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Scott Armstrong

Publications and source records attributed to Scott Armstrong.

At least 37 records · Page 2Linked to original sources
arXiv2021

Local laws and rigidity for Coulomb gases at any temperature

We study Coulomb gases in any dimension $d \geq 2$ and in a broad temperature regime. We prove local laws on the energy, separation and number of points down to the microscopic scale. These yield the existence of limiting point processes generalizing the Ginibre point process for arbitrary temperature and dimension. The local laws come together with a quantitative expansion of the free energy with a new explicit error rate in the case of a uniform background density. These estimates have explicit temperature dependence, allowing to treat regimes of very large or very small temperature, and exhibit a new minimal lengthscale for rigidity depending on the temperature. They apply as well to energy minimizers (formally zero temperature). The method is based on a bootstrap on scales and reveals the additivity of the energy modulo surface terms, via the introduction of subadditive and superadditive approximate energies.

math-ph↗
arXiv2021

Optimal unique continuation for periodic elliptic equations on large scales

We prove a quantitative, large-scale doubling inequality and large-scale three-ellipsoid inequality for solutions of uniformly elliptic equations with periodic coefficients. These estimates are optimal in terms of the minimal length scale on which they are valid, and are at least "almost" optimal in the prefactor constants--up to, at most, an iterated logarithm of the initial doubling ratio.

math.AP↗
arXiv2020

Thermal approximation of the equilibrium measure and obstacle problem

We consider the probability measure minimizing a free energy functional equal to the sum of a Coulomb interaction, a confinement potential and an entropy term, which arises in the statistical mechanics of Coulomb gases. In the limit where the inverse temperature $β$ tends to $\infty$ the entropy term disappears and the measure, which we call the "thermal equilibrium measure" tends to the well-known equilibrium measure, which can also be interpreted as a solution to the classical obstacle problem. We provide quantitative estimates on the convergence of the thermal equilibrium measure to the equilibrium measure in strong norms in the bulk of the latter, with a sequence of explicit correction terms in powers of $1/β$, as well as an analysis of the tail after the boundary layer of size $β^{-1/2}$.

math.AP↗
arXiv2020

Large-scale analyticity and unique continuation for periodic elliptic equations

We prove that a solution of an elliptic operator with periodic coefficients behaves on large scales like an analytic function, in the sense of approximation by polynomials with periodic corrections. Equivalently, the constants in the large-scale $C^{k,1}$ estimate scale exponentially in $k$, just as for the classical estimate for harmonic functions. As a consequence, we characterize entire solutions of periodic, uniformly elliptic equations which exhibit growth like $O(\exp(δ|x|))$ for small~$δ>0$. The large-scale analyticity also implies quantitative unique continuation results, namely a three-ball theorem with an optimal error term as well as a proof of the nonexistence of $L^2$ eigenfunctions at the bottom of the spectrum.

math.AP↗
arXiv2019

Higher-order linearization and regularity in nonlinear homogenization

We prove large-scale $C^\infty$ regularity for solutions of nonlinear elliptic equations with random coefficients, thereby obtaining a version of the statement of Hilbert's 19th problem in the context of homogenization. The analysis proceeds by iteratively improving three statements together: (i) the regularity of the homogenized Lagrangian $\bar{L}$, (ii) the commutation of higher-order linearization and homogenization, and (iii) large-scale $C^{0,1}$-type regularity for higher-order linearization errors. We consequently obtain a quantitative estimate on the scaling of linearization errors, a Liouville-type theorem describing the polynomially-growing solutions of the system of higher-order linearized equations, and an explicit (heterogenous analogue of the) Taylor series for an arbitrary solution of the nonlinear equations---with the remainder term optimally controlled. These results give a complete generalization to the nonlinear setting of the large-scale regularity theory in homogenization for linear elliptic equations.

math.AP↗
arXiv2019

$C^2$ regularity of the surface tension for the $\nabla ϕ$ interface model

We consider the $\nabla ϕ$ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least $C^{2,β}$ for some $β>0$. We also prove a fluctuation-dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite-volume surface tension to that of its infinite-volume limit.

math-ph↗
arXiv2019

Homogenization, linearization and large-scale regularity for nonlinear elliptic equations

We consider nonlinear, uniformly elliptic equations with random, highly oscillating coefficients satisfying a finite range of dependence. We prove that homogenization and linearization commute in the sense that the linearized equation (linearized around an arbitrary solution) homogenizes to the linearization of the homogenized equation (linearized around the corresponding solution of the homogenized equation). We also obtain a quantitative estimate on the rate of this homogenization. These results lead to a better understanding of differences of solutions to the nonlinear equation, which is of fundamental importance in quantitative homogenization. In particular, we obtain a large-scale $C^{0,1}$ estimate for differences of solutions---with optimal stochastic integrability. Using this estimate, we prove a large-scale $C^{1,1}$ estimate for solutions, also with optimal stochastic integrability. Each of these regularity estimates are new even in the periodic setting. As a second consequence of the large-scale regularity for differences, we improve the smoothness of the homogenized Lagrangian by showing that it has the same regularity as the heterogeneous Lagrangian, up to $C^{2,1}$.

math.AP↗
arXiv2018

Quantitative stochastic homogenization and large-scale regularity

This is a preliminary version of a book which presents the quantitative homogenization and large-scale regularity theory for elliptic equations in divergence-form. The self-contained 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 large-scale Lipschitz and higher regularity estimates and Liouville-type results, optimal quantitative estimates on the first-order correctors and their scaling limit to a Gaussian free field. There are several chapters containing new results, such as: quantitative estimates for the Dirichlet problem, including optimal quantitative estimates of the homogenization error and the two-scale expansion; optimal estimates for the homogenization of the parabolic and elliptic Green functions; and $W^{1,p}$-type estimates for two-scale expansions.

math.AP↗
arXiv2018

Quantitative stochastic homogenization and regularity theory of parabolic equations

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.

math.AP↗
arXiv2017

Optimal quantitative estimates in stochastic homogenization for elliptic equations in nondivergence form

We prove quantitative estimates for the stochastic homogenization of linear uniformly elliptic equations in nondivergence form. Under strong independence assumptions on the coefficients, we obtain optimal estimates on the subquadratic growth of the correctors with stretched exponential-type bounds in probability. Like the theory of Gloria and Otto \cite{GO1,GO2} for divergence form equations, the arguments rely on nonlinear concentration inequalities combined with certain estimates on the Green's functions and derivative bounds on the correctors. We obtain these analytic estimates by developing a $C^{1,1}$ regularity theory down to microscopic scale, which is of independent interest and is inspired by the $C^{0,1}$ theory introduced in the divergence form case by the first author and Smart \cite{AS2}.

math.AP↗
arXiv2017

Elliptic regularity and quantitative homogenization on percolation clusters

We establish quantitative homogenization, large-scale regularity and Liouville results for the random conductance model on a supercritical (Bernoulli bond) percolation cluster. The results are also new in the case that the conductivity is constant on the cluster. The argument passes through a series of renormalization steps: first, we use standard percolation results to find a large scale above which the geometry of the percolation cluster behaves (in a sense made precise) like that of Euclidean space. Then, following the work of Barlow, we find a succession of larger scales on which certain functional and elliptic estimates hold. This gives us the analytic tools to adapt the quantitative homogenization program of Armstrong and Smart to estimate the yet larger scale on which solutions on the cluster can be well-approximated by harmonic functions on $\mathbb{R}^d$. This is the first quantitative homogenization result in a porous medium and the harmonic approximation allows us to estimate the scale on which a higher-order regularity theory holds. The size of each of these random scales is shown to have at least a stretched exponential moment. As a consequence of this regularity theory, we obtain a Liouville-type result that states that, for each $k\in\mathbb{N}$, the vector space of solutions growing at most like $o(|x|^{k+1})$ as $|x|\to \infty$ has the same dimension as the set of harmonic polynomials of degree at most $k$, generalizing a result of Benjamini, Duminil-Copin, Kozma, and Yadin from $k\le1$ to $k\in\mathbb{N}$.

math.PR↗
arXiv2016

Scaling limits of energies and correctors

In stochastic homogenization of elliptic equations, the corrector plays a central role. Under a finite range of dependence assumption on the coefficient field, we show that the large-scale spatial averages of the corrector approach those of a variant of the Gaussian free field. In contrast to previous work, the argument does not rely on an underlying product structure of the probability measure. Instead, we rely on the additivity of energy quantities to show central limit theorems for these, and derive the large-scale behavior of the corrector as a consequence.

math.AP↗
arXiv2016

The additive structure of elliptic homogenization

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 \cite{AKM}: 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.

math.AP↗
arXiv2016

Quantitative analysis of boundary layers in periodic homogenization

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 and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.

math.AP↗
arXiv2016

Bounded correctors in almost periodic homogenization

We show that certain linear elliptic equations (and systems) in divergence form with almost periodic coefficients have bounded, almost periodic correctors. This is proved under a new condition we introduce which quantifies the almost periodic assumption and includes (but is not restricted to) the class of smooth, quasiperiodic coefficient fields which satisfy a Diophantine-type condition previously considered by Kozlov. The proof is based on a quantitative ergodic theorem for almost periodic functions combined with the new regularity theory recently introduced by the first author and Shen for equations with almost periodic coefficients. This yields control on spatial averages of the gradient of the corrector, which is converted into estimates on the size of the corrector itself via a multiscale Poincaré-type inequality.

math.AP↗
arXiv2016

Stochastic homogenization of quasilinear Hamilton-Jacobi equations and geometric motions

We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing geometric motions of level sets as well as a large class of viscous, non-convex Hamilton-Jacobi equations. The main results include the first proof of qualitative stochastic homogenization for such equations. We also present quantitative error estimates which give an algebraic rate of homogenization.

math.AP↗
arXiv2016

Mesoscopic higher regularity and subadditivity in elliptic homogenization

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 ($C^{k}$, $k \geq 1$) 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.

math.AP↗