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Antoine Gloria

Publications and source records attributed to Antoine Gloria.

At least 19 recordsLinked to original sources
arXiv2026

Batchelor's formula and infrared renormalization for sedimentation

We study the sedimentation of stationary random suspensions of rigid particles in Stokes flow. Batchelor's formula predicts the first dilute correction to the infinite-volume mean settling speed due to hydrodynamic interactions between suspended particles. A rigorous derivation has long been obstructed by the long-range nature of the Stokes flow, which gives rise to infrared divergences in the large-volume limit. In dimension $d>2$, for stationary suspensions satisfying quantitative decorrelation assumptions, we construct the infinite-volume mean settling speed and show that it governs the relative settling speed of particles in large containers, independently of the container shape. We then establish a renormalized cluster expansion of this mean settling speed in the dilute regime and compute it up to the two-particle term, thereby justifying Batchelor's formula. The proof is based on the infrared renormalization of hydrodynamic interactions. Infinite-volume observables are decomposed into an explicit singular part, carrying the non-integrable large-scale contribution, and a regular remainder controlled by elliptic estimates. The singular part is renormalized through counterterms that encode the diverging mean backflow generated by the suspension. At the level of the dilute cluster expansion, the renormalization is implemented cluster by cluster and the singular-regular decomposition is achieved through a finitary diagrammatic expansion of hydrodynamic interactions, inspired by the method of reflections, which isolates the leading divergent substructures and exposes the key cancellations.

math.AP↗
arXiv2026

Moment bounds on correctors for the degenerate random conductance model

We study the random conductance model on the lattice $\Z^d$, i.e. we consider a linear, finite-difference, divergence-form operator with random conductances $a$. We allow the conductances $a$ to be unbounded and degenerate. Assuming the conductances satisfy a spectral-gap inequality, we establish sharp bounds on the spatial growth of correctors, together with a quantitative relation between the stochastic integrability of the correctors and that of $a$.

math.PR↗
arXiv2025

Diffusivity of the Lorentz mirror walk in high dimensions

In the Lorentz mirror walk in dimension $d\geq 2$, mirrors are randomly placed on the vertices of $\mathbb{Z}^d$ at density $p\in[0,1]$. A light ray is then shot from the origin and deflected through the various mirrors in space. The object of study is the random trajectory obtained in this way, and it is of upmost interest to determine whether these trajectories are localized (finite) or delocalized (infinite). A folklore conjecture states that for $d=2$ these trajectories are finite for any density $p>0$, while in dimensions $d\geq 3$ and for $p>0$ small enough some trajectories are infinite. In this paper we prove that for all dimensions $d\geq 4$ and any small density $p$, the trajectories behave diffusively at all polynomial time scales $t\approx p^{-M}$ with $M>1$, and in particular, they do not close by this time.

math.PR↗
arXiv2025

Periodic homogenization and harmonic measures

Since the seminal work of Kenig and Pipher, the Dahlberg-Kenig-Pipher (DKP) condition on oscillations of the coefficient matrix became a standard threshold in the study of absolute continuity of the harmonic measure with respect to the Hausdorff measure on the boundary. It has been proved sufficient for absolute continuity in the domains with increasingly complex geometry, and known counterexamples show that in a certain sense it is necessary as well. In the present note, we introduce into the subject ideas from homogenization theory to exhibit a new class of operators for which the elliptic measure is well-behaved, featuring the coefficients violating the DKP condition, and on the contrary, oscillating so quickly, that the homogenization takes place.

math.AP↗
arXiv2025

Large-scale dispersive estimates for acoustic operators: homogenization meets localization

This work relates quantitatively homogenization to Anderson localization for acoustic operators in disordered media. By blending dispersive estimates for homogenized operators and quantitative homogenization of the wave equation, we derive large-scale dispersive estimates for waves in disordered media that we apply to the spreading of low-energy eigenstates. This gives a short and direct proof that the lower spectrum of the acoustic operator is purely absolutely continuous in case of periodic media, and it further provides new lower bounds on the localization length of possible eigenstates in case of quasiperiodic or random media.

math.AP↗
arXiv2025

Homogenization of the stochastic double-porosity model

This work is devoted to the homogenization of elliptic equations in high-contrast media in the so-called 'double-porosity' resonant regime, for which we solve two open problems of the literature. First, we prove qualitative stochastic homogenization under very weak conditions, which cover the case of inclusions that are not uniformly bounded or separated. Second, under stronger assumptions, we provide sharp error estimates for the two-scale expansion. The main difficulty is related to the loss of integrability of the control in the resonant zones.

math.AP↗
arXiv2024

Homogenization of the 2D Euler system: lakes and porous media

This work is devoted to the long-standing open problem of homogenization of 2D perfect incompressible fluid flows, such as the 2D Euler equations with impermeable inclusions modeling a porous medium, and such as the lake equations. The main difficulty is the homogenization of the transport equation for the associated fluid vorticity. In particular, a localization phenomenon for the vorticity could in principle occur, which would rule out the separation of scales. Our approach combines classical results from different fields to prevent such phenomena and to prove homogenization towards variants of the Euler and lake equations: we rely in particular on the homogenization theory for elliptic equations with stiff inclusions, on criteria for unique ergodicity of dynamical systems, and on complex analysis in form of extensions of the Radó-Kneser-Choquet theorem.

math.AP↗
arXiv2024

Quantitative homogenization for log-normal coefficients via Malliavin calculus: the one-dimensional case

The quantitative analysis of stochastic homogenization problems has been a very active field in the last fifteen years. Whereas the first results were motivated by applied questions (namely, the numerical approximation of homogenized coefficients), the more recent achievements in the field are much more analytically-driven and focus on the subtle interplay between PDE analysis (and in particular elliptic regularity theory) and probability (concentration, stochastic cancellations, scaling limits). The aim of this article is threefold. First we provide a complete and self-contained analysis for the popular example of log-normal coefficients with possibly fat tails in dimension $d=1$, establishing new results on the accuracy of the two-scale expansion and characterizing fluctuations (in the perspective of uncertainty quantification). Second, we work in a context where explicit formulas allow us to by-pass analytical difficulties and therefore mostly focus on the probabilistic side of the theory. Last, the one-dimensional setting gives intuition on the available results in higher dimension (provided results are correctly reformulated) to which we give precise entries to the recent literature.

math.AP↗
arXiv2024

Quantitative homogenization for log-normal coefficients

We establish quantitative homogenization results for the popular log-normal coefficients. Since the coefficients are neither bounded nor uniformly elliptic, standard proofs do not apply directly. Instead, we take inspiration from the approach developed for the nonlinear setting by the first two authors and capitalize on large-scale regularity results by Bella, Fehrmann, and Otto for degenerate coefficients in order to leverage an optimal control (in terms of scaling and stochastic integrability) of oscillations and fluctuations.

math.AP↗
arXiv2024

A spectral ansatz for the long-time homogenization of the wave equation

Consider the wave equation with heterogeneous coefficients in the homogenization regime. At large times, the wave interacts in a nontrivial way with the heterogeneities, giving rise to effective dispersive effects. The main achievement of the present work is a new ansatz for the long-time two-scale expansion inspired by spectral analysis. Based on this spectral ansatz, we extend and refine all previous results in the field, proving homogenization up to optimal timescales with optimal error estimates, and covering all the standard assumptions on heterogeneities (both periodic and stationary random settings).

math.AP↗
arXiv2023

The landscape function on $\mathbb R^d$

Consider the Schrödinger operator $-\triangle+λV$ with non-negative iid random potential $V$ of strength $λ>0$. We prove existence and uniqueness of the associated landscape function on the whole space, and show that its correlations decay exponentially. As a main ingredient we establish the (annealed and quenched) exponential decay of the Green function of $-\triangle+λV$ using Agmon's positivity method, rank-one perturbation in dimensions $d\ge 3$, and first-passage percolation in dimensions $d=1,2$.

math.PR↗
arXiv2023

Quantitative nonlinear homogenization: control of oscillations

Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with $p$-growth. This work is dedicated to a quantitative two-scale expansion result. By treating the range of exponents $2\le p <\infty$ in dimensions $d\le 3$, we are able to consider genuinely nonlinear elliptic equations and systems such as $-\nabla \cdot A(x)(1+|\nabla u|^{p-2})\nabla u=f$ (with $A$ random, non-necessarily symmetric) for the first time. When going from $p=2$ to $p>2$, the main difficulty is to analyze the associated linearized operator, whose coefficients are degenerate, unbounded, and depend on the random input $A$ via the solution of a nonlinear equation. One of our main achievements is the control of this intricate nonlinear dependence, leading to annealed Meyers' estimates for the linearized operator, which are key to the optimal quantitative two-scale expansion result we derive (this is also new in the periodic setting).

math.AP↗
arXiv2023

Homogenization of active suspensions and reduction of effective viscosity

We consider a suspension of active rigid particles (swimmers) in a steady Stokes flow, where particles are distributed according to a stationary ergodic random process, and we study its homogenization in the macroscopic limit. A key point in the model is that swimmers are allowed to adapt their propulsion to the surrounding fluid deformation: swimming forces are not prescribed a priori, but are rather obtained through the retroaction of the fluid. Qualitative homogenization of this nonlinear model requires an unusual proof that crucially relies on a semi-quantitative two-scale analysis. After introducing new correctors that accurately capture spatial oscillations created by swimming forces, we identify the contribution of the activity to the effective viscosity. In agreement with the physics literature, an analysis in the dilute regime shows that the activity of the particles can either increase or decrease the effective viscosity (depending on the swimming mechanism), which differs from the well-known case of passive suspensions.

math.AP↗
arXiv2022

Effective viscosity of semi-dilute suspensions

This review is devoted to the large-scale rheology of suspensions of rigid particles in Stokes fluid. After describing recent results on the definition of the effective viscosity of such systems in the framework of homogenization theory, we turn to our new results on the asymptotic expansion of the effective viscosity in the dilute regime. This includes an optimal proof of Einstein's viscosity formula for the first-order expansion, as well as the continuation of this expansion to higher orders. The essential difficulty originates in the long-range nature of hydrodynamic interactions: suitable renormalizations are needed and are captured by means of diagrammatic expansions.

math.AP↗
arXiv2022

The Clausius-Mossotti formula

In this note, we provide a short and robust proof of the Clausius-Mossotti formula for the effective conductivity in the dilute regime, together with an optimal error estimate. The proof makes no assumption on the underlying point process besides stationarity and ergodicity, and can be applied to dilute systems in many other contexts.

math.AP↗
arXiv2022

On Einstein's effective viscosity formula

In his PhD thesis, Einstein derived an explicit first-order expansion for the effective viscosity of a Stokes fluid with a suspension of small rigid particles at low density. His formal derivation relied on two implicit assumptions: (i) there is a scale separation between the size of the particles and the observation scale; and (ii) at first order, dilute particles do not interact with one another. In mathematical terms, the first assumption amounts to the validity of a homogenization result defining the effective viscosity tensor, which is now well understood. Next, the second assumption allowed Einstein to approximate this effective viscosity at low density by considering particles as being isolated. The rigorous justification is, in fact, quite subtle as the effective viscosity is a nonlinear nonlocal function of the ensemble of particles and as hydrodynamic interactions have borderline integrability. In the present memoir, we establish Einstein's effective viscosity formula in the most general setting. In addition, we pursue the low-density expansion to arbitrary order in form of a cluster expansion, where the summation of hydrodynamic interactions crucially requires suitable renormalizations. In particular, we justify a celebrated result by Batchelor and Green on the second-order correction and we explicitly describe all higher-order renormalizations for the first time. In some specific settings, we further address the summability of the whole cluster expansion. Our approach relies on a combination of combinatorial arguments, variational analysis, elliptic regularity, probability theory, and diagrammatic integration methods.

math.AP↗
arXiv2022

A short proof of Gevrey regularity for homogenized coefficients of the Poisson point process

In this short note we capitalize on and complete our previous results on the regularity of the homogenized coefficients for Bernoulli perturbations by addressing the case of the Poisson point process, for which the crucial uniform local finiteness assumption fails. In particular, we strengthen the qualitative regularity result first obtained in this setting by the first author to Gevrey regularity of order~2. The new ingredient is the independence of Poisson point processes, in a form recently used by Giunti, Gu, Mourrat, and Nitzschner.

math.PR↗