research

Recorded talks (selected)

• A PDE approach to mean-field disordered systems (24h lecture series). CRM-PIMS summer school in probability (2022). [abs] [links]

  The goal of statistical mechanics is to describe the large-scale behavior of collections of simple elements, often called spins, that interact through locally simple rules and are influenced by some amount of noise. We will focus on the situation where the local interactions are chosen at random, in which case the models are usually called "spin glasses". Such models are already surprisingly difficult to analyze when all spins interact with each other. In this course, we will revisit this analysis using tools from the theory of Hamilton-Jacobi equations.
  After an overview of the material covered in the course, we will start by focusing on the very simple Curie-Weiss model. Along the way, we will introduce analytical techniques related to the study of Hamilton-Jacobi equations, and use them to identify the limit free energy of the model. We will next transpose this strategy into a first class of disordered models coming from statistical inference. In terms of difficulty, these models are a useful bridge between the Curie-Weiss model and spin glasses. We will finally turn our attention to the latter models, in which infinite-dimensional Hamilton-Jacobi equations arise.
  
  Lecture videos
  Course materials
  Typed notes

• Quantitative homogenization of interacting particle systems. Webinar on stochastic analysis (2022). [abs] [link]

  I discuss a class of interacting particle systems in continuous space. Such models are known to "homogenize", in the sense that the behavior of the cloud of particles is approximately described by a partial differential equation over large scales. In the talk, I describe a first step towards making this result quantitative. The approach is inspired by recent developments in the homogenization of elliptic equations with random coefficients. Joint work with Arianna Giunti and Chenlin Gu.

• Rank-one matrix estimation and Hamilton-Jacobi equations (3h lecture series). Open online probability school (2020). [abs] [links]

  The main goal of the lectures is to present an approach to the study of disordered systems based on Hamilton-Jacobi equations. After giving some context and motivation, we carefully study the simple Curie-Weiss model using this approach. In the last lecture, we turn to the problem of estimating a large rank-one matrix, given noisy observations. This inference problem is known to have a phase transition, in the sense that the partial recovery of the original matrix is only possible if the signal-to-noise ratio exceeds a (non-zero) value. We discuss how the same approach allows to solve this problem as well.
  Lecture 1: [Youtube] [BIRS] [notes]
  Lecture 2: [Youtube] [BIRS] [notes]
  Lecture 3: [Youtube] [BIRS] [notes]
  Typed notes by Holden Lee [tex] [pdf]

• Mean-field disordered systems and Hamilton-Jacobi equations. One World probability seminar (2020). [abs] [link]

  I describe a connection between the limit free energy of mean-field disordered systems and certain Hamilton-Jacobi equations.

• Homogénéisation stochastique. Institut Henri Poincaré (2018). [abs] [link]

  An introduction to the topic of homogenization of divergence-form operators with random coefficients (in french).

• Describing the fluctuations in stochastic homogenisation. Banff International research station (2015). [abs] [link]

  Consider the solution of a divergence-form problem with random coefficients. Under suitable assumptions on the law of the coefficients, homogenisation theory ensures that as the correlation length of the random coefficients is sent to 0, this solution converges to the solution of a similar problem with constant, ”homogenised” coefficients. The problem of quantifying the error in this convergence has witnessed tremendous progress recently. The goal of this talk is to explain how one can go beyond error bounds, and describe precisely the statistics of the fluctuations in terms of a finite number of new effective parameters.

• The dynamic 𝜑⁴ model in the plane. Newton Institute (2015). [abs] [link]

  The dynamic 𝜑⁴₃model is a non-linear stochastic PDE which involves a cubic power of the solution. In dimensions 3 and less, solutions are expected to have the same local regularity as the linearised equation, for which the law of the Gaussian free field is invariant. Hence, in dimensions 2 and 3, some renormalisation needs to be performed in order to define the cubic power of the solution. In the (full) plane, I will explain how to do this and show that the stochastic PDE has a well-defined solution for all times. If time permits, I will also sketch a proof that the model is the scaling limit of a near-critical Ising model with long-range interactions. Joint work with Hendrik Weber.

Books

• (with Tomás Dominguez) Statistical mechanics of mean-field disordered systems: a Hamilton-Jacobi approach. [abs] [pdf]

  The goal of this book is to present new mathematical techniques for studying the behaviour of mean-field systems with disordered interactions. We mostly focus on certain problems of statistical inference in high dimension, and on spin glasses. The techniques we present aim to determine the free energy of these systems, in the limit of large system size, by showing that they asymptotically satisfy a Hamilton-Jacobi equation.
  The first chapter is a general introduction to statistical mechanics, discussing the concept of a Gibbs measure and the phenomenology of the Ising and Curie-Weiss models. Although non-disordered, the Curie-Weiss model will serve as a useful training ground to develop the Hamilton-Jacobi approach. We give a brief introduction to convex analysis and large deviation principles in Chapter 2, and identify the limit free energy of the Curie-Weiss model using these tools. In Chapter 3, we define the notion of viscosity solution to a Hamilton-Jacobi equation, and use it to recover the limit free energy of the Curie-Weiss model. We discover technical challenges to applying the same method to generalized versions of the Curie-Weiss model, and develop a new selection principle based on convexity to overcome these. We then turn to statistical inference in Chapter 4, focusing on the problem of recovering a large symmetric rank-one matrix from a noisy observation, and we see that the tools developed in the previous chapter apply to this setting as well. Chapter 5 is preparatory work for a discussion of the more challenging case of spin glasses. The first half of this chapter is a self-contained introduction to Poisson point processes, including limit theorems on extreme values of independent and identically distributed random variables, which we believe to be of independent interest. We finally turn to the setting of spin glasses in Chapter 6. For the Sherrington-Kirkpatrick model, we show how to relate the Parisi formula with the Hamilton-Jacobi approach. We conclude with a more informal discussion on the status of current research for more challenging models. A self-contained appendix collects a number of classical results in analysis and probability used throughout the book. Solutions to the exercises are also provided.
  We tried to make the book as self-contained as possible. In particular, no prior knowledge of partial differential equations is assumed.

• (with Scott Armstrong and Tuomo Kuusi) Quantitative stochastic homogenization and large-scale regularity. Grund. math. Wiss. vol. 352 (2019). [abs] [pdf]

  The focus of this book is the large-scale statistical behavior of solutions of divergence-form elliptic equations with random coefficients, which is closely related to the long-time asymptotics of reversible diffusions in random media and other basic models of statistical physics. Of particular interest is the quantification of the rate at which solutions converge to those of the limiting, homogenized equation in the regime of large scale separation, and the description of their fluctuations around this limit. This self-contained presentation gives a complete account of the essential ideas and fundamental results of this new theory of quantitative stochastic homogenization, including the latest research on the topic, and is supplemented with many new results. The book serves as an introduction to the subject for advanced graduate students and researchers working in partial differential equations, statistical physics, probability and related fields, as well as a comprehensive reference for experts in homogenization. Being the first text concerned primarily with stochastic (as opposed to periodic) homogenization and which focuses on quantitative results, its perspective and approach are entirely different from other books in the literature.

Expository papers

• An informal introduction to quantitative stochastic homogenization. J. Math. Phys. 60, 031506 (2019). [abs] [tex] [pdf]

  Divergence-form operators with random coefficients homogenize over large scales. Over the last decade, an intensive research effort focused on turning this asymptotic statement into quantitative estimates. The goal of this note is to review one approach for doing so based on the idea of renormalization. The discussion is highly informal, with pointers to mathematically precise statements.

• (with Hendrik Weber and Weijun Xu) Construction of 𝜑⁴₃diagrams for pedestrians. In From particle systems to partial differential equations IV, 1-46 (2017). [abs] [tex] [pdf]

  We aim to give a pedagogic and essentially self-contained presentation of the construction of various stochastic objects appearing in the dynamical 𝜑⁴₃model. The construction presented here is based on the use of paraproducts. The emphasis is on describing the stochastic objects themselves rather than introducing a solution theory for the equation.

Research papers

• (with Chenlin Gu and Maximilian Nitzschner) Quantitative equilibrium fluctuations for interacting particle systems. [abs] [tex] [pdf]

  We consider a class of interacting particle systems in continuous space of non-gradient type, which are reversible with respect to Poisson point processes with constant density. For these models, a rate of convergence was recently obtained for certain finite-volume approximations of the bulk diffusion matrix. Here, we show how to leverage this to obtain quantitative versions of a number of results capturing the large-scale fluctuations of these systems, such as the convergence of two-point correlation functions and the Green--Kubo formula.

• (with Hong-Bin Chen) On the free energy of vector spin glasses with non-convex interactions. [abs] [tex] [pdf]

  The limit free energy of spin-glass models with convex interactions can be represented as a variational problem involving an explicit functional. Models with non-convex interactions are much less well-understood, and simple variational formulas involving the same functional are known to be invalid in general. We show here that a slightly weaker property of the limit free energy does extend to non-convex models. Indeed, under the assumption that the limit free energy exists, we show that this limit can always be represented as a critical value of the said functional. Up to a small perturbation of the parameters defining the model, we also show that any subsequential limit of the law of the overlap matrix is a critical point of this functional. We believe that these results capture the fundamental conclusions of the non-rigorous replica method.

• Un-inverting the Parisi formula. Ann. Inst. Henri Poincaré Probab. Stat., to appear. [abs] [tex] [pdf]

  The free energy of any system can be written as the supremum of a functional involving an energy term and an entropy term. Surprisingly, the limit free energy of mean-field spin glasses is expressed as an infimum instead, a phenomenon sometimes called an inverted variational principle. Using a stochastic-control representation of the Parisi functional and convex duality arguments, we rewrite this limit free energy as a supremum over martingales in a Wiener space.

• (with Anastasia Kireeva) Breakdown of a concavity property of mutual information for non-Gaussian channels. Inf. Inference, to appear. [abs] [tex] [pdf]

  Let S and S̃ be two independent and identically distributed random variables, which we interpret as the signal, and let P₁ and P₂ be two communication channels. We can choose between two measurement scenarios: either we observe S through P₁ and P₂, and also S̃ through P₁ and P₂; or we observe S twice through P₁, and S̃ twice through P₂. In which of these two scenarios do we obtain the most information on the signal (S, S̃)? While the first scenario always yields more information when P₁ and P₂ are additive Gaussian channels, we give examples showing that this property does not extend to arbitrary channels. As a consequence of this result, we show that the continuous-time mutual information arising in the setting of community detection on sparse stochastic block models is not concave, even in the limit of large system size. This stands in contrast to the case of models with diverging average degree, and brings additional challenges to the analysis of the asymptotic behavior of this quantity.

• (with Tomás Dominguez) Mutual information for the sparse stochastic block model. Ann. Probab. 52 (2), 434-501 (2024). [abs] [tex] [pdf]

  We consider the problem of recovering the community structure in the stochastic block model with two communities. We aim to describe the mutual information between the observed network and the actual community structure in the sparse regime, where the total number of nodes diverges while the average degree of a given node remains bounded. Our main contributions are a conjecture for the limit of this quantity, which we express in terms of a Hamilton-Jacobi equation posed over a space of probability measures, and a proof that this conjectured limit provides a lower bound for the asymptotic mutual information. The well-posedness of the Hamilton-Jacobi equation is obtained in our companion paper. In the case when links across communities are more likely than links within communities, the asymptotic mutual information is known to be given by a variational formula. We also show that our conjectured limit coincides with this formula in this case.

• (with Tomás Dominguez) Infinite-dimensional Hamilton-Jacobi equations for statistical inference on sparse graphs. SIAM J. Math. Anal., to appear. [abs] [tex] [pdf]

  We study the well-posedness of an infinite-dimensional Hamilton-Jacobi equation posed on the set of non-negative measures and with a monotonic non-linearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton-Jacobi equations, and use the monotonicity of the non-linearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton-Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex non-linearity, we also provide a Hopf-Lax variational representation of the solution.

• (with Arianna Giunti, Chenlin Gu, and Maximilian Nitzschner) Smoothness of the diffusion coefficients for particle systems in continuous space. Comm. Contemp. Math. 25 (3), no. 2250027, 1-60 (2023). [abs] [tex] [pdf]

  For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively explicit descriptions of the derivatives of the diffusion matrix in terms of the corrector.

• (with Alex Dunlap) Local versions of sum-of-norms clustering. SIAM J. Math. Data Sci. 4 (4), 1250-1271 (2022). [abs] [tex] [pdf]

  Sum-of-norms clustering is a convex optimization problem whose solution can be used for the clustering of multivariate data. We propose and study a localized version of this method, and show in particular that it can separate arbitrarily close balls in the stochastic ball model. More precisely, we prove a quantitative bound on the error incurred in the clustering of disjoint connected sets. Our bound is expressed in terms of the number of datapoints and the localization length of the functional.

• (with Gabriel Brito Apolinario and Laurent Chevillard) Dynamical fractional and multifractal fields. J. Stat. Phys. 186 (1), no. 15, 1-35 (2022). [abs] [tex] [pdf]

  Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To reproduce the fractional, and more specifically multifractal, regularity nature of fully developed turbulence, these dynamical evolutions incorporate an homogenous pseudo-differential linear operator of degree 0 that takes care of transferring energy that is injected at large scales in the system, towards smaller scales according to a cascading mechanism. In the simplest situation which concerns the development of fractional regularity in a linear and Gaussian framework, we derive explicit predictions for the statistical behaviors of the solution at finite and infinite time. Doing so, we realize a cascading transfer of energy using linear, although non local, interactions. These evolutions can be seen as a stochastic version of recently proposed systems of forced waves intended to model the regime of weak wave turbulence in stratified and rotational flows. To include multifractal, i.e. intermittent, corrections to this picture, we get some inspiration from the Gaussian multiplicative chaos, which is known to be multifractal, to motivate the introduction of an additional quadratic interaction in these dynamical evolutions. Because the theoretical analysis of the obtained class of nonlinear SPDEs is much more demanding, we perform numerical simulations and observe the non-Gaussian and in particular skewed nature of their solution.

• (with Alex Dunlap) Sum-of-norms clustering does not separate nearby balls. J. Mach. Learn. Res. 25, 1-40 (2024). [abs] [tex] [pdf]

  Sum-of-norms clustering is a popular convexification of K-means clustering. We show that, if the dataset is made of a large number of independent random variables distributed according to the uniform measure on the union of two disjoint balls of unit radius, and if the balls are sufficiently close to one another, then sum-of-norms clustering will typically fail to recover the decomposition of the dataset into two clusters. As the dimension tends to infinity, this happens even when the distance between the centers of the two balls is taken to be as large as 2√2. In order to show this, we introduce and analyze a continuous version of sum-of-norms clustering, where the dataset is replaced by a general measure. In particular, we state and prove a local-global characterization of the clustering that seems to be new even in the case of discrete datapoints.

• (with Hong-Bin Chen and Jiaming Xia) Statistical inference of finite-rank tensors. Ann. Henri Lebesgue 5, 1161-1189 (2022). [abs] [tex] [pdf]

  We consider a general statistical inference model of finite-rank tensor products. For any interaction structure and any order of tensor products, we identify the limit free energy of the model in terms of a variational formula. Our approach consists of showing first that the limit free energy must be the viscosity solution to a certain Hamilton-Jacobi equation.

• (with Arianna Giunti and Chenlin Gu) Quantitative homogenization of interacting particle systems. Ann. Probab. 50 (5), 1885-1946 (2022). [abs] [tex] [pdf]

  For a class of interacting particle systems in continuous space, we show that finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. Our approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, we develop suitable modifications of the Caccioppoli and multiscale Poincaré inequalities, which are of independent interest.

• Free energy upper bound for mean-field vector spin glasses. Ann. Inst. Henri Poincaré Probab. Stat. 59 (3), 1143-1182 (2023). [abs] [tex] [pdf]

  We consider vector spin glasses whose energy function is a Gaussian random field with covariance given in terms of the matrix of scalar products. For essentially any model in this class, we give an upper bound for the limit free energy, which is expected to be sharp. The bound is expressed in terms of an infinite-dimensional Hamilton-Jacobi equation.

• Nonconvex interactions in mean-field spin glasses. Probab. Math. Phys. 2 (2), 281-339 (2021). [abs] [tex] [pdf]

  We propose a conjecture for the limit of mean-field spin glasses with a bipartite structure, and show that the conjectured limit is an upper bound. The conjectured limit is described in terms of the solution of an infinite-dimensional Hamilton-Jacobi equation. A fundamental difficulty of the problem is that the nonlinearity in this equation is not convex. We also question the possibility to characterize this conjectured limit in terms of a saddle-point problem.

• (with Dmitry Panchenko) Extending the Parisi formula along a Hamilton-Jacobi equation. Electron. J. Probab. 25, no. 23, 1-17 (2020). [abs] [tex] [pdf]

  We study the free energy of mixed p-spin spin glass models enriched with an additional magnetic field given by the canonical Gaussian field associated with a Ruelle probability cascade. We prove that this free energy converges to the Hopf-Lax solution of a certain Hamilton-Jacobi equation. Using this result, we give a new representation of the free energy of mixed p-spin models with soft spins.

• The Parisi formula is a Hamilton-Jacobi equation in Wasserstein space. Canad. J. Math. 74 (3), 607-629 (2022). [abs] [tex] [pdf]

  Parisi's formula is a self-contained description of the infinite-volume limit of the free energy of mean-field spin glass models. We show that this quantity can be recast as the solution of a Hamilton-Jacobi equation in the Wasserstein space of probability measures on the positive half-line.

• (with Antti Hannukainen and Harmen T. Stoppels) Computing homogenized coefficients via multiscale representation and hierarchical hybrid grids. ESAIM Math. Model. Numer. Anal. 55, S149-S185 (2021). [abs] [tex] [pdf] [github]

  We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement the method numerically using a finite-element method with hierarchical hybrid grids, which is a semi-implicit method allowing for significant gains in memory usage and execution time. Finally, we demonstrate the efficiency of our approach on two- and three-dimensional examples, for piecewise-constant coefficients with corner discontinuities. For moderate ellipticity contrast and for a precision of a few percentage points, our method allows to compute the homogenized coefficients on a laptop computer in a few seconds, in two dimensions, or in a few minutes, in three dimensions.

• Hamilton-Jacobi equations for finite-rank matrix inference. Ann. Appl. Probab. 30 (5), 2234-2260 (2020). [abs] [tex] [pdf]

  We compute the large-scale limit of the free energy associated with the problem of inference of a finite-rank matrix. The method follows the principle put forward in previous work which consists in identifying a suitable Hamilton-Jacobi equation satisfied by the limit free energy. We simplify the approach of previous work using a notion of weak solution of the Hamilton-Jacobi equation which is more convenient to work with and is applicable whenever the non-linearity in the equation is convex.

• (with Dallas Albritton, Scott Armstrong, and Matthew Novack) Variational methods for the kinetic Fokker-Planck equation. Anal. PDE, to appear. [abs] [tex] [pdf]

  We develop a functional analytic approach to the study of the Kramers and kinetic Fokker-Planck equations which parallels the classical H¹ theory of uniformly elliptic equations. In particular, we identify a function space analogous to H¹ and develop a well-posedness theory for weak solutions in this space. In the case of a conservative force, we identify the weak solution as the minimizer of a uniformly convex functional. We prove new functional inequalities of Poincaré and Hörmander type and combine them with basic energy estimates (analogous to the Caccioppoli inequality) in an iteration procedure to obtain the C^∞ regularity of weak solutions. We also use the Poincaré-type inequality to give an elementary proof of the exponential convergence to equilibrium for solutions of the kinetic Fokker-Planck equation which mirrors the classic dissipative estimate for the heat equation. Finally, we prove enhanced dissipation in a weakly collisional limit.

• Hamilton-Jacobi equations for mean-field disordered systems. Ann. Henri Lebesgue 4, 453-484 (2021). [abs] [tex] [pdf]

  We argue that Hamilton-Jacobi equations provide a convenient and intuitive approach for studying the large-scale behavior of mean-field disordered systems. This point of view is illustrated on the problem of inference of a rank-one matrix. We compute the large-scale limit of the free energy by showing that it satisfies an approximate Hamilton-Jacobi equation with asymptotically vanishing viscosity parameter and error term.

• (with Scott Armstrong, Tuomo Kuusi, and Antti Hannukainen) An iterative method for elliptic problems with rapidly oscillating coefficients. ESAIM Math. Model. Numer. Anal. 55 (1), 37-55 (2021). [abs] [tex] [pdf]

  We introduce a new iterative method for computing solutions of elliptic equations with random rapidly oscillating coefficients. Similarly to a multigrid method, each step of the iteration involves different computations meant to address different length scales. However, we use here the homogenized equation on all scales larger than a fixed multiple of the scale of oscillation of the coefficients. While the performance of standard multigrid methods degrades rapidly under the regime of large scale separation that we consider here, we show an explicit estimate on the contraction factor of our method which is independent of the size of the domain. We also present numerical experiments which confirm the effectiveness of the method.

• (with Scott Armstrong and Alexandre Bordas) Quantitative stochastic homogenization and regularity theory of parabolic equations. Anal. PDE 11 (8), 1945-2014 (2018). [abs] [tex] [pdf]

  We develop a quantitative theory of stochastic homogenization for linear, uniformly parabolic equations with coefficients depending on space and time. Inspired by recent works in the elliptic setting, our analysis is focused on certain subadditive quantities derived from a variational interpretation of parabolic equations. These subadditive quantities are intimately connected to spatial averages of the fluxes and gradients of solutions. We implement a renormalization-type scheme to obtain an algebraic rate for their convergence, which is essentially a quantification of the weak convergence of the gradients and fluxes of solutions to their homogenized limits. As a consequence, we obtain estimates of the homogenization error for the Cauchy-Dirichlet problem which are optimal in stochastic integrability. We also develop a higher regularity theory for solutions of the heterogeneous equation, including a uniform C^0,1-type estimate and a Liouville theorem of every finite order.

• Efficient methods for the estimation of homogenized coefficients. Found. Comp. Math. 19 (2), 435-483 (2019). [abs] [tex] [pdf]

  We introduce new methods to compute the homogenized coefficients of divergence-form operators with random coefficients. We focus on a discrete-space setting with i.i.d. coefficients, and investigate algorithms which take a sample of the random coefficient field as input. In order to produce an approximation of the homogenized coefficients at precision δ, any algorithm must perform at least of the order of δ^-2 operations. We present an algorithm that essentially achieves this lower bound, up to logarithmic factors. This improves upon the previously best known method by a factor of δ^-1/d. An additional new feature is that the method is cumulative: all computations done at a coarse precision remain useful if the estimate needs to be refined.

• (with Arianna Giunti and Yu Gu) Heat kernel upper bounds for interacting particle systems. Ann. Probab. 47 (2), 1056-1095 (2019). [abs] [tex] [pdf]

  We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

• (with Scott Armstrong, Tuomo Kuusi, and Christophe Prange) Quantitative analysis of boundary layers in periodic homogenization. Arch. Ration. Mech. Anal. 226 (2), 695-741 (2017). [abs] [tex] [pdf]

  We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three (at least) and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.

• (with Daniel Valesin) Spatial Gibbs random graphs. Ann. Appl. Probab. 28 (2), 751-789 (2018). [abs] [tex] [pdf]

  Many real-world networks of interest are embedded in physical space. We present a new random graph model aiming to reflect the interplay between the geometries of the graph and of the underlying space. The model favors configurations with small average graph distance between vertices, but adding an edge comes at a cost measured according to the geometry of the ambient physical space. In most cases, we identify the order of magnitude of the average graph distance as a function of the parameters of the model. As the proofs reveal, hierarchical structures naturally emerge from our simple modeling assumptions. Moreover, a critical regime exhibits an infinite number of discontinuous phase transitions.

• (with Arianna Giunti) Quantitative homogenization of degenerate random environments. Ann. Inst. Henri Poincaré Probab. Stat. 54 (1), 22-50 (2018). [abs] [tex] [pdf]

  We study discrete linear divergence-form operators with random coefficients, also known as the random conductance model. We assume that the conductances are bounded, independent and stationary; the law of a conductance may depend on the orientation of the associated edge. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive, in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector.

• (with Scott Armstrong and Tuomo Kuusi) The additive structure of elliptic homogenization. Invent. Math. 208 (3), 999-1154 (2017). [abs] [tex] [pdf]

  One of the principal difficulties in stochastic homogenization is transferring quantitative ergodic information from the coefficients to the solutions, since the latter are nonlocal functions of the former. In this paper, we address this problem in a new way, in the context of linear elliptic equations in divergence form, by showing that certain quantities associated to the energy density of solutions are essentially additive. As a result, we are able to prove quantitative estimates on the weak convergence of the gradients, fluxes and energy densities of the first-order correctors (under blow-down) which are optimal in both scaling and stochastic integrability. The proof of the additivity is a bootstrap argument, completing the program initiated in our previous work: using the regularity theory recently developed for stochastic homogenization, we reduce the error in additivity as we pass to larger and larger length scales. In the second part of the paper, we use the additivity to derive central limit theorems for these quantities by a reduction to sums of independent random variables. In particular, we prove that the first-order correctors converge, in the large-scale limit, to a variant of the Gaussian free field.

• (with Yu Gu) On generalized Gaussian free fields and stochastic homogenization. Electron. J. Probab. 22, no. 28, 1-21 (2017). [abs] [tex] [pdf]

  We study a generalization of the notion of Gaussian free field (GFF). Although the extension seems minor, we first show that a generalized GFF does not satisfy the spatial Markov property, unless it is a classical GFF. In stochastic homogenization, the scaling limit of the corrector is a possibly generalized GFF described in terms of an "effective fluctuation tensor" that we denote by Q. We prove an expansion of Q in the regime of small ellipticity ratio. This expansion shows that the scaling limit of the corrector is not necessarily a classical GFF, and in particular does not necessarily satisfy the Markov property.

• (with Hendrik Weber) The dynamic 𝜑⁴₃model comes down from infinity. Comm. Math. Phys. 356 (3), 673-753 (2017). [abs] [tex] [pdf]

  We prove an a priori bound for the dynamic 𝜑⁴₃model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blow-up of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the Krylov-Bogoliubov method. It thereby provides a new dynamic construction of the Euclidean 𝜑⁴₃field theory on finite volume. Our method is based on the local-in-time solution theory developed recently by Gubinelli, Imkeller, Perkowski and Catellier, Chouk. The argument relies entirely on deterministic PDE arguments (such as embeddings of Besov spaces and interpolation), which are combined to derive energy inequalities.

• (with Scott Armstrong and Tuomo Kuusi) Mesoscopic higher regularity and subadditivity in elliptic homogenization. Comm. Math. Phys. 347 (2), 315-361 (2016). [abs] [tex] [pdf]

  We introduce a new method for obtaining quantitative results in stochastic homogenization for linear elliptic equations in divergence form. Unlike previous works on the topic, our method does not use concentration inequalities (such as Poincaré or logarithmic Sobolev inequalities in the probability space) and relies instead on a higher (C^k, k ≥ 1) regularity theory for solutions of the heterogeneous equation, which is valid on length scales larger than a certain specified mesoscopic scale. This regularity theory, which is of independent interest, allows us to, in effect, localize the dependence of the solutions on the coefficients and thereby accelerate the rate of convergence of the expected energy of the cell problem by a bootstrap argument. The fluctuations of the energy are then tightly controlled using subadditivity. The convergence of the energy gives control of the scaling of the spatial averages of gradients and fluxes (that is, it quantifies the weak convergence of these quantities) which yields, by a new "multiscale" Poincaré inequality, quantitative estimates on the sublinearity of the corrector.

• (with Felix Otto) Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments. J. Funct. Anal. 270 (1), 201-228 (2016). [abs] [tex] [pdf]

  We introduce anchored versions of the Nash inequality. They allow to control the L² norm of a function by Dirichlet forms that are not uniformly elliptic. We then use them to provide heat kernel upper bounds for diffusions in degenerate static and dynamic random environments. As an example, we apply our results to the case of a random walk with degenerate jump rates that depend on an underlying exclusion process at equilibrium.

• (with Yu Gu) Scaling limit of fluctuations in stochastic homogenization. Multiscale Model. Simul. 14 (1), 452-481 (2016). [abs] [tex] [pdf]

  We investigate the global fluctuations of solutions to elliptic equations with random coefficients in the discrete setting. In dimension d ≥ 3 and for i.i.d. coefficients, we show that after a suitable scaling, these fluctuations converge to a Gaussian field that locally resembles a (generalized) Gaussian free field. The paper begins with a heuristic derivation of the result, which can be read independently and was obtained jointly with Scott Armstrong.

• (with James Nolen) Scaling limit of the corrector in stochastic homogenization. Ann. Appl. Probab. 27 (2), 944-959 (2017). [abs] [tex] [pdf]

  In the homogenization of divergence-form equations with random coefficients, a central role is played by the corrector. We focus on a discrete space setting and on dimension 3 and more. Completing the argument started in previous work, we identify the scaling limit of the corrector, which is akin to a Gaussian free field.

• (with Marco Furlan) A tightness criterion for random fields, with application to the Ising model. Electron. J. Probab. 22, no. 97, 1-29 (2017). [abs] [tex] [pdf]

  We present a criterion for a family of random distributions to be tight in local Hölder and Besov spaces of possibly negative regularity. We then apply this criterion to the magnetization field of the two-dimensional Ising model at criticality, answering a question of Camia, Garban and Newman.

• (with Hendrik Weber) Global well-posedness of the dynamic 𝜑⁴ model in the plane. Ann. Probab. 45 (4), 2398-2476 (2017). [abs] [tex] [pdf]

  We show global well-posedness of the dynamic 𝜑⁴ model in the plane. The model is a non-linear stochastic PDE that can only be interpreted in a "renormalised" sense. Solutions take values in suitable weighted Besov spaces of negative regularity.

• (with Paul de Buyer) Diffusive decay of the environment viewed by the particle. Electron. Commun. Probab. 20, no. 23, 1-12 (2015). [abs] [tex] [pdf]

  We prove an optimal diffusive decay of the environment viewed by the particle in random walk among random independent conductances, with, as a main assumption, finite second moment of the conductance. Our proof, using the analytic approach of Gloria, Neukamm and Otto, is very short and elementary.

• (with Scott Armstrong) Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219 (1), 255-348 (2016). [abs] [tex] [pdf]

  We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale L^∞-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched L² estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.

• (with Yu Gu) Pointwise two-scale expansion for parabolic equations with random coefficients. Probab. Theory Related Fields 166 (1), 585-618 (2016). [abs] [tex] [pdf]

  We investigate the first-order correction in the homogenization of linear parabolic equations with random coefficients. In dimension 3 and higher and for coefficients having a finite range of dependence, we prove a pointwise version of the two-scale expansion. A similar expansion is derived for elliptic equations in divergence form. The result is surprising, since it was not expected to be true without further symmetry assumptions on the law of the coefficients.

• (with Hendrik Weber) Convergence of the two-dimensional dynamic Ising-Kac model to 𝜑⁴₂. Comm. Pure Appl. Math. 70 (4), 717-812 (2017). [abs] [tex] [pdf]

  The Ising-Kac model is a variant of the ferromagnetic Ising model in which each spin variable interacts with all spins in a neighbourhood of radius 𝛾^-1 for 𝛾 ≪ 1 around its base point. We study the Glauber dynamics for this model on a discrete two-dimensional torus ℤ²/(2N+1)ℤ², for a system size N ≫ 𝛾^-1 and for an inverse temperature close to the critical value of the mean field model. We show that the suitably rescaled coarse-grained spin field converges in distribution to the solution of a non-linear stochastic partial differential equation.
  This equation is the dynamic version of the 𝜑⁴₂quantum field theory, which is formally given by a reaction diffusion equation driven by an additive space-time white noise. It is well-known that in two spatial dimensions, such equations are distribution valued and a Wick renormalisation has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation. We show that this need for renormalisation for the limiting equation is reflected in the discrete system by a shift of the critical temperature away from its mean field value.

• (with Daniel Valesin) Phase transition of the contact process on random regular graphs. Electron. J. Probab. 21, no. 31, 1-17 (2016). [abs] [tex] [pdf]

  We consider the contact process with infection rate λ on a random (d+1)-regular graph with n vertices, G_n. We study the extinction time τ (that is, the random amount of time until the infection disappears) as n is taken to infinity. We establish a phase transition depending on whether λ is smaller or larger than λ₁(T^d), the lower critical value for the contact process on the infinite, (d+1)-regular tree: if λ < λ₁(T^d), τ grows logarithmically with n, while if λ > λ₁(T^d), it grows exponentially with n. This result differs from the situation where, instead of G_n, the contact process is considered on the d-ary tree of finite height, since in this case, the transition is known to happen instead at the upper critical value for the contact process on T^d.

• (with Tom Mountford) Lyapunov exponents of random walks in small random potential: the upper bound. Electron. J. Probab. 20, no. 49, 1-18 (2015). [abs] [tex] [pdf]

  We consider the simple random walk on ℤ^d evolving in a random i.i.d. potential taking values in [0,+ ∞). The potential is not assumed integrable, and can be rescaled by a multiplicative factor λ > 0. Completing the work started in a companion paper, we give the asymptotic behaviour of the Lyapunov exponents for d ≥ 3, both annealed and quenched, as the scale parameter λ tends to zero.

• (with Michael Cranston, Tom Mountford, and Daniel Valesin) The contact process on finite homogeneous trees revisited. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2), 385-408 (2014). [abs] [tex] [pdf]

  We consider the contact process with infection rate λ on the d-ary tree of height n. We study the extinction time τ, that is, the random time it takes for the infection to disappear when the process is started from full occupancy. We prove two conjectures of Stacey regarding τ. Let λ₂ denote the upper critical value for the contact process on the infinite d-ary tree. First, if λ < λ₂, then τ divided by the height of the tree converges in probability, as n→∞, to a positive constant. Second, if λ > λ₂, then log E[τ] divided by the volume of the tree converges in probability to a positive constant, and τ/E[τ] converges in distribution to the exponential distribution of mean 1.

• (with Felix Otto) Correlation structure of the corrector in stochastic homogenization. Ann. Probab. 44 (5), 3207-3233 (2016). [abs] [tex] [pdf]

  Recently, the quantification of errors in the stochastic homogenization of divergence-form operators has witnessed important progress. Our aim now is to go beyond error bounds, and give precise descriptions of the effect of the randomness, in the large-scale limit. This paper is a first step in this direction. Our main result is to identify the correlation structure of the corrector, in dimension 3 and higher. This correlation structure is similar to, but different from that of a Gaussian free field.

• Significance level and positivity bias as causes for high rate of non-reproducible scientific results? [abs] [pdf]

  The high fraction of published results that turn out to be incorrect is a major concern of today's science. This paper contributes to the understanding of this problem in two independent directions. First, Johnson's recent claim that hypothesis testing with a significance level of α = 0.05 can alone lead to an unacceptably large proportion of false positive results is shown to be unfounded. Second, a way to quantify the effect of "positivity bias" (the tendency to consider only positive results as worthwhile) is introduced. We estimate the proportion of false positive results among positive results in terms of the significance level used and the positivity ratio. The latter quantity is the fraction of positive results over all results, be they positive or not, published or not. In particular, if one uses a significance level of α = 0.05, and produces 4 (possibly unpublished) negative results for every positive result, then the proportion of false positives among positive results can climb to a high 21%.

• First-order expansion of homogenized coefficients under Bernoulli perturbations. J. Math. Pures Appl. 103, 68-101 (2015). [abs] [tex] [pdf]

  Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter.

• (with Pierre Mathieu) Aging of asymmetric dynamics on the random energy model. Probab. Theory Related Fields 161 (1), 351-427 (2015). [abs] [tex] [pdf]

  We show aging of Glauber-type dynamics on the random energy model, in the sense that we obtain the scaling limits of the clock process and of the age process. The latter encodes the Gibbs weight of the configuration occupied by the dynamics. Both limits are expressed in terms of stable subordinators.

• (with Anne-Claire Egloffe, Antoine Gloria, and Thanh-Nhan Nguyen) Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth. IMA J. Numer. Anal. 35 (2), 499-545 (2015). [abs] [tex] [pdf]

  This article is concerned with numerical methods to approximate effective coefficients in stochastic homogenization of discrete linear elliptic equations, and their numerical analysis — which has been made possible by recent contributions on quantitative stochastic homogenization theory by two of us and by Otto. This article makes the connection between our theoretical results and computations. We give a complete picture of the numerical methods found in the literature, compare them in terms of known (or expected) convergence rates, and study them numerically. Two types of methods are presented: methods based on the corrector equation, and methods based on random walks in random environments. The numerical study confirms the sharpness of the analysis (which it completes by making precise the prefactors, next to the convergence rates), supports some of our conjectures, and calls for new theoretical developments.

• (with Tom Mountford) Lyapunov exponents of random walks in small random potential: the lower bound. Comm. Math. Phys. 323 (3), 1071-1120 (2013). [abs] [tex] [pdf]

  We consider the simple random walk on ℤ^d, d ≥ 3, evolving in a potential of the form βV, where (V(x), x ∈ ℤ^d) are i.i.d. random variables taking values in [0, + ∞), and β > 0. When the potential is integrable, the asymptotic behaviours as β tends to 0 of the associated quenched and annealed Lyapunov exponents are known (and coincide). Here, we do not assume such integrability, and prove a sharp lower bound on the annealed Lyapunov exponent for small β. The result can be rephrased in terms of the decay of the averaged Green function of the Anderson Hamiltonian -Δ + βV.

• Kantorovich distance in the martingale CLT and quantitative homogenization of parabolic equations with random coefficients. Probab. Theory Related Fields 160 (1-2), 279-314 (2014). [abs] [tex] [pdf]

  The article begins with a quantitative version of the martingale central limit theorem, in terms of the Kantorovich distance. This result is then used in the study of the homogenization of discrete parabolic equations with random i.i.d. coefficients. For smooth initial condition, the rescaled solution of such an equation, once averaged over the randomness, is shown to converge polynomially fast to the solution of the homogenized equation, with an explicit exponent depending only on the dimension. Polynomial rate of homogenization for the averaged heat kernel, with an explicit exponent, is then derived. Similar results for elliptic equations are also presented.

• (with Tom Mountford, Daniel Valesin, and Qiang Yao) Exponential extinction time of the contact process on finite graphs. Stochastic Process. Appl. 126 (7), 1974-2013 (2016). [abs] [tex] [pdf]

  We study the extinction time τ of the contact process on finite trees of bounded degree. We show that, if the infection rate is larger than the critical rate for the contact process on ℤ, then, uniformly over all trees of degree bounded by a given number, the expectation of τ grows exponentially with the number of vertices. Additionally, for any sequence of growing trees of bounded degree, τ divided by its expectation converges in distribution to the unitary exponential distribution. These also hold if one considers a sequence of graphs having spanning trees with uniformly bounded degree. Using these results, we consider the contact process on a random graph with vertex degrees following a power law. Improving a result of Chatterjee and Durrett (2009), we show that, for any infection rate, the extinction time for the contact process on this graph grows exponentially with the number of vertices.

• Lyapunov exponents, shape theorems and large deviations for the random walk in random potential. ALEA Lat. Am. J. Probab. Math. Stat. 9, 165-211 (2012). [abs] [tex] [pdf]

  We consider the simple random walk on ℤ^d evolving in a potential of independent and identically distributed random variables taking values in [0, + ∞]. We give optimal conditions for the existence of the quenched point-to-point Lyapunov exponent, and for different versions of a shape theorem. The method of proof applies as well to first-passage percolation, and builds up on an approach of Cox and Durrett (1981). The weakest form of shape theorem holds whenever the set of sites with finite potential percolates. Under this condition, we then show the existence of the quenched point-to-hyperplane Lyapunov exponent, and give a large deviation principle for the walk under the quenched weighted measure.

• A quantitative central limit theorem for the random walk among random conductances. Electron. J. Probab. 17, no. 97, 1-17 (2012). [abs] [tex] [pdf]

  We consider the random walk among random conductances on ℤ^d. We assume that the conductances are independent, identically distributed and uniformly bounded away from 0 and infinity. We obtain a quantitative version of the central limit theorem for this random walk, which takes the form of a Berry-Esseen estimate with speed t^-1/10 for d ≤ 2, and speed t^-1/5 for d ≥ 3, up to logarithmic corrections.

• On the rate of convergence in the martingale central limit theorem. Bernoulli 19 (2), 633-645 (2013). [abs] [tex] [pdf]

  Consider a discrete-time martingale, and let V² be its normalized quadratic variation. As V² approaches 1 and provided some Lindeberg condition is satisfied, the distribution of the rescaled martingale approaches the Gaussian distribution. For any p ≥ 1, Haeusler (1988) gives a bound on the rate of convergence in this central limit theorem, that is the sum of two terms, say A_p + B_p, where up to a constant, A_p = ‖V²-1‖_p^p/(2p+1). We discuss here the optimality of this term, focusing on the restricted class of martingales with bounded increments. In this context, Bolthausen (1982) sketches a strategy to prove optimality for p = 1. Here, we extend this strategy to any p ≥ 1, thus justifying the optimality of the term A_p. As a necessary step, we also provide a new bound on the rate of convergence in the central limit theorem for martingales with bounded increments that improves on the term B_p, generalizing another result of Bolthausen (1982).

• (with Antoine Gloria) Quantitative version of the Kipnis-Varadhan theorem and Monte-Carlo approximation of homogenized coefficients. Ann. Appl. Probab. 23 (4), 1544-1583 (2013). [abs] [tex] [pdf]

  This article is devoted to the analysis of a Monte-Carlo method to approximate effective coefficients in stochastic homogenization of discrete elliptic equations. We consider the case of independent and identically distributed coefficients, and adopt the point of view of the random walk in a random environment. Given some final time t > 0, a natural approximation of the homogenized coefficients is given by the empirical average of the final squared positions rescaled by t of n independent random walks in n independent environments. Relying on a quantitative version of the Kipnis-Varadhan theorem combined with estimates of spectral exponents obtained by an original combination of pde arguments and spectral theory, we first give a sharp estimate of the error between the homogenized coefficients and the expectation of the rescaled final position of the random walk in terms of t. We then complete the error analysis by quantifying the fluctuations of the empirical average in terms of n and t, and prove a large-deviation estimate. Our estimates are optimal, up to a logarithmic correction in dimension 2.

• On the delocalized phase of the random pinning model. Séminaire de probabilités 44, 401-407 (2012). [abs] [tex] [pdf]

  We consider the model of a directed polymer pinned to a line of i.i.d. random charges, and focus on the interior of the delocalized phase. We first show that in this region, the partition function remains bounded. We then prove that for almost every environment of charges, the probability that the number of contact points in [0,n] exceeds c log(n) tends to 0 as n tends to infinity. Our proofs rely on recent results of Birkner, Greven, den Hollander (2010) and Cheliotis, den Hollander (2010).

• (with Antoine Gloria) Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields 154 (1-2), 287-326 (2012). [abs] [tex] [pdf]

  This article deals with the numerical approximation of effective coefficients in stochastic homogenization of discrete linear elliptic equations. The originality of this work is the use of a well-known abstract spectral representation formula to design and analyze effective and computable approximations of the homogenized coefficients. In particular, we show that information on the edge of the spectrum of the generator of the environment viewed by the particle projected on the local drift yields bounds on the approximation error, and conversely. Combined with results by Otto and the first author in low dimension, and results by the second author in high dimension, this allows us to prove that for any dimension, there exists an explicit numerical strategy to approximate homogenized coefficients which converges at the rate of the central limit theorem.

• Scaling limit of the random walk among random traps on ℤ^d. Ann. Inst. Henri Poincaré Probab. Stat. 47 (3), 813-849 (2011). [abs] [tex] [pdf]

  Attributing a positive value τ_x to each x in ℤ^d, we investigate a nearest-neighbour random walk which is reversible for the measure with weights (τ_x), often known as "Bouchaud's trap model". We assume that these weights are independent, identically distributed and non-integrable random variables (with polynomial tail), and that d ≥ 5. We obtain the quenched subdiffusive scaling limit of the model, the limit being the fractional kinetics process. We begin our proof by expressing the random walk as a time change of a random walk among random conductances. We then focus on proving that the time change converges, under the annealed measure, to a stable subordinator. This is achieved using previous results concerning the mixing properties of the environment viewed by the time-changed random walk.

• Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47 (1), 294-327 (2011). [abs] [tex] [pdf]

  For the random walk among random conductances, we prove that the environment viewed by the particle converges to equilibrium polynomially fast in the variance sense, our main hypothesis being that the conductances are bounded away from zero. The basis of our method is the establishment of a Nash inequality, followed either by a comparison with the simple random walk or by a more direct analysis based on a martingale decomposition. As an example of application, we show that under certain conditions, our results imply an estimate of the speed of convergence of the mean square displacement of the walk towards its limit.

• Principal eigenvalue for the random walk among random traps on ℤ^d. Potential Anal. 33 (3), 227-247 (2010). [abs] [tex] [pdf] (first version [tex] [pdf])

  Let (τ_x, x ∊ ℤ^d) be i.i.d. random variables with heavy (polynomial) tails. Given a ∊ [0,1], we consider the Markov process defined by the jump rates τ_x^-(1-a) τ_y^a between two neighbours x and y in ℤ^d. We give the asymptotic behaviour of the principal eigenvalue of the generator of this process, with Dirichlet boundary condition. The prominent feature is a phase transition that occurs at some threshold depending on the dimension.

PhD students and postdocs

• Victor Issa (PhD, 2023 - )

• Alex Dunlap (postdoc, 2020 - 2023)

• Hong-Bin Chen (PhD, 2017 - 2022, jointly advised with Yuri Bakhtin)

• Chenlin Gu (PhD, 2018 - 2021)

• Paul Dario (PhD, 2016 - 2019, jointly advised with Scott Armstrong)

• Alexandre Bordas (PhD, 2015 - 2018, jointly advised with Scott Armstrong)

Theses

• Random PDEs: questions of regularity. Habilitation's thesis (2017). [pdf]

• Marches aléatoires réversibles en milieu aléatoire. Ph.D. thesis, advised by Pierre Mathieu and Alejandro Ramírez (2010). [abs] [tex] [pdf]

• Bachelor's and master's theses, advised by Thierry Bodineau and Gérard Ben Arous respectively. [pdf]