ABSTRACT Let be a bounded Lipschitz domain. For any we show that for any Dirichlet eigenvalue , it holds, where, is given explicitly. This reduces the ‐loss version of Pólya's conjecture to a computational problem. This estimate is based on quantitative estimates on the remainder of the Weyl law with explicit constants, which we give a new proof without using Neumann eigenvalues. Our arguments in deriving such uniform estimates yield also, in all dimensions , classes of domains that may even have rather irregular shapes or boundaries but satisfy Pólya's conjecture. Another key observation is that on strip‐tiling domains (and therefore any triangles for instance) one actually has better eigenvalue estimates than Pólya conjectured.
We prove that the Yang-Mills (YM) measure for the trivial principal bundle over the two-dimensional torus, with any connected, compact structure group, is invariant for the associated renormalised Langevin dynamic. Our argument relies on a combination of regularity structures, lattice gauge-fixing and Bourgain's method for invariant measures. Several corollaries are presented including a gauge-fixed decomposition of the YM measure into a Gaussian free field and an almost Lipschitz remainder, and a proof of universality for the YM measure that we derive from a universality for the Langevin dynamic for a wide class of discrete approximations. The latter includes standard lattice gauge theories associated to Wilson, Villain and Manton actions. An important step in the argument, which is of independent interest, is a proof of uniqueness for the mass renormalisation of the gauge-covariant continuum Langevin dynamic, which allows us to identify the limit of discrete approximations. This latter result relies on Euler estimates for singular SPDEs and for Young ODEs arising from Wilson loops.
This paper is devoted to the quantitative homogenization of multiscale elliptic operator , where , , and . We assume that is 1-periodic in each and real analytic. Classically, the method of reiterated homogenization has been applied to study this multiscale elliptic operator, which leads to a convergence rate limited by the ratios . In the present paper, under the assumption of real analytic coefficients, we introduce the so-called multiscale correctors and more accurate effective operators, and improve the ratio part of the convergence rate to . This convergence rate is optimal in the sense that cannot be replaced by a larger constant. As a byproduct, the uniform Lipschitz estimate is established under a mild double-log scale-separation condition.
We verify the existence of full replica symmetry breaking (FRSB) for the Sherrington-Kirkpatrick (SK) model and determine the structure of its Parisi measure slightly beyond the high temperature regime. More specifically, we prove that the support of the Parisi measure for the SK model consists of an interval starting at the origin slightly beyond the high temperature regime.
Batchelor predicted that a passive scalar with diffusivity , advected by a smooth fluid velocity, should typically have Fourier mass distributed as for . For a broad class of velocity fields, we give a quantitative lower bound for a version of this prediction summed over constant width annuli in Fourier space. This improves on previously known results, which require the prediction to be summed over the whole ball.
We consider the liquid drop model with a positive background density in the thermodynamic limit. We prove a two-term asymptotics for the ground state energy per unit volume in the dilute limit. Our proof justifies the expectation that optimal configurations consist of droplets of unit size that arrange themselves according to minimizers for the Jellium problem for point particles. In particular, we provide the first rigorous derivation of what is known as the gnocchi phase in astrophysics.
It follows from the work of Kapovitch and Wilking that a closed manifold with nonnegative Ricci curvature has a uniformly almost nilpotent fundamental group. Leftover questions and conjectures, have asked if in this context the fundamental group is actually uniformly almost abelian. The main goal of this work is to construct examples with uniformly positive Ricci curvature whose fundamental groups cannot be uniformly virtually abelian.
In this paper, we consider the Toda lattice at thermal equilibrium, meaning that its variables and are independent Gaussian and Gamma random variables, respectively. This model can be thought of a dense collection of many "quasiparticles" that act as solitons. We establish a law of large numbers for the trajectory of these quasiparticles, showing that they travel with approximately constant velocities, which are explicit. Our proof is based on a direct analysis of the asymptotic scattering relation, an equation that approximately governs the dynamics of quasiparticles locations. This makes use of a regularization argument that essentially linearizes this relation, together with concentration estimates for the Toda lattice's (random) Lax matrix.
We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be . We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.
In this paper, we prove global existence of weak solutions, their regularization, and relaxation for large data for a broad class of Fokker-Planck-Alignment models, which appear in collective dynamics. The main feature of these results, as opposed to previously known ones, is the lack of regularity or no-vacuum requirements on the initial data. With a particular application to the classical kinetic Cucker-Smale model, we demonstrate that any bounded data with finite higher moment, , , gives rise to a global instantly smooth solution, satisfying entropy equality and relaxing exponentially fast. The results are achieved through the use of a new thickness-based renormalization procedure, which circumvents the problem of degenerate diffusion in nonperturbative regime.
Generalizing previous results of Arezzo-Pacard-Singer, Seyyedali-Székelyhidi and Hallam, we prove the invariance under smooth blowups of the class of weighted extremal Kähler manifolds, modulo a log-concavity assumption on the first weight. Through recent work of Di Nezza-Jubert-Lahdili and Han-Liu, this is obtained as a consequence of a general uniform coercivity estimate for the (relative, weighted) Mabuchi energy on the blowup, which applies more generally to any equivariant resolution of singularities of Fano type of a compact Kähler klt space whose Mabuchi energy is assumed to be coercive.
ABSTRACT We study the high‐dimensional dynamics of online stochastic gradient descent (SGD) for the multi‐spiked tensor model. This multi‐index model arises from the tensor principal component analysis (PCA) problem with multiple spikes, where the goal is to estimate the unknown signal vectors within the ‐dimensional unit sphere through maximum likelihood estimation from noisy observations of a ‐tensor. We determine the number of samples and the conditions on the signal‐to‐noise ratios (SNRs) required to efficiently recover the unknown spikes from natural random initializations. We show that full recovery of all spikes is possible provided a number of sample scaling as , matching the algorithmic threshold identified in the rank‐one case. Our results are obtained through a detailed analysis of a low‐dimensional system that describes the evolution of the correlations between the estimators and the spikes, while sharply controlling the noise in the dynamics. We find the spikes are recovered sequentially in a process we term “sequential elimination”: once a correlation exceeds a critical threshold, all correlations sharing a row or column index become sufficiently small, allowing the next correlation to grow and become macroscopic. The order in which correlations become macroscopic depends on their initial values and the corresponding SNRs, leading to either exact recovery or recovery of a permutation of the spikes. In the matrix case, when , if the SNRs are sufficiently separated, we achieve exact recovery of the spikes, whereas equal SNRs lead to recovery of the subspace spanned by them.
We study the steady states of the Euler equations on the periodic channel or annulus. We show that if these flows are laminar (layered by closed non-contractible streamlines which foliate the domain), then they must be either parallel or circular flows. We also show that a large subset of these shear flows are isolated from non-shear stationary states. For Poiseuille flow, , our result shows that all stationary solutions in a sufficiently small neighbourhood are shear flows. We then show that if with , then in any neighbourhood, there exist smooth non-shear steady states, traveling waves, and quasiperiodic solutions of any number of non-commensurate frequencies. This proves the rigidity near Poiseuille is sharp. Finally, we prove that on general compact doubly connected domains, laminar steady Euler flows with constant velocity on the boundary must also be either parallel or circular, and the domain a periodic channel or an annulus. This shows that laminar free boundary Euler solutions must have Euclidean symmetry.
The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in is obtained explicitly for generic rational initial data . An explicit asymptotic wave profile is given, in terms of the branches of the multivalued solution of the inviscid Burgers equation with initial data , such that the solution of the Benjamin-Ono equation with dispersion parameter and initial data satisfies in the locally uniform sense as , provided a discriminant inequality holds implying that certain caustic curves in the -plane are avoided. In some cases, this convergence implies strong convergence. The asymptotic profile is consistent with the modulated multiphase wave solutions described by Dobrokhotov and Krichever.
ABSTRACT We study the ground state energy of a gas of spin fermions with repulsive short‐range interactions. We derive an upper bound that agrees, at low density , with the Huang–Yang conjecture. The latter captures the first three terms in an asymptotic low‐density expansion, and in particular the Huang–Yang correction term of order . Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi‐bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero‐energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe–Goldstone equation.
The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high-dimensional settings. However, existing analyses of the algorithm for strongly log-concave distributions suggest that, as the dimension of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the error for the full set of variables, the behavior for a small number of variables can be significantly better: A number of iterations proportional to , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all -marginals. We refer to this effect as delocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log-concave distributions with certain sparse interactions. Our analysis relies on a novel metric to measure convergence. A key technical challenge we address is the lack of a one-step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting.
We provide a rigorous justification of the linearized Boltzmann and Landau equations for interacting particle systems with long-range interaction. The result shows that for a system of Hamiltonian particles governed by truncated power law potentials of the form near (with the effective radius of the particles), the covariance of the equilibrium fluctuations converges to solutions of kinetic equations in appropriate scaling limits and , corresponding to a low density regime. We prove that in Dimension 3, for , the limiting system approaches the uncut-off linearized Boltzmann equation for the scaling . The Coulomb singularity appears as a threshold value. Kinetic scaling limits with universally converge to the linearized Landau equation, and we prove the onset of the Coulomb logarithm for .
The joint moments of the derivatives of the characteristic polynomial of a random unitary matrix, and also a variant of the characteristic polynomial that is real on the unit circle, in the large matrix size limit, have been studied intensively in the past 25 years, partly in relation to conjectural connections to the Riemann -function and Hardy's function. We completely settle the most general version of the problem of convergence of these joint moments, after they are suitably rescaled, for an arbitrary number of derivatives and with arbitrary positive real exponents. Our approach relies on a hidden, higher-order exchangeable structure, that of an exchangeable array, which, as far as we know, had never been used before in the study of characteristic polynomials of random matrices. We then develop a systematic method, based on a class of Hankel determinants shifted by partitions, that allows us for the first time to give an exact representation of all these joint moments, for finite matrix size, in terms of derivatives of - Painlev & eacute; V transcendents. As an application, we can also represent all the joint moments of power sum linear statistics of a certain determinantal point process behind this problem in terms of derivatives of -Painlev & eacute; III' transcendents. This gives an efficient way to compute all these quantities explicitly. Our methods can be used to obtain analogous results for a number of other models sharing the same features.
We show that any open set that is a finite distance away from a Lipschitz subgraph will become a Lipschitz subgraph after flowing under fractional mean curvature flow for a finite, universal time. Our proof is quantitative and inherently nonlocal, as the corresponding statement is false for classical mean curvature flow. This is the first regularizing effect proven for weak solutions to nonlocal curvature flow.
We provide a complete solution to the problem of infinite quantum signal processing (QSP) for the class of Szeg & odblac; functions, which are functions that satisfy a logarithmic integrability condition and include almost any function that allows for a QSP representation. We do so by introducing a new algorithm called the Riemann-Hilbert-Weiss algorithm, which can compute any individual phase factor independent of all other phase factors. Our algorithm is also the first provably stable numerical algorithm for computing phase factors of any arbitrary Szeg & odblac; function. The proof of stability involves solving a Riemann-Hilbert factorization problem in nonlinear Fourier analysis using elements of spectral theory.
