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Persistence and ball exponents for Gaussian stationary processes Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-29 Naomi D. Feldheim, Ohad N. Feldheim, Sumit Mukherjee
Consider a real Gaussian stationary process , indexed on either or and admitting a spectral measure . We study , the persistence exponent of . We show that, if has a positive density at the origin, then the persistence exponent exists; moreover, if has an absolutely continuous component, then if and only if this spectral density at the origin is finite. We further establish continuity of in , in (under
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Boundary conditions and universal finite‐size scaling for the hierarchical |φ|4$|\varphi |^4$ model in dimensions 4 and higher Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-29 Emmanuel Michta, Jiwoon Park, Gordon Slade
We analyse and clarify the finite‐size scaling of the weakly‐coupled hierarchical ‐component model for all integers in all dimensions , for both free and periodic boundary conditions. For , we prove that for a volume of size with periodic boundary conditions the infinite‐volume critical point is an effective finite‐volume critical point, whereas for free boundary conditions the effective critical point
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Boundary statistics for the six‐vertex model with DWBC Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-24 Vadim Gorin, Karl Liechty
We study the behavior of configurations in the symmetric six‐vertex model with weights in the square with Domain Wall Boundary Conditions as . We prove that when , configurations near the boundary have fluctuations of order and are asymptotically described by the GUE‐corners process of random matrix theory. On the other hand, when , the fluctuations are of finite order and configurations are asymptotically
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On classification of global dynamics for energy‐critical equivariant harmonic map heat flows and radial nonlinear heat equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-18 Kihyun Kim, Frank Merle
We consider the global dynamics of finite energy solutions to energy‐critical equivariant harmonic map heat flow (HMHF) and radial nonlinear heat equation (NLH). It is known that any finite energy equivariant solutions to (HMHF) decompose into finitely many harmonic maps (bubbles) separated by scales and a body map, as approaching to the maximal time of existence. Our main result for (HMHF) gives a
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Maximum of the characteristic polynomial of i.i.d. matrices Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-08 Giorgio Cipolloni, Benjamin Landon
We compute the leading order asymptotic of the maximum of the characteristic polynomial for i.i.d. matrices with real or complex entries. In particular, this result is new even for real Ginibre matrices, which was left as an open problem in Lambert et al. Electron. J. Probab. 29 (2024); the complex Ginibre case was covered in Lambert, Comm. Math Phys. 378 (2020). These are the first universality results
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The porous medium equation: Large deviations and gradient flow with degenerate and unbounded diffusion Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-05 Benjamin Gess, Daniel Heydecker
The problem of deriving a gradient flow structure for the porous medium equation which is thermodynamic, in that it arises from the large deviations of some microscopic particle system is studied. To this end, a rescaled zero‐range process with jump rate is considered, and its hydrodynamic limit and dynamical large deviations are shown in the presence of both degenerate and unbounded diffusion. The
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Uniqueness on average of large isoperimetric sets in noncompact manifolds with nonnegative Ricci curvature Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-04-05 Gioacchino Antonelli, Marco Pozzetta, Daniele Semola
Let be a complete Riemannian manifold which is not isometric to , has nonnegative Ricci curvature, Euclidean volume growth, and quadratic Riemann curvature decay. We prove that there exists a set with density 1 at infinity such that for every there is a unique isoperimetric set of volume in ; moreover, its boundary is strictly volume preserving stable. The latter result cannot be improved to uniqueness
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First‐order Sobolev spaces, self‐similar energies and energy measures on the Sierpiński carpet Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-02-18 Mathav Murugan, Ryosuke Shimizu
For any , we construct ‐energies and the corresponding ‐energy measures on the Sierpiński carpet. A salient feature of our Sobolev space is the self‐similarity of energy. An important motivation for the construction of self‐similar energy and energy measures is to determine whether or not the Ahlfors regular conformal dimension is attained on the Sierpiński carpet. If the Ahlfors regular conformal
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Stability of perfectly matched layers for Maxwell's equations in rectangular solids Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-02-11 Laurence Halpern, Jeffrey Rauch
Perfectly matched layers are extensively used to compute approximate solutions for Maxwell's equations in using a bounded computational domain, usually a rectangular solid. A smaller rectangular domain of interest is surrounded by layers designed to absorb outgoing waves in perfectly reflectionless manner. On the boundary of the computational domain, an absorbing boundary condition is imposed that
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On the Read‐Shockley energy for grain boundaries in 2D polycrystals Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-02-11 Martino Fortuna, Adriana Garroni, Emanuele Spadaro
In the 50's Read and Shockley proposed a formula for the energy of small angle grain boundaries in polycrystals based on linearized elasticity and an ansatz on the distribution of incompatibilities of the lattice at the interface between two grains. The logarithmic scaling of this formula has been rigorously justified without any ansatz on the geometry of dislocations only recently in an article by
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Analysis of density matrix embedding theory around the non‐interacting limit Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-02-04 Eric Cancès, Fabian M. Faulstich, Alfred Kirsch, Eloïse Letournel, Antoine Levitt
This article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground‐state density matrix is a fixed‐point of the DMET map for non‐interacting systems, (ii) there exists a unique physical solution in the weakly‐interacting regime, and (iii) DMET is exact up to first order in the coupling parameter
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Special Lagrangian pair of pants Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-01-30 Yang Li
We construct special Lagrangian pair of pants in general dimensions, inside the cotangent bundle of with the Euclidean structure, building upon earlier topological ideas of Matessi. The construction uses a combination of PDE and geometric measure theory.
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Localized and degenerate controls for the incompressible Navier–Stokes system Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-01-29 Vahagn Nersesyan, Manuel Rissel
We consider the global approximate controllability of the two‐dimensional incompressible Navier–Stokes system driven by a physically localized and degenerate force. In other words, the fluid is regulated via four scalar controls that depend only on time and appear as coefficients in an effectively constructed driving force supported in a given subdomain. Our idea consists of squeezing low mode controls
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Phase transition of parabolic Ginzburg–Landau equation with potentials of high‐dimensional wells Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-01-28 Yuning Liu
In this work, we study the co‐dimensional one interface limit and geometric motions of parabolic Ginzburg–Landau systems with potentials of high‐dimensional wells. The main result generalizes the one by Lin et al. (Comm. Pure Appl. Math. 65 (2012), no. 6, 833–888) to a dynamical case. In particular combining modulated energy methods and weak convergence methods, we derive the limiting harmonic heat
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Polynomial lower bound on the effective resistance for the one‐dimensional critical long‐range percolation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2025-01-27 Jian Ding, Zherui Fan, Lu‐Jing Huang
In this work, we study the critical long‐range percolation (LRP) on , where an edge connects and independently with probability 1 for and with probability for some fixed . Viewing this as a random electric network where each edge has a unit conductance, we show that with high probability the effective resistances from the origin 0 to and from the interval to (conditioned on no edge joining and ) both
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A flow‐type scaling limit for random growth with memory Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-12-30 Amir Dembo, Kevin Yang
We study a stochastic Laplacian growth model, where a set grows according to a reflecting Brownian motion in stopped at level sets of its boundary local time. We derive a scaling limit for the leading‐order behavior of the growing boundary (i.e., “interface”). It is given by a geometric flow‐type pde. It is obtained by an averaging principle for the reflecting Brownian motion. We also show that this
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A dual‐space multilevel kernel‐splitting framework for discrete and continuous convolution Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-12-12 Shidong Jiang, Leslie Greengard
We introduce a new class of multilevel, adaptive, dual‐space methods for computing fast convolutional transformations. These methods can be applied to a broad class of kernels, from the Green's functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual‐space multilevel kernel‐splitting)
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On the isoperimetric Riemannian Penrose inequality Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-12-06 Luca Benatti, Mattia Fogagnolo, Lorenzo Mazzieri
We prove that the Riemannian Penrose inequality holds for asymptotically flat 3‐manifolds with nonnegative scalar curvature and connected horizon boundary, provided the optimal decay assumptions are met, which result in the mass being a well‐defined geometric invariant. Our proof builds on a novel interplay between the Hawking mass and a potential‐theoretic version of it, recently introduced by Agostiniani
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Randomly pivoted Cholesky: Practical approximation of a kernel matrix with few entry evaluations Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-12-04 Yifan Chen, Ethan N. Epperly, Joel A. Tropp, Robert J. Webber
The randomly pivoted Cholesky algorithm (RPCholesky) computes a factorized rank‐ approximation of an positive‐semidefinite (psd) matrix. RPCholesky requires only entry evaluations and additional arithmetic operations, and it can be implemented with just a few lines of code. The method is particularly useful for approximating a kernel matrix. This paper offers a thorough new investigation of the empirical
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Hydrodynamic large deviations of TASEP Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-11-22 Jeremy Quastel, Li‐Cheng Tsai
We consider the large deviations from the hydrodynamic limit of the Totally Asymmetric Simple Exclusion Process (TASEP). This problem was studied by Jensen and Varadhan and was shown to be related to entropy production in the inviscid Burgers equation. Here we prove the full large deviation principle. Our method relies on the explicit formula of Matetski, Quastel, and Remenik for the transition probabilities
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On the derivation of the homogeneous kinetic wave equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-11-21 Charles Collot, Pierre Germain
The nonlinear Schrödinger equation in the weakly nonlinear regime with random Gaussian fields as initial data is considered. The problem is set on the torus in any dimension greater than two. A conjecture in statistical physics is that there exists a kinetic time scale depending on the frequency localization of the data and on the strength of the nonlinearity, on which the expectation of the squares
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On the stability of Runge–Kutta methods for arbitrarily large systems of ODEs Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-11-20 Eitan Tadmor
We prove that Runge–Kutta (RK) methods for numerical integration of arbitrarily large systems of Ordinary Differential Equations are linearly stable. Standard stability arguments—based on spectral analysis, resolvent condition or strong stability, fail to secure the stability of RK methods for arbitrarily large systems. We explain the failure of different approaches, offer a new stability theory based
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Mean‐field limit of non‐exchangeable systems Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-11-16 Pierre‐Emmanuel Jabin, David Poyato, Juan Soler
This paper deals with the derivation of the mean‐field limit for multi‐agent systems on a large class of sparse graphs. More specifically, the case of non‐exchangeable multi‐agent systems consisting of non‐identical agents is addressed. The analysis does not only involve PDEs and stochastic analysis but also graph theory through a new concept of limits of sparse graphs (extended graphons) that reflect
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The α$\alpha$‐SQG patch problem is illposed in C2,β$C^{2,\beta }$ and W2,p$W^{2,p}$ Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-11-16 Alexander Kiselev, Xiaoyutao Luo
We consider the patch problem for the ‐(surface quasi‐geostrophic) SQG system with the values and being the 2D Euler and the SQG equations respectively. It is well‐known that the Euler patches are globally wellposed in non‐endpoint Hölder spaces, as well as in , spaces. In stark contrast to the Euler case, we prove that for , the ‐SQG patch problem is strongly illposed in every Hölder space with .
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Semiconvexity estimates for nonlinear integro‐differential equations Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-11-15 Xavier Ros‐Oton, Clara Torres‐Latorre, Marvin Weidner
In this paper we establish for the first time local semiconvexity estimates for fully nonlinear equations and for obstacle problems driven by integro‐differential operators with general kernels. Our proof is based on the Bernstein technique, which we develop for a natural class of nonlocal operators and consider to be of independent interest. In particular, we solve an open problem from Cabré‐Dipierro‐Valdinoci
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Rectifiability, finite Hausdorff measure, and compactness for non‐minimizing Bernoulli free boundaries Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-10-14 Dennis Kriventsov, Georg S. Weiss
While there are numerous results on minimizers or stable solutions of the Bernoulli problem proving regularity of the free boundary and analyzing singularities, much less is known about critical points of the corresponding energy. Saddle points of the energy (or of closely related energies) and solutions of the corresponding time‐dependent problem occur naturally in applied problems such as water waves
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On the Pólya conjecture for the Neumann problem in planar convex domains Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-10-10 N. Filonov
Denote by the counting function of the spectrum of the Neumann problem in the domain on the plane. G. Pólya conjectured that . We prove that for convex domains . Here is the first zero of the Bessel function .
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Uniqueness of the blow‐down limit for a triple junction problem Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-10-09 Zhiyuan Geng
We prove the uniqueness of blow‐down limit at infinity for an entire minimizing solution of a planar Allen–Cahn system with a triple‐well potential. Consequently, can be approximated by a triple junction map at infinity. The proof exploits a careful analysis of energy upper and lower bounds, ensuring that the diffuse interface remains within a small neighborhood of the approximated triple junction
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Smooth asymptotics for collapsing Calabi–Yau metrics Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-10-09 Hans‐Joachim Hein, Valentino Tosatti
We prove that Calabi–Yau metrics on compact Calabi–Yau manifolds whose Kähler classes shrink the fibers of a holomorphic fibration have a priori estimates of all orders away from the singular fibers. To this end, we prove an asymptotic expansion of these metrics in terms of powers of the fiber diameter, with ‐order remainders that satisfy uniform ‐estimates with respect to a collapsing family of background
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Integral formulation of Klein–Gordon singular waveguides Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-10-07 Guillaume Bal, Jeremy Hoskins, Solomon Quinn, Manas Rachh
We consider the analysis of singular waveguides separating insulating phases in two‐space dimensions. The insulating domains are modeled by a massive Schrödinger equation and the singular waveguide by appropriate jump conditions along the one‐dimensional interface separating the insulators. We present an integral formulation of the problem and analyze its mathematical properties. We also implement
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On minimizers in the liquid drop model Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-10-07 Otis Chodosh, Ian Ruohoniemi
We prove that round balls of volume uniquely minimize in Gamow's liquid drop model.
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On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-09-11 Yu Deng, Alexandru D. Ionescu, Fabio Pusateri
Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear
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Convergence to the planar interface for a nonlocal free‐boundary evolution Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-09-05 Felix Otto, Richard Schubert, Maria G. Westdickenberg
We capture optimal decay for the Mullins–Sekerka evolution, a nonlocal, parabolic free boundary problem from materials science. Our main result establishes convergence of BV solutions to the planar profile in the physically relevant case of ambient space dimension three. Far from assuming small or well‐prepared initial data, we allow for initial interfaces that do not have graph structure and are not
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Asymptotics of block Toeplitz determinants with piecewise continuous symbols Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-08-28 Estelle Basor, Torsten Ehrhardt, Jani A. Virtanen
We determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of
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Global regularity for critical SQG in bounded domains Comm. Pure Appl. Math. (IF 3.1) Pub Date : 2024-07-24 Peter Constantin, Mihaela Ignatova, Quoc‐Hung Nguyen
We prove the existence and uniqueness of global smooth solutions of the critical dissipative SQG equation in bounded domains in . We introduce a new methodology of transforming the single nonlocal nonlinear evolution equation in a bounded domain into an interacting system of extended nonlocal nonlinear evolution equations in the whole space. The proof then uses the method of the nonlinear maximum principle
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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