research
Book

(with S. Armstrong, T. Kuusi) Quantitative stochastic homogenization and largescale regularity. (Updated August 7, 2018) [abs] [pdf]
This is a preliminary version of a book which presents the quantitative homogenization and largescale regularity theory for elliptic equations in divergenceform. The selfcontained presentation gives new and simplified proofs of the core results proved in the last several years, including the algebraic convergence rate for the variational subadditive quantities, the largescale Lipschitz and higher regularity estimates and Liouvilletype results, optimal quantitative estimates on the firstorder correctors and their scaling limit to a Gaussian free field. The last chapter contains new results on the homogenization of the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the twoscale expansion.
Expository papers

An informal introduction to quantitative stochastic homogenization. Proceedings of ICMP 2018. [abs] [tex] [pdf]
Divergenceform 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 H. Weber, W. Xu) Construction of 𝜑^{4}_{3}diagrams for pedestrians. In From particle systems to partial differential equations IV, 146 (2017). [abs] [tex] [pdf]
We aim to give a pedagogic and essentially selfcontained presentation of the construction of various stochastic objects appearing in the dynamical 𝜑^{4}_{3}model. 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

HamiltonJacobi equations for meanfield disordered systems. [abs] [tex] [pdf]
We argue that HamiltonJacobi equations provide a convenient and intuitive approach for studying the largescale behavior of meanfield disordered systems. This point of view is illustrated on the problem of inference of a rankone matrix. We compute the largescale limit of the free energy by showing that it satisfies an approximate HamiltonJacobi equation with asymptotically vanishing viscosity parameter and error term. 
(with S. Armstrong, T. Kuusi, A. Hannukainen) An iterative method for elliptic problems with rapidly oscillating coefficients. [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 S. Armstrong, A. Bordas) Quantitative stochastic homogenization and regularity theory of parabolic equations. Anal. PDE, to appear. [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 renormalizationtype 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 CauchyDirichlet 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., to appear. [abs] [tex] [pdf]
We introduce new methods to compute the homogenized coefficients of divergenceform operators with random coefficients. We focus on a discretespace 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 A. Giunti, Y. Gu) Heat kernel upper bounds for interacting particle systems. Ann. Probab., to appear. [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 offdiagonal estimates of CarneVaropoulos type. 
(with S. Armstrong, T. Kuusi, C. Prange) Quantitative analysis of boundary layers in periodic homogenization. Arch. Ration. Mech. Anal. 226 (2), 695741 (2017). [abs] [tex] [pdf]
We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergenceform 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 D. Valesin) Spatial Gibbs random graphs. Ann. Appl. Probab. 28 (2), 751789 (2018). [abs] [tex] [pdf]
Many realworld 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 A. Giunti) Quantitative homogenization of degenerate random environments. Ann. Inst. Henri Poincaré Probab. Stat. 54 (1), 2250 (2018). [abs] [tex] [pdf]
We study discrete linear divergenceform 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 S. Armstrong, T. Kuusi) The additive structure of elliptic homogenization. Invent. Math. 208 (3), 9991154 (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 firstorder correctors (under blowdown) 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 firstorder correctors converge, in the largescale limit, to a variant of the Gaussian free field. 
(with Y. Gu) On generalized Gaussian free fields and stochastic homogenization. Electron. J. Probab. 22, no. 28, 121 (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 H. Weber) The dynamic 𝜑^{4}_{3}model comes down from infinity. Comm. Math. Phys. 356 (3), 673753 (2017). [abs] [tex] [pdf]
We prove an a priori bound for the dynamic 𝜑^{4}_{3}model on the torus which is independent of the initial condition. In particular, this bound rules out the possibility of finite time blowup of the solution. It also gives a uniform control over solutions at large times, and thus allows to construct invariant measures via the KrylovBogoliubov method. It thereby provides a new dynamic construction of the Euclidean 𝜑^{4}_{3}field theory on finite volume. Our method is based on the localintime 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 S. Armstrong, T. Kuusi) Mesoscopic higher regularity and subadditivity in elliptic homogenization. Comm. Math. Phys. 347 (2), 315361 (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 F. Otto) Anchored Nash inequalities and heat kernel bounds for static and dynamic degenerate environments. J. Funct. Anal. 270 (1), 201228 (2016). [abs] [tex] [pdf]
We introduce anchored versions of the Nash inequality. They allow to control the L^{2} 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 Y. Gu) Scaling limit of fluctuations in stochastic homogenization. Multiscale Model. Simul. 14 (1), 452481 (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 J. Nolen) Scaling limit of the corrector in stochastic homogenization. Ann. Appl. Probab. 27 (2), 944959 (2017). [abs] [tex] [pdf]
In the homogenization of divergenceform 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 M. Furlan) A tightness criterion for random fields, with application to the Ising model. Electron. J. Probab. 22, no. 97, 129 (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 twodimensional Ising model at criticality, answering a question of Camia, Garban and Newman. 
(with H. Weber) Global wellposedness of the dynamic 𝜑^{4} model in the plane. Ann. Probab. 45 (4), 23982476 (2017). [abs] [tex] [pdf]
We show global wellposedness of the dynamic 𝜑^{4} model in the plane. The model is a nonlinear stochastic PDE that can only be interpreted in a "renormalised" sense. Solutions take values in suitable weighted Besov spaces of negative regularity. 
(with P. de Buyer) Diffusive decay of the environment viewed by the particle. Electron. Commun. Probab. 20, no. 23, 112 (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 S. Armstrong) Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219 (1), 255348 (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 largescale L^{∞}type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a oneparameter family of mixing assumptions, allowing for very weak mixing with nonintegrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched L^{2} 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 divergenceform equations. 
(with Y. Gu) Pointwise twoscale expansion for parabolic equations with random coefficients. Probab. Theory Related Fields 166 (1), 585618 (2016). [abs] [tex] [pdf]
We investigate the firstorder 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 twoscale 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 H. Weber) Convergence of the twodimensional dynamic IsingKac model to 𝜑^{4}_{2}. Comm. Pure Appl. Math. 70 (4), 717812 (2017). [abs] [tex] [pdf]
The IsingKac 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 twodimensional torus ℤ^{2}/(2N+1)ℤ^{2}, 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 coarsegrained spin field converges in distribution to the solution of a nonlinear stochastic partial differential equation.
This equation is the dynamic version of the 𝜑^{4}_{2}quantum field theory, which is formally given by a reaction diffusion equation driven by an additive spacetime white noise. It is wellknown that in two spatial dimensions, such equations are distribution valued and a Wick renormalisation has to be performed in order to define the nonlinear 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 D. Valesin) Phase transition of the contact process on random regular graphs. Electron. J. Probab. 21, no. 31, 117 (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 λ_{1}(T^{d}), the lower critical value for the contact process on the infinite, (d+1)regular tree: if λ < λ_{1}(T^{d}), τ grows logarithmically with n, while if λ > λ_{1}(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 dary 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 T. Mountford) Lyapunov exponents of random walks in small random potential: the upper bound. Electron. J. Probab. 20, no. 49, 118 (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 M. Cranston, T. Mountford, D. Valesin) The contact process on finite homogeneous trees revisited. ALEA Lat. Am. J. Probab. Math. Stat. 11 (2), 385408 (2014). [abs] [tex] [pdf]
We consider the contact process with infection rate λ on the dary 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 λ_{2} denote the upper critical value for the contact process on the infinite dary tree. First, if λ < λ_{2}, then τ divided by the height of the tree converges in probability, as n→∞, to a positive constant. Second, if λ > λ_{2}, 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 F. Otto) Correlation structure of the corrector in stochastic homogenization. Ann. Probab. 44 (5), 32073233 (2016). [abs] [tex] [pdf]
Recently, the quantification of errors in the stochastic homogenization of divergenceform 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 largescale 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 nonreproducible 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%. 
Firstorder expansion of homogenized coefficients under Bernoulli perturbations. J. Math. Pures Appl. 103, 68101 (2015). [abs] [tex] [pdf]
Divergenceform 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 firstorder expansion of the homogenized coefficients, as a function of the perturbation parameter. 
(with P. Mathieu) Aging of asymmetric dynamics on the random energy model. Probab. Theory Related Fields 161 (1), 351427 (2015). [abs] [tex] [pdf]
We show aging of Glaubertype 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 A.C. Egloffe, A. Gloria, T.N. Nguyen) Random walk in random environment, corrector equation, and homogenized coefficients: from theory to numerics, back and forth. IMA J. Numer. Anal. 35 (2), 499545 (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 T. Mountford) Lyapunov exponents of random walks in small random potential: the lower bound. Comm. Math. Phys. 323 (3), 10711120 (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 (12), 279314 (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 T. Mountford, D. Valesin, Q. Yao) Exponential extinction time of the contact process on finite graphs. Stochastic Process. Appl. 126 (7), 19742013 (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, 165211 (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 pointtopoint Lyapunov exponent, and for different versions of a shape theorem. The method of proof applies as well to firstpassage 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 pointtohyperplane 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, 117 (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 BerryEsseen 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), 633645 (2013). [abs] [tex] [pdf]
Consider a discretetime martingale, and let V^{2} be its normalized quadratic variation. As V^{2} 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} = ‖V^{2}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 A. Gloria) Quantitative version of the KipnisVaradhan theorem and MonteCarlo approximation of homogenized coefficients. Ann. Appl. Probab. 23 (4), 15441583 (2013). [abs] [tex] [pdf]
This article is devoted to the analysis of a MonteCarlo 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 KipnisVaradhan 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 largedeviation 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, 401407 (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 A. Gloria) Spectral measure and approximation of homogenized coefficients. Probab. Theory Related Fields 154 (12), 287326 (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 wellknown 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), 813849 (2011). [abs] [tex] [pdf]
Attributing a positive value τ_{x} to each x in ℤ^{d}, we investigate a nearestneighbour 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 nonintegrable 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 timechanged random walk. 
Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47 (1), 294327 (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), 227247 (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}^{(1a)} 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.
Theses

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

Marches aléatoires réversibles en milieu aléatoire. Ph.D. thesis, supervised by P. Mathieu and A. Ramírez (May 2010). [abs] [tex] [pdf]

Bachelor's and master's theses, supervised resp. by T. Bodineau and G. Ben Arous. [pdf]