# J-PSI (Working group of young researchers: statistical physics and interactions)

Organizers : Grégoire Ferré (gregoire.ferre.AT.enpc.fr) and Julien Roussel (julien.roussel.AT.enpc.fr)

Our goal is to provide accessible PhD courses on mathematical statistical physics in a broad sense. The talks will be given by PhD students or young researchers, and aim at presenting a subject of general interest, such as models, methods of proof or numerical techniques. In particular, we want to foster the interactions between different but correlated disciplines, including stochastic processes, large deviations, numerical analysis or physics. This working group is supported by the SMAI via the BOUM grant, and by the IHP which accepted to host us.

The J-PSI working group is now over, many thanks to all participants.

# Past event

### Tuesday, 19th of June, Room Jacques-Louis Lions, Inria Paris

Convergence to equilibrium in statistical physics

We planned a one-day event to end the activity of this working group, including talks from confirmed researchers and a poster session. Lunch was provided.

Access: INRIA Paris, 2 rue Simone Iff, 75012, Building C, ground floor, room Jacques-Louis Lions.

### Schedule:

09:00-09:25  Welcome / Coffee

09:25-09:30  Introductory remarks: presentation of the J-PSI

We will introduce the concept of Poincaré and log-Sobolev inequalities, some applications to Gibbs measures and Markov processes, and some few connections with random matrix theory.

11:00-11:30  Coffee break

11:30-12:15  David Aristoff (Colorado State University) Weighted ensemble: a new particle method for variance reduction

Interacting particle methods are splitting methods that can be used to efficiently sample from high dimensional distributions. Such distributions can arise in stochastic molecular dynamics and other complex systems. We are interested in particle methods that do not change the underlying trajectory dynamics. Instead a population of trajectories evolves without bias, and periodically some “favorable” trajectories are copied while others are killed. This resampling procedure leads to unbiased estimators that can have much lower variance than naive Monte Carlo. We describe a new framework for minimizing variance based on a “weighted ensemble”, an evolving ensemble of particles and weights. This framework is different from the standard Feynman-Kac/Gibbs Boltzmann resampling. We show how coarse graining techniques can be used to guide the weighted ensemble to minimize variance in a near-optimal way. Potential applications include computation of rare event probabilities and large mean hitting times.

12:15-02:00  Provided lunch

Chinese hotpot: Chongqing, 145 Avenue Daumesnil, 75012 Paris

02:00-02:45 Pierre-André Zitt (LAMA, UPEM) Non reversible samplers

In order to get numerical samples from a target probability measure, the Monte Carlo Markov Chain method (MCMC) consists in devising a stochastic process that leaves the target measure invariant. It is now “common knowledge” that non-reversible processes often perform better at this task than reversible ones: we will give an overview of recent results on various non reversible samplers and give positive results and open questions regarding this claim.

02:45-03:15  Coffee break

03:15-04:00  Max Fathi (Institut de Mathématiques de Toulouse) Stein kernels and the CLT

Stein’s method is a set of techniques for estimating distances between probability measures via well-chosen integration by parts formulas. I will discuss a few ways of combining those techniques with ideas from functional analysis and PDEs, and present some applications to rates of convergence in the central limit theorem.

04:00-04:15  Discussion

04:15-05:00 Arnaud Guillin (Université Blaise Pascal) Kinetic Fokker-Planck equation and convergence to equilibrium

The goal is here to present two different approaches to study the long time behavior of the kinetic Fokker-Planck equation. The first one will be very analytic (modified entropy and adaptive log-Sobolev) whereas the second will be fundamentally more probabilistic (Wasserstein and coupling).

# Past sessions

## Assaf Shapira (LPSM, UPMC)

Kinetically constrained models and the bootstrap percolation

Kinetically constraint models are a family of interacting particle systems studied in order to model glassy materials. In particular, they can explain how time scales diverge in systems with a very simple disordered equilibrium. A closely related family of models is the bootstrap percolation, that could be used in order to understand many of the properties of kinetically constrained models. I will present both processes and the relation between them, and show some results on the divergence of time scales.

## Nathan Noiry (Modal’X, Paris Nanterre)

Sparse Random Covariance Matrices

One of the striking phenomena in random matrices theory is the universal behavior of the eigenvalues. Roughly speaking, the spectrum of large random matrices converge towards deterministic laws that only depend on the symmetries of the model. In the covariance setting, the limiting law was given by Marchenko and Pastur in 1967. I will give an idea of the proof based on the method of moments, which involves combinatorics on planar rooted trees. A possible generalization concerns the sparse setting where many entries of the matrices are set to zero. I will present a recent result concerning this model, which has some links with the spectrum of sparse random graphs.

## Manon Baudel (Cermics, ENPC)

Metastability and Spectral Theory for non-reversible Markov processes

Metastable systems are characterized by the existence of separated time scales and quasi-invariant subspaces. On short time scale, the system reaches a local equilibrium and stays in a limited part of the available phase space. On longer time scale, one will observe transitions from one local equilibrium to another one.
We show the connection between the dynamical behavior of non-reversible Markov processes and the existence of small eigenvalues of the corresponding generators.
Under a non-degeneracy condition, to each metastable set corresponds a real simple eigenvalue of the generator which is connected to the inverse of the mean exit time from this set up to small errors. The proofs rely on the approximation of the system by a process on a reduced state space.

## Qiming Du (LPSM, UPMC)

Coalescent tree typed measures in Feynman-Kac interacting particle systems

The Feynman-Kac model is the core part of the mathematical language to analysis the genetic typed algorithms such as Particle Filters (or Sequential Monte Carlo methods) and
Multilevel Splitting methods. In the first part, I will introduce some basic notations and results such as consistency and CLT theorem of the classic Feynman-Kac models, alongs with a toy example to illustrate the procedure of rare events estimation of Langevin processes by Multilevel splitting methods.
In the second part, I present some recent results in variance estimation of the Feynman-Kac models by tracing the genealogical information of the associated mean field particle systems, where the coalescent tree typed measure plays a key role in the whole story. Some numeric results will also be presented in the last part.

## Yann Chiffaudel (LPMA, Paris VII)

Diffusion in the model of mirrors

The 2D model of mirrors has been introduced 30 years ago, and yet few results are known up to now. I propose a generalization of this model in dimension higher than two, and I will present an analytical and numerical study of the model and a related relevant one. Key words: kinematic random walk, random environment, diffusion coefficient, permutation model, self-avoiding random walk, perturbative approach.

## Jérôme Tubiana (LPT, ENS)

Compositional representations in restricted Boltzmann machines: theory and applications

Restricted Boltzmann Machines (RBM) form a family of probability distributions simple yet powerful for modeling high-dimensional, complex data sets. Besides learning a generative model, they also extract features, producing a graded and distributed representation of data. However, not all variants of RBM perform equally well, and little theoretical arguments exist for these empirical observations. By analyzing an ensemble of RBMs with random weights using statistical physics tools, we characterize the structural conditions (statistics of weights, choice of non-linearity…) allowing the emergence of such efficient representations.

Lastly, we present a new application of RBMs: the analysis of protein sequences alignments. We show that RBMs extract high-order patterns of coevolution that arise from the structural and functional constraints of the protein family. These patterns can be recombined to generate artificial protein sequences with prescribed chemical properties.

## Nicolas Brosse (CMAP, École Polytechnique)

The Tamed Unadjusted Langevin Algorithm or how to calm down the impatient

In this talk, we consider the problem of sampling from a probability measure $\pi$ having a density on $\mathbb{R}^d$ proportional to $x\mapsto \mathrm{e}^{-U(x)}$. The Euler discretization of the Langevin stochastic differential equation (SDE) is known to be unstable, when the potential $U$ is superlinear. Based on previous works on the taming of superlinear drift coefficients for SDEs, we introduce the Tamed Unadjusted Langevin Algorithm (TULA) and obtain non-asymptotic bounds in $V$-total variation norm and Wasserstein distance of order $2$ between the iterates of TULA and  $\pi$, as well as weak error bounds. Some numerical experiments based on a double well potential and on the Ginzburg-Landau model illustrate our results.
This is joint work with Alain Durmus, Eric Moulines and Sotirios
Sabanis (arXiv:1710.05559).

## Raphael Ducatez (CEREMADE, University Paris Dauphine)

Product of random matrices and Anderson localization in dimension one

We present general results concerning products of random matrices. Under general hypotheses, it is possible to show that the norm behaves like a sum of independent and identically distributed random variables, and that we have a “strong law of large numbers” theorem, as well as a central limit theorem, or convergence towards a brownian motion.

We apply next these results to the one dimensional Anderson model that was introduced in physics in order to describe an electron in a conductor with impurities. The conductivity of the material depends on the particular features of the eigenvectors of the Hamiltonian that we can derive from products of random matrices.

### Tuesday, 16th of January, 3pm, Room 201, IHP

Explicit non equivalence of dynamical ensemble from non-ergodicity

The first part of the talk will be dedicated to a quick introduction to dynamical ensembles. Considering Markov processes, they are the ensemble of trajectories respecting various conditions. In particular, an equivalence has been proved at the level of large deviations between the ensemble of trajectories with a fixed value of some observable and the ensemble where only the mean value of this observable is fixed. (Chetrite, R. & Touchette, H. Nonequilibrium Markov Processes Conditioned on Large Deviations doi:10.1007/s00023-014-0375-8). Then, in a second part, I will speak of the conditions of validity of this equivalence, and focus on the physical implications of the non equivalence case, via the study of the fully connected Ising model.

## Upanshu Sharma (Cermics, ENPC)

Large deviations and Fokker-Planck equations: more than meets the eye

In recent years it has been shown that the variational-evolution structure of many well-known evolution equations (such as gradient flows) is intimately related to the large deviations of underlying particle systems. A natural question then would be, ‘Can we use this structure to study asymptotic problems arising in these equations?’. In this talk I will show that indeed this is the case for Fokker-Planck equations. These ideas will be illustrated by means of a classical example: the overdamped limit of the Langevin equation.

## Boris Nectoux (CERMICS,  ENPC)

Mean exit time from a domain in the small temperature regime

This talk is concerned with the obtention of sharp asymptotic estimates of the mean exit time from a domain in the small temperature regime for the overdamped Langevin process.

The first part of the talk is dedicated to the study of one dimensional examples for which the mean exit time from a domain can be computed  using the Dynkin’s formula and Laplace’s method. In higher dimensions, many techniques have been developped in the past few years to study mean exit time from a domain for elliptic diffusion processes (such as techniques from large deviations, potential theoretic approach, semi-classical analysis…).

In the second part of the talk,  we explain how the potential theoretic approach developped by Bovier et al. combined with leveling results obtained by M.V. Day can be used to get  sharp asymptotic equivalent in the small temperature regime of the mean exit time from a bounded domain with only one attractor for the overdamped Langevin process [1].

[1] B. N, Sharp estimate of the mean exit time of a bounded domain in the zero white noise limit, 2017.

## Simon Coste (LPMA, Paris VII)

The Alon-Friedman second eigenvalue theorem

We are interested in the spectrum of d-regular graphs when the size of the graph grows to infinity, and especially in the greatest non-trivial eigenvalue lambda_2. By the classical Alon-Boppana bound, lambda_2 is essentially greater than 2sqrt(d-1) ; finding d-regular graphs with the corresponding minimal eigenvalue had turned out to be very hard and explicit constructions of such graphs (called Ramanujan graphs) are rare. However, Alon conjectured in 1986 that most d-regular are nearly Ramanujan, meaning that the second nontrivial eigenvalue of a random d-regular graph with many vertices should be very close to 2sqrt(d-1). This was a challenging problem for more than twenty years, until the first proof by Friedman in 2005. We will present this result and a generalization to directed graphs.

## Paul Melotti (UPMC)

Dimer models and limit shapes

The dimer model has been introduced in the 60’s to study the absorption of diatomic molecules upon the surface of a cristal.It is also used to model valence shell electrons in some materials, or for paving problems.Throughout the years, this model has shown a surprisingly rich analytical and combinatorial structure. It provides an example of two dimensional “integrable” model: several quantities, such as n-points correlations or Gibbs measures, can be computed explicitly. In this talk, I will introduce some techniques for studying this model (Kasteleyn matrices, height function, variational principle) and show the phenomenon of “limit forms”, a famous example of which being the theorem of Arctic circle for the Aztec diamond.

## Nikolas Nüksen (Imperial College London)

Construction of optimal Langevin samplers

Drawing samples from a given probability distribution is an important problem in computational statistics and molecular dynamics. The talk will start with a characterisation of the class of diffusion processes that are ergodic with respect to a fixed probability measure, allowing to approach the sampling problem by using long-time averages. I will then introduce different criteria (spectral gap, asymptotic variance, large deviations,…) to compare different dynamics within this class (with the objective of reducing the computational cost). A particular example will be the comparison between reversible and nonreversible systems.

## Marielle Simon (MEPHYSTO, INRIA Lille)

Interpolation between standard and anomalous diffusion of energy

Over the last few years, anomalous behaviors have been observed for one-dimensional chains of oscillators. The rigorous derivation of such behaviors from deterministic systems of Newtonian particles is very challenging, due to the existence of conservation laws, which impose very poor ergodic properties to the dynamical system. A possible way out of this lack of ergodicity is to introduce stochastic models, in such a way that the qualitative behaviour of the system is not modified. One starts with a chain of oscillators with a Hamiltonian dynamics, and then adds a stochastic which keeps the fundamental conservation laws (energy, momentum and stretch, usually).
One may first investigate the macroscopic evolution of the fluctuation field (around equilibrium), associated to the conserved quantities.For the unpinned harmonic chain where the velocities of particles can randomly change sign (and therefore the only conserved quantities of the dynamics are the energy and the stretch), it is known that, under a diffusive space-time scaling, the energy profile evolves following a non-linear diffusive equation involving the stretch. Recently it has been shown that in the case of one-dimensional harmonic oscillators with noise that preserves the momentum, the scaling limit of the energy fluctuations is ruled by the fractional heat equation.Finally, one tries to understand the transition regime for the energy fluctuations. Let us consider the same harmonic Hamiltonian dynamics, but now perturbed by two stochastic noises: both perturbations conserve the energy, but only the first one preserves the momentum. If the second one is null, the momentum is conserved, the energy transport is superdiffusive and described by a Lévy process governed by a fractional Laplacian. Otherwise, the volume conservation is destroyed, and the energy normally diffuses. What happens when the intensity of the second noise vanishes with the size of the chain? In this case, we can show that the limit of the energy fluctuation field depends on the evanescent speed of the random perturbation, we recover the two very different regimes for the energy transport, and we prove the existence of a crossover between the normal diffusion regime and the fractional superdiffusion regime.

Large deviations for the empirical measure of Coulomb gas and random matrices

A Coulomb gas is a system of n interacting particles, which can be derived from the study of the spectrum of random matrices (GUE, Ginibre…). We will see how one can prove a large deviation principle for the empirical measure of theses gas, and we will investigate its consequences (limit of empirical measures, concentration, variational formula for the limit measure…).

## Julien Reygner (CERMICS, ENPC)

Large deviations for empirical measures

The first part of this talk will be dedicated to a quick introduction to the general theory of large deviations. Then, applications will be given to the study of empirical measures of large particle systems, either independent or weakly interacting. The purpose is to illustrate such notions as Sanov’s Theorem, or the Dawson-Gärtner theory for McKean-Vlasov particle systems.

## Romain Poncet (CMAP, Ecole Polytechnique)

Metropolis-Hastings algorithm and Langevin sampler for Gibbs measures

Cet exposé prend la forme d’un mini-cours de présentation des algorithmes d’échantillonnage de type Langevin. Ces méthodes permettent de simuler une chaîne de Markov ergodique par rapport à une mesure (connue à une constante multiplicative près). Ce sont typiquement des mesures de Gibbs. Cette chaîne de Markov permet de pouvoir estimer, par une méthode de Monte-Carlo, des quantités moyennes associées à ces mesure.Dans ce mini-cours, nous rappellerons les résultats élémentaires sur les chaînes de Markov afin de permettre à tout le monde de suivre la présentation. Nous présenterons ensuite l’algorithme de Metropolis-Hastings ainsi que des résultats fondamentaux sur la dynamique de Langevin afin de pouvoir décrire l’algorithme Metropolis-Adjusted Langevin Algorithm (MALA). Dans une dernière partie je présenterai une méthode de réduction de variance pour MALA, issue de mes travaux de thèse, exploitant une classe de dynamiques de Langevin dites non-réversibles.