Communications on Pure and Applied Mathematics最新文献

{"title":"Complex analytic dependence on the dielectric permittivity in ENZ materials: The photonic doping example","authors":"Robert V. Kohn, Raghavendra Venkatraman","doi":"10.1002/cpa.22138","DOIUrl":"10.1002/cpa.22138","url":null,"abstract":"<p>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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Ω</mi>\u0000 <mo>×</mo>\u0000 <mi>R</mi>\u0000 </mrow>\u0000 <annotation>$Omega times mathbb {R}$</annotation>\u0000 </semantics></math> is affected by the presence of a “dopant” <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>⊂</mo>\u0000 <mi>Ω</mi>\u0000 </mrow>\u0000 <annotation>$D subset Omega$</annotation>\u0000 </semantics></math> in which the dielectric permittivity is not near zero. Mathematically, this reduces to analysis of a 2D Helmholtz equation <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>div</mi>\u0000 <mspace></mspace>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>a</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∇</mo>\u0000 <mi>u</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 <msup>\u0000 <mi>k</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mi>u</mi>\u0000 <mo>=</mo>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 <annotation>$mathrm{div}, (a(x)nabla u) + k^2 u = f$</annotation>\u0000 </semantics></math> with a piecewise-constant, complex valued coefficient <i>a</i> that is nearly infinite (say <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>=</mo>\u0000 <mfrac>\u0000 <mn>1</mn>\u0000 <mi>δ</mi>\u0000 </mfrac>\u0000 </mrow>\u0000 <annotation>$a = frac{1}{delta }$</annotation>\u0000 </semantics></math> with <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>δ</mi>\u0000 <mo>≈</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$delta approx 0$</annotation>\u0000 </semantics></math>) in <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Ω</mi>\u0000 <mo>∖</mo>\u0000 <mover>\u0000 <mi>D</mi>\u0000 <mo>¯</mo>\u0000 </mover>\u0000 </mrow>\u0000 <annotation>$Omega setminus overline{D}$</annotation>\u0000 </semantics></math>. We show (under suitable hypotheses) that the solution <i>u</i> depends analytically on δ near 0, and we give a simple PDE characterization of the terms in its Taylor expansion. For the appli","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1278-1301"},"PeriodicalIF":3.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911078","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Landscape complexity beyond invariance and the elastic manifold","authors":"Gérard Ben Arous,&nbsp;Paul Bourgade,&nbsp;Benjamin McKenna","doi":"10.1002/cpa.22146","DOIUrl":"10.1002/cpa.22146","url":null,"abstract":"<p>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 the Larkin mass, confirming formulas of Fyodorov and Le Doussal. One essential, dynamical, step of the proof also applies to a general signal-to-noise model of soft spins in an anisotropic well, for which we prove a negative-second-moment threshold distinguishing positive from zero complexity. A universal near-critical behavior appears within this phase portrait, namely quadratic near-critical vanishing of the complexity of total critical points, and cubic near-critical vanishing of the complexity of local minima. These two models serve as a paradigm of complexity calculations for Gaussian landscapes exhibiting few distributional symmetries, that is, beyond the invariant setting. The two main inputs for the proof are determinant asymptotics for non-invariant random matrices from our companion paper (Ben Arous, Bourgade, McKenna 2022), and the atypical convexity and integrability of the limiting variational problems.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1302-1352"},"PeriodicalIF":3.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135487689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Phase diagram and topological expansion in the complex quartic random matrix model","authors":"Pavel Bleher,&nbsp;Roozbeh Gharakhloo,&nbsp;Kenneth T-R McLaughlin","doi":"10.1002/cpa.22164","DOIUrl":"10.1002/cpa.22164","url":null,"abstract":"<p>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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>N</mi>\u0000 <mi>j</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>g</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {N}_j(g)$</annotation>\u0000 </semantics></math> of 4-valent connected graphs with <i>j</i> vertices on a compact Riemann surface of genus <i>g</i>. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus <i>g</i>, we calculate the leading term in the asymptotics of <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>N</mi>\u0000 <mi>j</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>g</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$mathcal {N}_j(g)$</annotation>\u0000 </semantics></math> as the number of vertices tends to infinity. Using the theory of quadratic differentials, we characterize the critical contours in the complex parameter plane where phase transitions in the quartic model take place, thereby proving a result of David. These phase transitions are of the following four types: (a) one-cut to two-cut through the splitting of the cut at the origin, (b) two-cut to three-cut through the birth of a new cut at the origin, (c) one-cut to three-cut through the splitting of the cut at two symmetric points, and (d) one-cut to three-cut through the birth of two symmetric cuts.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1405-1485"},"PeriodicalIF":3.0,"publicationDate":"2023-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22164","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134911088","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Magnetic slowdown of topological edge states","authors":"Guillaume Bal,&nbsp;Simon Becker,&nbsp;Alexis Drouot","doi":"10.1002/cpa.22154","DOIUrl":"10.1002/cpa.22154","url":null,"abstract":"<p>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 at speed one, here they experience slowdown, dispersion, and Aharonov–Bohm effects. Several numerical simulations illustrate our results.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1235-1277"},"PeriodicalIF":3.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135879004","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Compressive phase retrieval: Optimal sample complexity with deep generative priors","authors":"Paul Hand,&nbsp;Oscar Leong,&nbsp;Vladislav Voroninski","doi":"10.1002/cpa.22155","DOIUrl":"10.1002/cpa.22155","url":null,"abstract":"<p>Advances in compressive sensing (CS) provided reconstruction algorithms of sparse signals from linear measurements with optimal sample complexity, but natural extensions of this methodology to nonlinear inverse problems have been met with potentially fundamental sample complexity bottlenecks. In particular, tractable algorithms for compressive phase retrieval with sparsity priors have not been able to achieve optimal sample complexity. This has created an open problem in compressive phase retrieval: under generic, phaseless linear measurements, are there tractable reconstruction algorithms that succeed with optimal sample complexity? Meanwhile, progress in machine learning has led to the development of new data-driven signal priors in the form of generative models, which can outperform sparsity priors with significantly fewer measurements. In this work, we resolve the open problem in compressive phase retrieval and demonstrate that generative priors can lead to a fundamental advance by permitting optimal sample complexity by a tractable algorithm. We additionally provide empirics showing that exploiting generative priors in phase retrieval can significantly outperform sparsity priors. These results provide support for generative priors as a new paradigm for signal recovery in a variety of contexts, both empirically and theoretically. The strengths of this paradigm are that (1) generative priors can represent some classes of natural signals more concisely than sparsity priors, (2) generative priors allow for direct optimization over the natural signal manifold, which is intractable under sparsity priors, and (3) the resulting non-convex optimization problems with generative priors can admit benign optimization landscapes at optimal sample complexity, perhaps surprisingly, even in cases of nonlinear measurements.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1147-1223"},"PeriodicalIF":3.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135936118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Thermodynamic limit of the first Lee-Yang zero","authors":"Jianping Jiang,&nbsp;Charles M. Newman","doi":"10.1002/cpa.22159","DOIUrl":"10.1002/cpa.22159","url":null,"abstract":"&lt;p&gt;We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;annotation&gt;${mathbb {R}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of finite-volume singularities in &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;${mathbb {C}}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. For the Ising model defined on a finite &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Λ&lt;/mi&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Lambda subset mathbb {Z}^d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; at inverse temperature &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$beta ge 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and external field &lt;i&gt;h&lt;/i&gt;, let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Λ&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$alpha _1(Lambda ,beta )$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the modulus of the first zero (that closest to the origin) of its partition function (in the variable &lt;i&gt;h&lt;/i&gt;). We prove that &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;Λ&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$alpha _1(Lambda ,beta )$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; decreases to &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;β&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$alpha _1(mathbb {Z}^d,beta )$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; as Λ increases to &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1224-1234"},"PeriodicalIF":3.0,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136024171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Fermi isospectrality for discrete periodic Schrödinger operators","authors":"Wencai Liu","doi":"10.1002/cpa.22161","DOIUrl":"10.1002/cpa.22161","url":null,"abstract":"&lt;p&gt;Let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Γ&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;⊕&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;⊕&lt;/mi&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mi&gt;⊕&lt;/mi&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Gamma =q_1mathbb {Z}oplus q_2 mathbb {Z}oplus ldots oplus q_dmathbb {Z}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;q&lt;/mi&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;/msub&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$q_lin mathbb {Z}_+$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mn&gt;2&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mtext&gt;…&lt;/mtext&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$l=1,2,ldots ,d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, are pairwise coprime. Let &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$Delta +V$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; be the discrete Schrödinger operator, where Δ is the discrete Laplacian on &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {Z}^d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and the potential &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;V&lt;/mi&gt;\u0000 &lt;mo&gt;:&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;mo&gt;→&lt;/mo&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$V:mathbb {Z}^drightarrow mathbb {C}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is Γ-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1126-1146"},"PeriodicalIF":3.0,"publicationDate":"2023-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136072215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A scaling limit of the parabolic Anderson model with exclusion interaction","authors":"Dirk Erhard,&nbsp;Martin Hairer","doi":"10.1002/cpa.22145","DOIUrl":"10.1002/cpa.22145","url":null,"abstract":"&lt;p&gt;We consider the (discrete) parabolic Anderson model &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;/&lt;/mo&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;Δ&lt;/mi&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;ξ&lt;/mi&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$partial u(t,x)/partial t=Delta u(t,x) +xi _t(x) u(t,x)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;t&lt;/mi&gt;\u0000 &lt;mo&gt;≥&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$tge 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;x&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;Z&lt;/mi&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$xin mathbb {Z}^d$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, where the ξ-field is &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathbb {R}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-valued and plays the role of a dynamic random environment, and Δ is the discrete Laplacian. We focus on the case in which ξ is given by a properly rescaled symmetric simple exclusion process under which it converges to an Ornstein–Uhlenbeck process. Scaling the Laplacian diffusively and restricting ourselves to a torus, we show that in dimension &lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;d&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mn&gt;3&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$d=3$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; upon considering a suitably renormalised version of the above equation, the sequence of solutions converges in law. As a by-product of our main result we obtain precise asymptotics for the survival probability of a si","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"1065-1125"},"PeriodicalIF":3.0,"publicationDate":"2023-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136071502","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Pure gravity traveling quasi-periodic water waves with constant vorticity","authors":"Massimiliano Berti,&nbsp;Luca Franzoi,&nbsp;Alberto Maspero","doi":"10.1002/cpa.22143","DOIUrl":"10.1002/cpa.22143","url":null,"abstract":"<p>We prove the existence of small amplitude time quasi-periodic solutions of the <i>pure gravity</i> water waves equations  with <i>constant vorticity</i>, for a bidimensional fluid over a flat bottom delimited by a space periodic free interface. Using a Nash-Moser implicit function iterative scheme we construct traveling nonlinear waves which pass through each other slightly deforming and retaining forever a quasiperiodic structure. These solutions exist for any fixed value of depth and gravity and restricting the vorticity parameter to a Borel set of asymptotically full Lebesgue measure.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"990-1064"},"PeriodicalIF":3.0,"publicationDate":"2023-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136108415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Critical local well-posedness for the fully nonlinear Peskin problem","authors":"Stephen Cameron,&nbsp;Robert M. Strain","doi":"10.1002/cpa.22139","DOIUrl":"10.1002/cpa.22139","url":null,"abstract":"<p>We study the problem where a one-dimensional elastic string is immersed in a two-dimensional steady Stokes fluid. This is known as the Stokes immersed boundary problem and also as the Peskin problem. We consider the case with equal viscosities and with a fully non-linear tension law; this model has been called the fully nonlinear Peskin problem. In this case we prove local in time wellposedness for arbitrary initial data in the scaling critical Besov space <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msubsup>\u0000 <mover>\u0000 <mi>B</mi>\u0000 <mo>̇</mo>\u0000 </mover>\u0000 <mrow>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>T</mi>\u0000 <mo>;</mo>\u0000 <msup>\u0000 <mi>R</mi>\u0000 <mn>2</mn>\u0000 </msup>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$dot{B}^{3/2}_{2,1}(mathbb {T}; mathbb {R}^2)$</annotation>\u0000 </semantics></math>. We additionally prove the optimal higher order smoothing effects for the solution. To prove this result we derive a new formulation of the boundary integral equation that describes the parametrization of the string, and we crucially utilize a new cancelation structure.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 2","pages":"901-989"},"PeriodicalIF":3.0,"publicationDate":"2023-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43194065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}