Communications on Pure and Applied Mathematics最新文献_第10页

Phase diagram and topological expansion in the complex quartic random matrix model 复四次随机矩阵模型的相图与拓扑展开

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-14 DOI: 10.1002/cpa.22164

Pavel Bleher, Roozbeh Gharakhloo, Kenneth T-R McLaughlin

{"title":"Phase diagram and topological expansion in the complex quartic random matrix model","authors":"Pavel Bleher, Roozbeh Gharakhloo, Kenneth T-R McLaughlin","doi":"10.1002/cpa.22164","DOIUrl":"10.1002/cpa.22164","url":null,"abstract":"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 j vertices on a compact Riemann surface of genus g. We explicitly evaluate these numbers for Riemann surfaces of genus 0,1,2, and 3. Also, for a Riemann surface of an arbitrary genus g, 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.","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}

引用次数: 4

Magnetic slowdown of topological edge states 拓扑边缘态的磁减速

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-12 DOI: 10.1002/cpa.22154

Guillaume Bal, Simon Becker, Alexis Drouot

引用次数: 5

Compressive phase retrieval: Optimal sample complexity with deep generative priors 压缩相位检索:具有深度生成先验的最优样本复杂度

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-11 DOI: 10.1002/cpa.22155

Paul Hand, Oscar Leong, Vladislav Voroninski

{"title":"Compressive phase retrieval: Optimal sample complexity with deep generative priors","authors":"Paul Hand, Oscar Leong, Vladislav Voroninski","doi":"10.1002/cpa.22155","DOIUrl":"10.1002/cpa.22155","url":null,"abstract":"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.","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}

引用次数: 5

Thermodynamic limit of the first Lee-Yang zero 热力学极限第一李杨零

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-11 DOI: 10.1002/cpa.22159

Jianping Jiang, Charles M. Newman

{"title":"Thermodynamic limit of the first Lee-Yang zero","authors":"Jianping Jiang, Charles M. Newman","doi":"10.1002/cpa.22159","DOIUrl":"10.1002/cpa.22159","url":null,"abstract":"We complete the verification of the 1952 Yang and Lee proposal that thermodynamic singularities are exactly the limits in <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>${mathbb {R}}$</annotation>\u0000 </semantics></math> of finite-volume singularities in <math>\u0000 <semantics>\u0000 <mi>C</mi>\u0000 <annotation>${mathbb {C}}$</annotation>\u0000 </semantics></math>. For the Ising model defined on a finite <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Λ</mi>\u0000 <mo>⊂</mo>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$Lambda subset mathbb {Z}^d$</annotation>\u0000 </semantics></math> at inverse temperature <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>β</mi>\u0000 <mo>≥</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$beta ge 0$</annotation>\u0000 </semantics></math> and external field h, let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>α</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Λ</mi>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$alpha _1(Lambda ,beta )$</annotation>\u0000 </semantics></math> be the modulus of the first zero (that closest to the origin) of its partition function (in the variable h). We prove that <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>α</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>Λ</mi>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$alpha _1(Lambda ,beta )$</annotation>\u0000 </semantics></math> decreases to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>α</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <mo>,</mo>\u0000 <mi>β</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$alpha _1(mathbb {Z}^d,beta )$</annotation>\u0000 </semantics></math> as Λ increases to <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\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}

引用次数: 1

Fermi isospectrality for discrete periodic Schrödinger operators 离散周期Schrödinger算符的费米等谱性

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-10 DOI: 10.1002/cpa.22161

Wencai Liu

{"title":"Fermi isospectrality for discrete periodic Schrödinger operators","authors":"Wencai Liu","doi":"10.1002/cpa.22161","DOIUrl":"10.1002/cpa.22161","url":null,"abstract":"Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Γ</mi>\u0000 <mo>=</mo>\u0000 <msub>\u0000 <mi>q</mi>\u0000 <mn>1</mn>\u0000 </msub>\u0000 <mi>Z</mi>\u0000 <mi>⊕</mi>\u0000 <msub>\u0000 <mi>q</mi>\u0000 <mn>2</mn>\u0000 </msub>\u0000 <mi>Z</mi>\u0000 <mi>⊕</mi>\u0000 <mtext>…</mtext>\u0000 <mi>⊕</mi>\u0000 <msub>\u0000 <mi>q</mi>\u0000 <mi>d</mi>\u0000 </msub>\u0000 <mi>Z</mi>\u0000 </mrow>\u0000 <annotation>$Gamma =q_1mathbb {Z}oplus q_2 mathbb {Z}oplus ldots oplus q_dmathbb {Z}$</annotation>\u0000 </semantics></math>, where <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>q</mi>\u0000 <mi>l</mi>\u0000 </msub>\u0000 <mo>∈</mo>\u0000 <msub>\u0000 <mi>Z</mi>\u0000 <mo>+</mo>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$q_lin mathbb {Z}_+$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>l</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mtext>…</mtext>\u0000 <mo>,</mo>\u0000 <mi>d</mi>\u0000 </mrow>\u0000 <annotation>$l=1,2,ldots ,d$</annotation>\u0000 </semantics></math>, are pairwise coprime. Let <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>Δ</mi>\u0000 <mo>+</mo>\u0000 <mi>V</mi>\u0000 </mrow>\u0000 <annotation>$Delta +V$</annotation>\u0000 </semantics></math> be the discrete Schrödinger operator, where Δ is the discrete Laplacian on <math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^d$</annotation>\u0000 </semantics></math> and the potential <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>V</mi>\u0000 <mo>:</mo>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 <mo>→</mo>\u0000 <mi>C</mi>\u0000 </mrow>\u0000 <annotation>$V:mathbb {Z}^drightarrow mathbb {C}$</annotation>\u0000 </semantics></math> is Γ-periodic. We prove three rigidity theorems for discrete periodic Schrödinger operators in any dimension <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\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}

引用次数: 9

A scaling limit of the parabolic Anderson model with exclusion interaction 具有排斥相互作用的抛物型Anderson模型的标度极限

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-10 DOI: 10.1002/cpa.22145

Dirk Erhard, Martin Hairer

{"title":"A scaling limit of the parabolic Anderson model with exclusion interaction","authors":"Dirk Erhard, Martin Hairer","doi":"10.1002/cpa.22145","DOIUrl":"10.1002/cpa.22145","url":null,"abstract":"We consider the (discrete) parabolic Anderson model <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>∂</mi>\u0000 <mi>u</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>,</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>/</mo>\u0000 <mi>∂</mi>\u0000 <mi>t</mi>\u0000 <mo>=</mo>\u0000 <mi>Δ</mi>\u0000 <mi>u</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>,</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>+</mo>\u0000 <msub>\u0000 <mi>ξ</mi>\u0000 <mi>t</mi>\u0000 </msub>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mi>u</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>,</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$partial u(t,x)/partial t=Delta u(t,x) +xi _t(x) u(t,x)$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>t</mi>\u0000 <mo>≥</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$tge 0$</annotation>\u0000 </semantics></math>, <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>x</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>d</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$xin mathbb {Z}^d$</annotation>\u0000 </semantics></math>, where the ξ-field is <math>\u0000 <semantics>\u0000 <mi>R</mi>\u0000 <annotation>$mathbb {R}$</annotation>\u0000 </semantics></math>-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 <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>=</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$d=3$</annotation>\u0000 </semantics></math> 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}

引用次数: 5

Pure gravity traveling quasi-periodic water waves with constant vorticity 具有恒定涡度的纯重力行准周期水波

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-09 DOI: 10.1002/cpa.22143

Massimiliano Berti, Luca Franzoi, Alberto Maspero

引用次数: 23

Critical local well-posedness for the fully nonlinear Peskin problem 全非线性Peskin问题的临界局部适定性

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-08 DOI: 10.1002/cpa.22139

Stephen Cameron, Robert M. Strain

{"title":"Critical local well-posedness for the fully nonlinear Peskin problem","authors":"Stephen Cameron, Robert M. Strain","doi":"10.1002/cpa.22139","DOIUrl":"10.1002/cpa.22139","url":null,"abstract":"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.","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}

引用次数: 1

Constrained deformations of positive scalar curvature metrics, II 正标量曲率度量的约束变形，2

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-07 DOI: 10.1002/cpa.22153

Alessandro Carlotto, Chao Li

引用次数: 4

Convergence of the self-dual U(1)-Yang–Mills–Higgs energies to the ( n − 2 ) $(n-2)$ -area functional 自对偶U(1)‐Yang-Mills-Higgs能量收敛到(n−2)$(n-2)$‐面积泛函

IF 3 1区数学

Communications on Pure and Applied Mathematics Pub Date : 2023-09-07 DOI: 10.1002/cpa.22150

Davide Parise, Alessandro Pigati, Daniel Stern

引用次数: 0