean field spin glasses

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Orateur: Prof. Antonio Auffinger (Northwestern University)

mer. 4 mai

27 Rigorous analysis of pure states of spin glass models on sparse random graphs

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs).
Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Orateur: Prof. Amin Coja-Oghlan (Goethe University Frankfurt)

talk_cortona2.pdf

10:30 Break

28 Rigorous analysis of the complexity of the landscape of spin glass models

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent).
The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

Orateur: Prof. Gerard Ben Arous (New York University)

12:30 Lunch

29 The low temperature properties of the Gibbs measure in mean field spin glasses

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Orateur: Prof. Antonio Auffinger (Northwestern University)

15:00 Break

30 The low temperature properties of the Gibbs measure in mean field spin glasses

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Orateur: Prof. Antonio Auffinger (Northwestern University)

jeu. 5 mai

31 Rigorous analysis of the complexity of the landscape of spin glass models

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent).
The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

Orateur: Prof. Gerard Ben Arous (New York University)

10:30 Break

32 Rigorous analysis of pure states of spin glass models on sparse random graphs

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs).
Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Orateur: Prof. Amin Coja-Oghlan (Goethe University Frankfurt)

talk_cortona3.pdf

12:30 Lunch

33 Rigorous analysis of the complexity of the landscape of spin glass models

The energy landscape of mean field spin glasses displays a large (exponential in the dimension of the problem) number of minima and saddles. The geometrical properties of such random functions in high dimensions are of fundamental importance to understand the behavior of local algorithms that try to find optima in such landscapes (for example gradient descent).
The aim of these lectures we will be to discuss how these properties can be analyzed adapting the Kac-Rice formula to disordered systems and how this brings Random Matrix Theory in the problem.

Orateur: Prof. Gerard Ben Arous (New York University)

15:00 Break

34 The low temperature properties of the Gibbs measure in mean field spin glasses

Spin glasses at low temperature, display a non-ergodic phase characterized by an exponential number of pure states. The properties of the corresponding Gibbs measure have been characterized through the heuristic replica method culminated in the celebrated Parisi formula for low temperature spin glasses. The scope of these lectures will be to describe and analyze rigorously the properties of the Parisi formula for several spin glass models.

Orateur: Prof. Antonio Auffinger (Northwestern University)

ven. 6 mai

35 Rigorous analysis of pure states of spin glass models on sparse random graphs

Disordered models on Bethe lattices, i.e., the random d-regular graph, emerge naturally both in the context of the mean field treatment of spin glasses as well as in computer science where they arise as generic random instances of constraint satisfaction problems (CSPs).
Their analysis has been performed in the physics literature through the non-rigorous cavity method technique developed in the context of statistical physics of spin glasses. These lectures are devoted to show how this method can be turned into a rigorous probabilistic tool to analyze the Gibbs measure of such models and how this can be used to establish a series of rigorous results on prototypical CSPs.

Orateur: Prof. Amin Coja-Oghlan (Goethe University Frankfurt)

talk_cortona4.pdf

10:30 Break

36 TBA - Silvio Franz

TBA

Orateur: Prof. Silvio Franz (Université Paris-Saclay, LPTMS)

12:30 Lunch - end of the program