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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 : 20241014
Dennis Kriventsov, Georg S. WeissWhile 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

On the Pólya conjecture for the Neumann problem in planar convex domains Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20241010
N. FilonovDenote 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 .

Uniqueness of the blow‐down limit for a triple junction problem Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20241009
Zhiyuan GengWe 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


Smooth asymptotics for collapsing Calabi–Yau metrics Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20241009
Hans‐Joachim Hein, Valentino TosattiWe 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

Integral formulation of Klein–Gordon singular waveguides Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20241007
Guillaume Bal, Jeremy Hoskins, Solomon Quinn, Manas RachhWe 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

On minimizers in the liquid drop model Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20241007
Otis Chodosh, Ian RuohoniemiWe prove that round balls of volume uniquely minimize in Gamow's liquid drop model.

On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240911
Yu Deng, Alexandru D. Ionescu, Fabio PusateriOur 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


Convergence to the planar interface for a nonlocal free‐boundary evolution Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240905
Felix Otto, Richard Schubert, Maria G. WestdickenbergWe 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

Asymptotics of block Toeplitz determinants with piecewise continuous symbols Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240828
Estelle Basor, Torsten Ehrhardt, Jani A. VirtanenWe 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


Global regularity for critical SQG in bounded domains Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240724
Peter Constantin, Mihaela Ignatova, Quoc‐Hung NguyenWe 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


Almost sharp lower bound for the nodal volume of harmonic functions Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240529
Alexander Logunov, Lakshmi Priya M. E., Andrea SartoriThis 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

Multiplicative chaos measures from thick points of log‐correlated fields Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240517
Janne Junnila, Gaultier Lambert, Christian WebbWe 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

Twisted Kähler–Einstein metrics in big classes Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240517
Tamás Darvas, Kewei ZhangWe 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.

Allen–Cahn solutions with triple junction structure at infinity Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240517
Étienne Sandier, Peter SternbergWe 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

The Calogero–Moser derivative nonlinear Schrödinger equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240506
Patrick Gérard, Enno LenzmannWe 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

Infinite‐width limit of deep linear neural networks Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240506
Lénaïc Chizat, Maria Colombo, Xavier Fernández‐Real, Alessio FigalliThis 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

Leapfrogging vortex rings for the three‐dimensional incompressible Euler equations Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240506
Juan Dávila, Manuel del Pino, Monica Musso, Juncheng WeiA 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


Wegner estimate and upper bound on the eigenvalue condition number of non‐Hermitian random matrices Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240503
László Erdős, Hong Chang JiWe 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

Pearcey universality at cusps of polygonal lozenge tilings Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240430
Jiaoyang Huang, Fan Yang, Lingfu ZhangWe 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

Inhomogeneous turbulence for the Wick Nonlinear Schrödinger equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240427
Zaher Hani, Jalal Shatah, Hui ZhuWe 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

The threshold energy of low temperature Langevin dynamics for pure spherical spin glasses Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240419
Mark SellkeWe 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


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 : 20240309
Zhenhua LiuGiven 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


Delta‐convex structure of the singular set of distance functions Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240307
Tatsuya Miura, Minoru TanakaFor 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

Non‐degenerate minimal submanifolds as energy concentration sets: A variational approach Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240228
Guido De Philippis, Alessandro PigatiWe 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

A Liouville‐type theorem for cylindrical cones Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240223
Nick Edelen, Gábor SzékelyhidiSuppose 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

Diameter estimates in Kähler geometry Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20240222
Bin Guo, Duong H. Phong, Jian Song, Jacob SturmDiameter 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

Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231221
Ujan Gangopadhyay, Subhroshekhar Ghosh, Kin Aun TanGibbsian 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

Arnold diffusion in Hamiltonian systems on infinite lattices Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231217
Filippo Giuliani, Marcel GuardiaWe consider a system of infinitely many penduli on an mdimensional 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 nexttonearest neighbor or long range as long as it is strongly decaying

Chord measures in integral geometry and their Minkowski problems Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231211
Erwin Lutwak, Dongmeng Xi, Deane Yang, Gaoyong ZhangTo the families of geometric measures of convex bodies (the area measures of AleksandrovFenchelJessen, 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

Overcrowding and separation estimates for the Coulomb gas Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231204
Eric ThomaWe 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 highdensity Jancovici–Lebowitz–Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly subGaussian tails for charge excess in dimension 2. The existence of microscopic

The anisotropic minmax theory: Existence of anisotropic minimal and CMC surfaces Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231201
Guido De Philippis, Antonio De RosaWe 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 minmax 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

Critical sets of solutions of elliptic equations in periodic homogenization Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231120
Fanghua Lin, Zhongwei ShenIn this paper we study critical sets of solutions u ε $u_\varepsilon$ of secondorder elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the firstorder correctors, we show that the ( d − 2 ) $(d2)$ dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period ε, provided that doubling indices for

Infinite order phase transition in the slow bond TASEP Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231120
Sourav Sarkar, Allan Sly, Lingfu ZhangIn 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 wellknown question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations

Subquadratic harmonic functions on CalabiYau manifolds with maximal volume growth Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231116
ShihKai ChiuOn a complete CalabiYau 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 ConlonHein. We prove this result by proving a Liouvilletype theorem for harmonic 1forms, which follows from a new local L 2 $L^2$ estimate of the exterior derivative.

Global solutions of the compressible EulerPoisson equations with large initial data of spherical symmetry Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231113
GuiQiang G. Chen, Lin He, Yong Wang, Difan YuanWe are concerned with a global existence theory for finiteenergy solutions of the multidimensional EulerPoisson 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 selfconsistent gravitational field for gaseous stars.

Quantitative homogenization of principal Dirichlet eigenvalue shape optimizers Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231113
William M. FeldmanWe 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

A dynamical approach to the study of instability near Couette flow Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231108
Hui Li, Nader Masmoudi, Weiren ZhaoIn 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

The maximum of logcorrelated Gaussian fields in random environment Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231102
Florian Schweiger, Ofer ZeitouniWe 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 randomlyshifted Gumbel distribution. We prove that

Wellposedness of stochastic heat equation with distributional drift and skew stochastic heat equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231030
Siva Athreya, Oleg Butkovsky, Khoa Lê, Leonid MytnikWe study stochastic reaction–diffusion equation

Integrability of SLE via conformal welding of random surfaces Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231019
Morris Ang, Nina Holden, Xin SunWe demonstrate how to obtain integrability results for the SchrammLoewner evolution (SLE) from Liouville conformal field theory (LCFT) and the matingoftrees 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

On the incompressible limit for a tumour growth model incorporating convective effects Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231016
Noemi David, Markus SchmidtchenIn 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 selfpropulsion. The main result of this work is the incompressible limit of this model which builds a bridge between

LogSobolev inequality for the φ24 and φ34 measures Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231016
Roland Bauerschmidt, Benoit DagallierThe continuum φ24$\varphi ^4_2$ and φ34$\varphi ^4_3$ measures are shown to satisfy a logSobolev 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

LogSobolev inequality for near critical Ising models Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231016
Roland Bauerschmidt, Benoit DagallierFor general ferromagnetic Ising models whose coupling matrix has bounded spectral radius, we show that the logSobolev constant satisfies a simple bound expressed only in terms of the susceptibility of the model. This bound implies very generally that the logSobolev constant is uniform in the system size up to the critical point (including on lattices), without using any mixing conditions. Moreover

Magnetic helicity, weak solutions and relaxation of ideal MHD Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231008
Daniel Faraco, Sauli Lindberg, László SzékelyhidiWe revisit the issue of conservation of magnetic helicity and the WoltjerTaylor 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

Soft RiemannHilbert problems and planar orthogonal polynomials Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231008
Haakan HedenmalmRiemannHilbert 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. Matrixvalued RiemannHilbert problems were considered by Deift et al. in the 1990s with a noncommutative adaptation of the steepest

Local laws and a mesoscopic CLT for βensembles Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231008
Luke PeilenWe study the statistical mechanics of the loggas, or βensemble, 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 loggas. Simultaneously, we exhibit a control on fluctuations

Hölder regularity of the Boltzmann equation past an obstacle Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231006
Chanwoo Kim, Donghyun LeeRegularity 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 hardsphere molecule, which undergoes the elastic reflection in the intermolecular collision and the contact with the boundary of a convex obstacle

Conformal covariance of connection probabilities and fields in 2D critical percolation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231005
Federico CamiaFitting 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 welldefined scaling limit for every n ≥ 2 $n \ge 2$ . Moreover, the limiting functions P n ( x 1 , … , x n

Multiplicity1 minmax minimal hypersurfaces in manifolds with positive Ricci curvature Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231005
Costante BellettiniWe address the oneparameter 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

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 : 20231005
Tobias Grafke, Tobias Schäfer, Eric VandenEijndenFreidlinWentzell 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

Stationary measure for the open KPZ equation Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231005
Ivan Corwin, Alisa KnizelWe 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

Highdimensional limit theorems for SGD: Effective dynamics and critical scaling Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231004
Gérard Ben Arous, Reza Gheissari, Aukosh JagannathWe study the scaling limits of stochastic gradient descent (SGD) with constant stepsize in the highdimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finitedimensional 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 stepsize. It yields both ballistic

Sinekernel determinant on two large intervals Comm. Pure Appl. Math. (IF 3.1) Pub Date : 20231004
Benjamin Fahs, Igor KrasovskyWe 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.