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Almost sharp lower bound for the nodal volume of harmonic functions Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-29 Alexander Logunov, Lakshmi Priya M. E., Andrea Sartori
This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let be a real‐valued harmonic function in with and . We prove where the doubling index is a notion of growth defined by This gives an almost sharp lower bound for the Hausdorff measure of the zero set of , which is conjectured to be linear in . The new ingredients of the article are
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Multiplicative chaos measures from thick points of log‐correlated fields Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-17 Janne Junnila, Gaultier Lambert, Christian Webb
We prove that multiplicative chaos measures can be constructed from extreme level sets or thick points of the underlying logarithmically correlated field. We develop a method which covers the whole subcritical phase and only requires asymptotics of suitable exponential moments for the field. As an application, we establish that these estimates hold for the logarithm of the absolute value of the characteristic
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Twisted Kähler–Einstein metrics in big classes Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-17 Tamás Darvas, Kewei Zhang
We prove existence of twisted Kähler–Einstein metrics in big cohomology classes, using a divisorial stability condition. In particular, when is big, we obtain a uniform Yau–Tian–Donaldson (YTD) existence theorem for Kähler–Einstein (KE) metrics. To achieve this, we build up from scratch the theory of Fujita–Odaka type delta invariants in the transcendental big setting, using pluripotential theory.
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Allen–Cahn solutions with triple junction structure at infinity Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-17 Étienne Sandier, Peter Sternberg
We construct an entire solution to the elliptic system where is a ‘triple‐well’ potential. This solution is a local minimizer of the associated energy in the sense that minimizes the energy on any compact set among competitors agreeing with outside that set. Furthermore, we show that along subsequences, the ‘blowdowns’ of given by approach a minimal triple junction as . Previous results had assumed
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The Calogero–Moser derivative nonlinear Schrödinger equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-06 Patrick Gérard, Enno Lenzmann
We study the Calogero–Moser derivative nonlinear Schrödinger NLS equation posed on the Hardy–Sobolev space with suitable . By using a Lax pair structure for this ‐critical equation, we prove global well‐posedness for and initial data with sub‐critical or critical ‐mass . Moreover, we prove uniqueness of ground states and also classify all traveling solitary waves. Finally, we study in detail the class
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Infinite‐width limit of deep linear neural networks Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-06 Lénaïc Chizat, Maria Colombo, Xavier Fernández‐Real, Alessio Figalli
This paper studies the infinite‐width limit of deep linear neural networks (NNs) initialized with random parameters. We obtain that, when the number of parameters diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear NN. Moreover, even if the weights remain random, we get their precise law along the
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Leapfrogging vortex rings for the three‐dimensional incompressible Euler equations Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-06 Juan Dávila, Manuel del Pino, Monica Musso, Juncheng Wei
A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three‐dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging
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Wegner estimate and upper bound on the eigenvalue condition number of non‐Hermitian random matrices Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-05-03 László Erdős, Hong Chang Ji
We consider non‐Hermitian random matrices of the form , where is a general deterministic matrix and consists of independent entries with zero mean, unit variance, and bounded densities. For this ensemble, we prove (i) a Wegner estimate, that is, that the local density of eigenvalues is bounded by and (ii) that the expected condition number of any bulk eigenvalue is bounded by ; both results are optimal
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Pearcey universality at cusps of polygonal lozenge tilings Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-04-30 Jiaoyang Huang, Fan Yang, Lingfu Zhang
We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases
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Inhomogeneous turbulence for the Wick Nonlinear Schrödinger equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-04-27 Zaher Hani, Jalal Shatah, Hui Zhu
We introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self‐interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial
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The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-04-19 Mark Sellke
We study the Langevin dynamics for spherical ‐spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound
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Every finite graph arises as the singular set of a compact 3‐D calibrated area minimizing surface Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-03-09 Zhenhua Liu
Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6‐manifold with the third Betti number , we construct a calibrated 3‐dimensional homologically area minimizing surface on equipped in a smooth metric , so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly twisted
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Delta‐convex structure of the singular set of distance functions Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-03-07 Tatsuya Miura, Minoru Tanaka
For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean
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Non‐degenerate minimal submanifolds as energy concentration sets: A variational approach Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-02-28 Guido De Philippis, Alessandro Pigati
We prove that every non‐degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg–Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet–Jerrard–Sternberg. The same proof applies also to the ‐Yang–Mills–Higgs
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A Liouville‐type theorem for cylindrical cones Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-02-23 Nick Edelen, Gábor Székelyhidi
Suppose that is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), , and a complete embedded minimal hypersurface of lying to one side of . If the density at infinity of is less than twice the density of , then we show that , where is the Hardt–Simon foliation of . This extends a result of L. Simon, where an additional smallness assumption is required for
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Diameter estimates in Kähler geometry Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-02-22 Bin Guo, Duong H. Phong, Jian Song, Jacob Sturm
Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for estimates for the Monge–Ampère equation, with a key improvement allowing degeneracies of the volume form of codimension strictly greater than one. As a consequence, we solve the long‐standing problem of uniform diameter bounds
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Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-12-21 Ujan Gangopadhyay, Subhroshekhar Ghosh, Kin Aun Tan
Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated
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Arnold diffusion in Hamiltonian systems on infinite lattices Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-12-17 Filippo Giuliani, Marcel Guardia
We consider a system of infinitely many penduli on an m-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying
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Chord measures in integral geometry and their Minkowski problems Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-12-11 Erwin Lutwak, Dongmeng Xi, Deane Yang, Gaoyong Zhang
To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their
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Overcrowding and separation estimates for the Coulomb gas Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-12-04 Eric Thoma
We prove several results for the Coulomb gas in any dimension d ≥ 2 $d \ge 2$ that follow from isotropic averaging, a transport method based on Newton's theorem. First, we prove a high-density Jancovici–Lebowitz–Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in dimension 2. The existence of microscopic
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The anisotropic min-max theory: Existence of anisotropic minimal and CMC surfaces Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-12-01 Guido De Philippis, Antonio De Rosa
We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth 3–dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension
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Critical sets of solutions of elliptic equations in periodic homogenization Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-20 Fanghua Lin, Zhongwei Shen
In this paper we study critical sets of solutions u ε $u_\varepsilon$ of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first-order correctors, we show that the ( d − 2 ) $(d-2)$ -dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε, provided that doubling indices for
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Infinite order phase transition in the slow bond TASEP Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-20 Sourav Sarkar, Allan Sly, Lingfu Zhang
In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to 1−ε$1-\varepsilon$ for some small ε>0$\varepsilon >0$. Janowsky and Lebowitz posed the well-known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations
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Subquadratic harmonic functions on Calabi-Yau manifolds with maximal volume growth Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-16 Shih-Kai Chiu
On a complete Calabi-Yau manifold M $M$ with maximal volume growth, a harmonic function with subquadratic polynomial growth is the real part of a holomorphic function. This generalizes a result of Conlon-Hein. We prove this result by proving a Liouville-type theorem for harmonic 1-forms, which follows from a new local L 2 $L^2$ estimate of the exterior derivative.
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Global solutions of the compressible Euler-Poisson equations with large initial data of spherical symmetry Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-13 Gui-Qiang G. Chen, Lin He, Yong Wang, Difan Yuan
We are concerned with a global existence theory for finite-energy solutions of the multidimensional Euler-Poisson equations for both compressible gaseous stars and plasmas with large initial data of spherical symmetry. One of the main challenges is the strengthening of waves as they move radially inward towards the origin, especially under the self-consistent gravitational field for gaseous stars.
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Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-13 William M. Feldman
We apply new results on free boundary regularity to obtain a quantitative convergence rate for the shape optimizers of the first Dirichlet eigenvalue in periodic homogenization. We obtain a linear (with logarithmic factors) convergence rate for the optimizing eigenvalue. Large scale Lipschitz free boundary regularity of almost minimizers is used to apply the optimal L2$L^2$ homogenization theory in
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A dynamical approach to the study of instability near Couette flow Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-08 Hui Li, Nader Masmoudi, Weiren Zhao
In this paper, we obtain the optimal instability threshold of the Couette flow for Navier–Stokes equations with small viscosity ν > 0 $\nu >0$ , when the perturbations are in the critical spaces H x 1 L y 2 $H^1_xL_y^2$ . More precisely, we introduce a new dynamical approach to prove the instability for some perturbation of size ν 1 2 − δ 0 $\nu ^{\frac{1}{2}-\delta _0}$ with any small δ 0 > 0 $\delta
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The maximum of log-correlated Gaussian fields in random environment Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-11-02 Florian Schweiger, Ofer Zeitouni
We study the distribution of the maximum of a large class of Gaussian fields indexed by a box VN⊂Zd$V_N\subset \mathbb {Z}^d$ and possessing logarithmic correlations up to local defects that are sufficiently rare. Under appropriate assumptions that generalize those in Ding et al., we show that asymptotically, the centered maximum of the field has a randomly-shifted Gumbel distribution. We prove that
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Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-30 Siva Athreya, Oleg Butkovsky, Khoa Lê, Leonid Mytnik
We study stochastic reaction–diffusion equation
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Integrability of SLE via conformal welding of random surfaces Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-19 Morris Ang, Nina Holden, Xin Sun
We demonstrate how to obtain integrability results for the Schramm-Loewner evolution (SLE) from Liouville conformal field theory (LCFT) and the mating-of-trees framework for Liouville quantum gravity (LQG). In particular, we prove an exact formula for the law of a conformal derivative of a classical variant of SLE called SLEκ(ρ−;ρ+)$\operatorname{SLE}_\kappa (\rho _-;\rho _+)$. Our proof is built
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On the incompressible limit for a tumour growth model incorporating convective effects Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-16 Noemi David, Markus Schmidtchen
In this work we study a tissue growth model with applications to tumour growth. The model is based on that of Perthame, Quirós, and Vázquez proposed in 2014 but incorporates the advective effects caused, for instance, by the presence of nutrients, oxygen, or, possibly, as a result of self-propulsion. The main result of this work is the incompressible limit of this model which builds a bridge between
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Log-Sobolev inequality for the φ24 and φ34 measures Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-16 Roland Bauerschmidt, Benoit Dagallier
The continuum φ24$\varphi ^4_2$ and φ34$\varphi ^4_3$ measures are shown to satisfy a log-Sobolev inequality uniformly in the lattice regularisation under the optimal assumption that their susceptibility is bounded. In particular, this applies to all coupling constants in any finite volume, and uniformly in the volume in the entire high temperature phases of the φ24$\varphi ^4_2$ and φ34$\varphi ^4_3$ models
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Log-Sobolev inequality for near critical Ising models Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-16 Roland Bauerschmidt, Benoit Dagallier
For general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the log-Sobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the log-Sobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover
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Magnetic helicity, weak solutions and relaxation of ideal MHD Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-08 Daniel Faraco, Sauli Lindberg, László Székelyhidi
We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics (MHD) in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations
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Soft Riemann-Hilbert problems and planar orthogonal polynomials Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-08 Haakan Hedenmalm
Riemann-Hilbert problems are jump problems for holomorphic functions along given interfaces. They arise in various contexts, for example, in the asymptotic study of certain nonlinear partial differential equations and in the asymptotic analysis of orthogonal polynomials. Matrix-valued Riemann-Hilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest
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Local laws and a mesoscopic CLT for β-ensembles Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-08 Luke Peilen
We study the statistical mechanics of the log-gas, or β-ensemble, for general potential and inverse temperature. By means of a bootstrap procedure, we prove local laws on the next order energy that are valid down to microscopic length scales. To our knowledge, this is the first time that this kind of a local quantity has been controlled for the log-gas. Simultaneously, we exhibit a control on fluctuations
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Hölder regularity of the Boltzmann equation past an obstacle Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-06 Chanwoo Kim, Donghyun Lee
Regularity and singularity of the solutions according to the shape of domains is a challenging research theme in the Boltzmann theory. In this paper, we prove an Hölder regularity in Cx,v0,12−$C^{0,\frac{1}{2}-}_{x,v}$ for the Boltzmann equation of the hard-sphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle
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Conformal covariance of connection probabilities and fields in 2D critical percolation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-05 Federico Camia
Fitting percolation into the conformal field theory framework requires showing that connection probabilities have a conformally invariant scaling limit. For critical site percolation on the triangular lattice, we prove that the probability that n vertices belong to the same open cluster has a well-defined scaling limit for every n ≥ 2 $n \ge 2$ . Moreover, the limiting functions P n ( x 1 , … , x n
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Multiplicity-1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-05 Costante Bellettini
We address the one-parameter minmax construction for the Allen–Cahn energy that has recently lead to a new proof of the existence of a closed minimal hypersurface in an arbitrary compact Riemannian manifold Nn+1$N^{n+1}$ with n≥2$n\ge 2$ (Guaraco's work, relying on works by Hutchinson, Tonegawa, and Wickramasekera when sending the Allen–Cahn parameter to 0). We obtain the following result: if the Ricci
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Sharp asymptotic estimates for expectations, probabilities, and mean first passage times in stochastic systems with small noise Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-05 Tobias Grafke, Tobias Schäfer, Eric Vanden-Eijnden
Freidlin-Wentzell theory of large deviations can be used to compute the likelihood of extreme or rare events in stochastic dynamical systems via the solution of an optimization problem. The approach gives exponential estimates that often need to be refined via calculation of a prefactor. Here it is shown how to perform these computations in practice. Specifically, sharp asymptotic estimates are derived
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Stationary measure for the open KPZ equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-05 Ivan Corwin, Alisa Knizel
We provide the first construction of stationary measures for the open KPZ equation on the spatial interval [0,1] with general inhomogeneous Neumann boundary conditions at 0 and 1 depending on real parameters u and v, respectively. When u+v≥0$u+v\ge 0$, we uniquely characterize the constructed stationary measures through their multipoint Laplace transform, which we prove is given in terms of a stochastic
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High-dimensional limit theorems for SGD: Effective dynamics and critical scaling Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-04 Gérard Ben Arous, Reza Gheissari, Aukosh Jagannath
We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the dimension goes to infinity. Our approach allows one to choose the summary statistics that are tracked, the initialization, and the step-size. It yields both ballistic
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Sine-kernel determinant on two large intervals Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-04 Benjamin Fahs, Igor Krasovsky
We consider the probability of two large gaps (intervals without eigenvalues) in the bulk scaling limit of the Gaussian Unitary Ensemble of random matrices. We determine the multiplicative constant in the asymptotics. We also provide the full explicit asymptotics (up to decreasing terms) for the transition between one and two large gaps.
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Almost monotonicity formula for H-minimal Legendrian surfaces in the Heisenberg group Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-03 Tristan Rivière
We prove an almost monotonicity formula for H-minimal Legendrian Surfaces (also called contact stationary Legendrian immersions or Hamiltonian stationary immersions) in the Heisenberg Group H2${\mathbb {H}}^2$. From this formula we deduce a Bernstein-Liouville type theorem for H-minimal Legendrian Surfaces. We also present some possible range of applications of this formula.
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Directed mean curvature flow in noisy environment Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-03 Andris Gerasimovičs, Martin Hairer, Konstantin Matetski
We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity
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The forbidden region for random zeros: Appearance of quadrature domains Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-10-02 Alon Nishry, Aron Wennman
Our main discovery is a surprising interplay between quadrature domains on the one hand, and the zeros of the Gaussian Entire Function (GEF) on the other. Specifically, consider the GEF conditioned on the rare hole event that there are no zeros in a given large Jordan domain. We show that in the natural scaling limit, a quadrature domain enclosing the hole emerges as a forbidden region, where the zero
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Optimal regularity for supercritical parabolic obstacle problems Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-29 Xavier Ros-Oton, Clara Torres-Latorre
We study the obstacle problem for parabolic operators of the type ∂t+L$\partial _t + L$, where L is an elliptic integro-differential operator of order 2s, such as (−Δ)s$(-\Delta )^s$, in the supercritical regime s∈(0,12)$s \in (0,\frac{1}{2})$. The best result in this context was due to Caffarelli and Figalli, who established the Cx1,s$C^{1,s}_x$ regularity of solutions for the case L=(−Δ)s$L = (-\Delta
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Spectrum of random d-regular graphs up to the edge Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-28 Jiaoyang Huang, Horng-Tzer Yau
Consider the normalized adjacency matrices of random d-regular graphs on N vertices with fixed degree d⩾3$d\geqslant 3$. We prove that, with probability 1−N−1+ε$1-N^{-1+\varepsilon }$ for any ε>0$\varepsilon >0$, the following two properties hold as N→∞$N \rightarrow \infty$ provided that d⩾3$d\geqslant 3$: (i) The eigenvalues are close to the classical eigenvalue locations given by the Kesten–McKay
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Erratum for “Global Identifiability of Differential Models” Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-22 Hoon Hong, Alexey Ovchinnikov, Gleb Pogudin, Chee Yap
We are grateful to Peter Thompson for pointing out an error in [1, Lemma 3.5, p. 1848]. The original proof worked only under the assumption that θ̂$\hat{\theta }$ is a vector of constants. However, some of the components of θ̂$\hat{\bm{\theta }}$ could be the states of the dynamic under consideration, and the lemma was used in such a setup (i.e., with θ̂$\hat{\bm{\theta }}$ involving states) later
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Discrete honeycombs, rational edges, and edge states Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-22 Charles L. Fefferman, Sonia Fliss, Michael I. Weinstein
Consider the tight binding model of graphene, sharply terminated along an edge l parallel to a direction of translational symmetry of the underlying period lattice. We classify such edges l into those of “zigzag type” and those of “armchair type”, generalizing the classical zigzag and armchair edges. We prove that zero energy / flat band edge states arise for edges of zigzag type, but never for those
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An upper Minkowski dimension estimate for the interior singular set of area minimizing currents Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-18 Anna Skorobogatova
We show that for an area minimizing m-dimensional integral current T of codimension at least two inside a sufficiently regular Riemannian manifold, the upper Minkowski dimension of the interior singular set is at most m − 2 $m-2$ . This provides a strengthening of the existing ( m − 2 ) $(m-2)$ -dimensional Hausdorff dimension bound due to Almgren and De Lellis & Spadaro. As a by-product of the proof
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Logarithmic cotangent bundles, Chern-Mather classes, and the Huh-Sturmfels involution conjecture Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-15 Laurenţiu G. Maxim, Jose Israel Rodriguez, Botong Wang, Lei Wu
Using compactifications in the logarithmic cotangent bundle, we obtain a formula for the Chern classes of the pushforward of Lagrangian cycles under an open embedding with normal crossing complement. This generalizes earlier results of Aluffi and Wu-Zhou. The first application of our formula is a geometric description of Chern-Mather classes of an arbitrary very affine variety, generalizing earlier
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Landscape complexity beyond invariance and the elastic manifold Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-14 Gérard Ben Arous, Paul Bourgade, Benjamin McKenna
This paper characterizes the annealed, topological complexity (both of total critical points and of local minima) of the elastic manifold. This classical model of a disordered elastic system captures point configurations with self-interactions in a random medium. We establish the simple versus glassy phase diagram in the model parameters, with these phases separated by a physical boundary known as
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Phase diagram and topological expansion in the complex quartic random matrix model Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-14 Pavel Bleher, Roozbeh Gharakhloo, Kenneth T-R McLaughlin
We use the Riemann–Hilbert approach, together with string and Toda equations, to study the topological expansion in the quartic random matrix model. The coefficients of the topological expansion are generating functions for the numbers N j ( g ) $\mathcal {N}_j(g)$ of 4-valent connected graphs with j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann
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Global minimizers of a large class of anisotropic attractive-repulsive interaction energies in 2D Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-14 José A. Carrillo, Ruiwen Shu
We study a large family of Riesz-type singular interaction potentials with anisotropy in two dimensions. Their associated global energy minimizers are given by explicit formulas whose supports are determined by ellipses under certain assumptions. More precisely, by parameterizing the strength of the anisotropic part we characterize the sharp range in which these explicit ellipse-supported configurations
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Complex analytic dependence on the dielectric permittivity in ENZ materials: The photonic doping example Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-14 Robert V. Kohn, Raghavendra Venkatraman
Motivated by the physics literature on “photonic doping” of scatterers made from “epsilon-near-zero” (ENZ) materials, we consider how the scattering of time-harmonic TM electromagnetic waves by a cylindrical ENZ region Ω × R $\Omega \times \mathbb {R}$ is affected by the presence of a “dopant” D ⊂ Ω $D \subset \Omega$ in which the dielectric permittivity is not near zero. Mathematically, this reduces
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Magnetic slowdown of topological edge states Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2023-09-12 Guillaume Bal, Simon Becker, Alexis Drouot
We study the propagation of wavepackets along curved interfaces between topological, magnetic materials. Our Hamiltonian is a massive Dirac operator with a magnetic potential. We construct semiclassical wavepackets propagating along the curved interface as adiabatic modulations of straight edge states under constant magnetic fields. While in the magnetic-free case, the wavepackets propagate coherently