arXiv subjects
Publications and source records attributed to Mircea Petrache.
We study random perturbations of quasi-periodic uniformly discrete sets in the $d$-dimensional euclidean space. By means of Diffraction Theory, we find conditions under which a quasi-periodic set $X$ can be almost surely recovered from its random perturbations. This extends the recent periodic case result of Yakir from "Recovery the lattice from its random perturbations" IMRN 2020 (arXiv:200201508).
We prove discrete-to-continuum convergence of interaction energies defined on lattices in the Euclidean space (with interactions beyond nearest neighbours) to a crystalline perimeter, and we discuss the possible Wulff shapes obtainable in this way. Exploiting the "multigrid construction" of quasiperiodic tilings (which is an extension of De Bruijn's "pentagrid" construction of Penrose tilings) we adapt the same techniques to also find the macroscopical homogenized perimeter when we microscopically rescale a given quasiperiodic tiling.
We introduce and study the unconstrained polarization (or Chebyshev) problem which requires to find an $N$-point configuration that maximizes the minimum value of its potential over a set $A$ in $p$-dimensional Euclidean space. This problem is compared to the constrained problem in which the points are required to belong to the set $A$. We find that for Riesz kernels $1/|x-y|^s$ with $s>p-2$ the optimum unconstrained configurations concentrate close to the set $A$ and based on this fundamental fact we recover the same asymptotic value of the polarization as for the more classical constrained problem on a class of $d$-rectifiable sets. We also investigate the new unconstrained problem in special cases such as for spheres and balls. In the last section we formulate some natural open problems and conjectures.
We develop criteria based on a calibration argument via discrete PDE and semidiscrete optimal transport, for finding sharp isoperimetric inequalities of the form $(\sharp Ω)^{d-1} \le C (\sharp \overrightarrow{\partialΩ})^d$ where $Ω$ is a subset of vertices of a graph and $\overrightarrow{\partialΩ}$ is the oriented edge-boundary of $Ω$, as well as the optimum isoperimetric shapes $Ω$. The method is a discrete counterpart to Optimal Transport and ABP method proofs valid in the continuum, and answers a question appearing in Hamamuki \cite{hamamuki}, extending that work valid for rectangular grids, to a larger class of graphs, including graphs dual to simplicial meshes of equal volume. We also connect the problem to the theory Voronoi tessellations and of Aleksandrov solutions from semidiscrete optimal transport. The role of the geometric-arithmetic inequality that was used in previous works in the continuum case and in the $\mathbb Z^d$-graph case is now played by a geometric cell-optimization constant, where the optimization problem is like in Minkowski's proof of his classical theorem for convex polyhedra. Finally, we study the optimal constant in the related discrete Neumann boundary problem, and present a series of possible directions for a further classification of discrete edge-isoperimetric constants and shapes.
We consider two-dimensional zero-temperature systems of $N$ particles to which we associate an energy of the form $$ \mathcal{E}[V](X):=\sum_{1\le i \sqrt{2}$, in which case ${\bar{\mathcal E}_{\mathrm{sq}}[V]}=-4$. To the best of our knowledge, this is the first proof of crystallization to the square lattice for a two-body interaction energy.
The Cohn-Kumar conjecture states that the triangular lattice in dimension 2, the $E_8$ lattice in dimension 8, and the Leech lattice in dimension 24 are universally minimizing in the sense that they minimize the total pair interaction energy of infinite point configurations for all completely monotone functions of the squared distance. This conjecture was recently proved by Cohn-Kumar-Miller-Radchenko-Viazovska in dimensions 8 and 24. We explain in this note how the conjecture implies the minimality of the same lattices for the Coulomb and Riesz renormalized energies as well as jellium and periodic jellium energies, hence settling the question of their minimization in dimensions 8 and 24.
Starting with the work of Preiss on the geometry of measures, the classification of uniform measures in $\mathbb R^d$ has remained open, except for $d=1$ and for compactly supported measures in $d=2$, and for codimension $1$. In this paper we study $1$-dimensional measures in $\mathbb R^d$ for all $d$ and classify uniform measures with connected $1$-dimensional support, which turn out to be homogeneous measures. We provide as well a partial classification of general uniform measures of dimension $1$ in the absence of the connected support hypothesis.
We investigate the minimization of the energy per point $E\_f$ among $d$-dimensional Bravais lattices, depending on the choice of pairwise potential equal to a radially symmetric function $f(|x|^2)$. We formulate criteria for minimality and non-minimality of some lattices for $E\_f$ at fixed scale based on the sign of the inverse Laplace transform of $f$ when $f$ is a superposition of exponentials, beyond the class of completely monotone functions. We also construct a family of non-completely monotone functions having the triangular lattice as the unique minimizer of $E\_f$ at any scale. For Lennard-Jones type potentials, we reduce the minimization problem among all Bravais lattices to a minimization over the smaller space of unit-density lattices and we establish a link to the maximum kissing problem. New numerical evidence for the optimality of particular lattices for all the exponents are also given. We finally design one-well potentials $f$ such that the square lattice has lower energy $E\_f$ than the triangular one. Many open questions are also presented.
We consider two sharp next-order asymptotics problems, namely the asymptotics for the minimum energy for optimal point configurations and the asymptotics for the many-marginals Optimal Transport, in both cases with Riesz costs with inverse power-law long range interactions. The first problem describes the ground state of a Coulomb or Riesz gas, while the second appears as a semiclassical limit of the Density Functional Theory energy modelling a quantum version of the same system. Recently the second-order term in these expansions was precisely described, and corresponds respectively to a Jellium and to a Uniform Electron Gas model. The present work shows that for inverse-power-law interactions with power $s\in [d-2,d)$ in $d$ dimensions, the two problems have the same minimum value asymptotically. For the Coulomb case in $d=3$, our result verifies the physicists' long-standing conjecture regarding the equality of the second-order terms for these two problems. Furthermore, our work implies that, whereas minimum values are equal, the minimizers may be different. Moreover, provided that the crystallization hypothesis in $d=3$ holds, which is analogous to Abrikosov's conjecture in $d=2$, then our result verifies the physicists' conjectured $\approx 1.4442$ lower bound on the famous Lieb-Oxford constant. Our work also rigorously confirms some of the predictions formulated by physicists, regarding the optimal value of the Uniform Electron Gas second-order asymptotic term. We also show that on the whole range $s\in(0,d)$, the Uniform Electron Gas second-order constant is continuous in $s$.
In the setting of optimal transport with $N\ge 2$ marginals, a necessary condition for transport plans to be optimal is that they are $c$-cyclically monotone. For $N=2$ there exist several proofs that in very general settings $c$-cyclical monotoncity is also sufficient for optimality, while for $N\ge 3$ this is only known under strong conditions on $c$. Here we give a counterexample which shows that $c$-cylclical monotonicity is in general not sufficient for optimality if $N\ge 3$. Comparison with the $N=2$ case shows how the main proof strategies valid for the case $N=2$ might fail for $N\ge 3$. We leave open the question of what is the optimal condition on $c$ under which $c$-cyclical monotonicity is sufficient for optimality. The new concept of an $N$-flow seems to be helpful for understanding the counterexample: our construction is based on the absence of finite-support $N$-cycles in the set where our counterexample cost $c$ is finite. To follow this idea we formulate a Smirnov-type decomposition for $N$-flows.
Motivated by a problem arising from Density Functional Theory, we provide the sharp next-order asymptotics for a class of multimarginal optimal transport problems with cost given by singular, long-range pairwise interaction potentials. More precisely, we consider an $N$-marginal optimal transport problem with $N$ equal marginals supported on $\mathbb R^d$ and with cost of the form $\sum_{i\neq j}|x_i-x_j|^{-s}$. In this setting we determine the second-order term in the $N\to\infty$ asymptotic expansion of the minimum energy, for the long-range interactions corresponding to all exponents $0<s<d$. We also prove a small oscillations property for this second-order energy term. Our results can be extended to a larger class of models than power-law-type radial costs, such as non-rotationally-invariant costs. The key ingredient and main novelty in our proofs is a robust extension and simplification of the Fefferman-Gregg decomposition (Fefferman 1985, Gregg 1989), extended here to our class of kernels, and which provides a unified method valid across our full range of exponents. Our first result generalizes a recent work of Lewin, Lieb and Seiringer (2017), who dealt with the second-order term for the Coulomb case $s=1,d=3$, by different methods.
The space of Sobolev connections, as it has been introduced for studying the variation of Yang-Mills Lagrangian in the critical dimension $4$, happens not to be weakly sequentially complete in dimension larger than $4$. This is a major obstruction for studying the variations of this important Lagrangian in high dimensions. The present paper generalizes the previous result of the authors from 'The resolution of the Yang-Mills Plateau problem in super-critical dimensions', Adv. Math. 316 (arxiv:1306.2010), valid in $5$ dimensions to arbitrary dimension and introduces a space of so called 'weak connections' for which we prove the weak sequential closure under Yang-Mills energy control. We also establish a strong approximation property of any weak connection by smooth connections away from codimension $5$ polyhedral sets. This last property is used in a subsequent work in preparation for establishing the partial regularity property for general stationary Yang-Mills weak connections.
We consider the thin-film equation $\partial_t h + \nabla \cdot \left(h^2 \nabla Δh\right) = 0$ in physical space dimensions (i.e., one dimension in time $t$ and two lateral dimensions with $h$ denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier-Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the $1+1$-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porous-medium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity.
We consider the minimization of theta functions $θ\_Λ(α)=\sum\_{p\inΛ}e^{-πα|p|^2}$ amongst lattices $Λ\subset \mathbb R^d$, by reducing the dimension of the problem, following as a motivation the case $d=3$, where minimizers are supposed to be either the BCC or the FCC lattices. A first way to reduce dimension is by considering layered lattices, and minimize either among competitors presenting different sequences of repetitions of the layers, or among competitors presenting different shifts of the layers with respect to each other. The second case presents the problem of minimizing theta functions also on translated lattices, namely minimizing $(L,u)\mapsto θ\_{L+u}(α)$. Another way to reduce dimension is by considering lattices with a product structure or by successively minimizing over concentric layers. The first direction leads to the question of minimization amongst orthorhombic lattices, whereas the second is relevant for asymptotics questions, which we study in detail in two dimensions.
In this work we consider an optimal transport problem with coefficients in a normed Abelian group $G$, and extract a purely intrinsic condition on $G$ that guarantees that the optimal transport (or the corresponding minimum filling) is not branching. The condition turns out to be equivalent to the nonbranching of minimum fillings in geodesic metric spaces. We completely characterize finitely generated normed groups and finite-dimensional normed vector spaces of coefficients that induce nonbranching optimal transport plans. We also provide a complete classification of normed groups for which the optimal transport plans, besides being nonbranching, have acyclic support. This seems to initiate a new geometric classifications of certain normed groups. In the nonbranching case we also provide a global version of calibration, i.e. a generalization of Monge-Kantorovich duality.
For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in the case of Coulomb gases we show that the energy concentration is precisely determined by the macroscopic density of points, independently of the scale. This uses the "jellium energy" which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy. As a corollary, we obtain sharp error bounds on the discrepancy between the number of points and its expected average of optimal point configurations for Coulomb gases, extending previous results valid only for $2$-dimensional log-gases. For Riesz gases with interaction potentials $g(x)=|x|^{-s}, s\in]\min\{0,d-2\},d[$ and one-dimensional log-gases, we prove the same equidistribution result under an extra hypothesis on the decay of the localized energy, which we conjecture to hold for minimizing configurations. In this case we use the Caffarelli-Silvestre description of the non-local fractional Laplacians in $\mathbb R^d$ to localize the problem.
We show that for a metric space with an even number of points there is a 1-Lipschitz map to a tree-like space with the same matching number. This result gives the first basic version of an unoriented Kantorovich duality. The study of the duality gives a version of global calibrations for 1-chains with coefficients in $\mathbb Z_2$. Finally we extend the results to infinite metric spaces and present a notion of "matching dimension" which arises naturally.
We study the minimization problem for the Yang-Mills energy under fixed boundary connection in supercritical dimension $n\geq 5$. We define the natural function space A_{G} in which to formulate this problem in analogy to the space of integral currents used for the classical Plateau problem. The space A_{G} can be also interpreted as a space of weak connections on a "real measure theoretic version" of reflexive sheaves from complex geometry. We prove the weak closure result which ensures the existence of energy-minimizing weak connections in A_{G}. We then prove that any weak connection from A_{G} can be obtained as a L^2-limit of classical connections over bundles with defects. This approximation result is then extended to a Morrey analogue. We prove the optimal regularity result for Yang-Mills local minimizers. On the way to prove this result we establish a Coulomb gauge extraction theorem for weak curvatures with small Yang-Mills density. This generalizes to the general framework of weak $L^2$ curvatures previous works of Meyer-Rivière and Tao-Tian in which respectively a strong approximability property and an admissibility property were assumed in addition.