Communications on Pure and Applied Mathematics最新文献

{"title":"Non-degenerate minimal submanifolds as energy concentration sets: A variational approach","authors":"Guido De Philippis,&nbsp;Alessandro Pigati","doi":"10.1002/cpa.22193","DOIUrl":"10.1002/cpa.22193","url":null,"abstract":"<p>We prove that every non-degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg–Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet–Jerrard–Sternberg. The same proof applies also to the <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>U</mi>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$U(1)$</annotation>\u0000 </semantics></math>-Yang–Mills–Higgs and to the Allen–Cahn–Hilliard energies. While for the latter energies gluing methods are also effective, in general dimension our proof is by now the only available one in the Ginzburg–Landau setting, where the weaker energy concentration is the main technical difficulty.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3581-3627"},"PeriodicalIF":3.0,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139987545","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 Liouville-type theorem for cylindrical cones","authors":"Nick Edelen,&nbsp;Gábor Székelyhidi","doi":"10.1002/cpa.22192","DOIUrl":"10.1002/cpa.22192","url":null,"abstract":"&lt;p&gt;Suppose that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;msubsup&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;/msubsup&gt;\u0000 &lt;mo&gt;⊂&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;+&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbf {C}_0^n subset mathbb {R}^{n+1}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is a smooth strictly minimizing and strictly stable minimal hypercone (such as the Simons cone), &lt;span&gt;&lt;/span&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;0&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$l ge 0$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, and &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; a complete embedded minimal hypersurface of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&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;mi&gt;l&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msup&gt;\u0000 &lt;annotation&gt;$mathbb {R}^{n+1+l}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; lying to one side of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$mathbf {C}= mathbf {C}_0 times mathbb {R}^l$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. If the density at infinity of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;annotation&gt;$M$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is less than twice the density of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;C&lt;/mi&gt;\u0000 &lt;annotation&gt;$mathbf {C}$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, then we show that &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;M&lt;/mi&gt;\u0000 &lt;mo&gt;=&lt;/mo&gt;\u0000 &lt;mi&gt;H&lt;/mi&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;/mrow&gt;\u0000 &lt;mo&gt;×&lt;/mo&gt;\u0000 &lt;msup&gt;\u0000 &lt;mi&gt;R&lt;/mi&gt;\u0000 &lt;mi&gt;l&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3557-3580"},"PeriodicalIF":3.0,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139938269","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":"Diameter estimates in Kähler geometry","authors":"Bin Guo,&nbsp;Duong H. Phong,&nbsp;Jian Song,&nbsp;Jacob Sturm","doi":"10.1002/cpa.22196","DOIUrl":"10.1002/cpa.22196","url":null,"abstract":"<p>Diameter estimates for Kähler metrics are established which require only an entropy bound and no lower bound on the Ricci curvature. The proof builds on recent PDE techniques for <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>L</mi>\u0000 <mi>∞</mi>\u0000 </msup>\u0000 <annotation>$L^infty$</annotation>\u0000 </semantics></math> 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 and Gromov–Hausdorff convergence of the Kähler–Ricci flow, for both finite-time and long-time solutions.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3520-3556"},"PeriodicalIF":3.0,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139928843","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":"Approximate Gibbsian structure in strongly correlated point fields and generalized Gaussian zero ensembles","authors":"Ujan Gangopadhyay,&nbsp;Subhroshekhar Ghosh,&nbsp;Kin Aun Tan","doi":"10.1002/cpa.22187","DOIUrl":"10.1002/cpa.22187","url":null,"abstract":"&lt;p&gt;Gibbsian structure in random point fields has been a classical tool for studying their spatial properties. However, exact Gibbs property is available only in a relatively limited class of models, and it does not adequately address many random fields with a strongly dependent spatial structure. In this work, we provide a very general framework for approximate Gibbsian structure for strongly correlated random point fields, including those with a highly singular spatial structure. These include processes that exhibit strong spatial &lt;i&gt;rigidity&lt;/i&gt;, in particular, a certain one-parameter family of analytic Gaussian zero point fields, namely the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-GAFs, that are known to demonstrate a wide range of such spatial behavior. Our framework entails conditions that may be verified via finite particle approximations to the process, a phenomenon that we call an approximate Gibbs property. We show that these enable one to compare the spatial conditional measures in the infinite volume limit with Gibbs-type densities supported on appropriate singular manifolds, a phenomenon we refer to as a generalized Gibbs property. Our work provides a general mechanism to rigorously understand the limiting behavior of spatial conditioning in strongly correlated point processes with growing system size. We demonstrate the scope and versatility of our approach by showing that a generalized Gibbs property holds with a logarithmic pair potential for the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-GAFs for any value of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;. In this vein, we settle in the affirmative an open question regarding the existence of point processes with any specified level of rigidity. In particular, for the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;annotation&gt;$alpha$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-GAF zero process, we establish the &lt;i&gt;level of rigidity&lt;/i&gt; to be exactly &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⌊&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/mfrac&gt;\u0000 &lt;mo&gt;⌋&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$lfloor frac{1}{alpha} rfloor$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, a fortiori demonstrating the phenomenon of spatial &lt;i&gt;tolerance&lt;/i&gt; subject to the local conservation of &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⌊&lt;/mo&gt;\u0000 &lt;mfrac&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;mi&gt;α&lt;/mi&gt;\u0000 &lt;/mfrac&gt;\u0000 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3427-3519"},"PeriodicalIF":3.0,"publicationDate":"2023-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138840301","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":"Arnold diffusion in Hamiltonian systems on infinite lattices","authors":"Filippo Giuliani,&nbsp;Marcel Guardia","doi":"10.1002/cpa.22191","DOIUrl":"10.1002/cpa.22191","url":null,"abstract":"<p>We consider a system of infinitely many penduli on an <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>-dimensional lattice with a weak coupling. For any prescribed path in the lattice, for suitable couplings, we construct orbits for this Hamiltonian system of infinite degrees of freedom which transfer energy between nearby penduli along the path. We allow the weak coupling to be next-to-nearest neighbor or long range as long as it is strongly decaying. The transfer of energy is given by an Arnold diffusion mechanism which relies on the original V. I Arnold approach: to construct a sequence of hyperbolic invariant quasi-periodic tori with transverse heteroclinic orbits. We implement this approach in an infinite dimensional setting, both in the space of bounded <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^m$</annotation>\u0000 </semantics></math>-sequences and in spaces of decaying <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>Z</mi>\u0000 <mi>m</mi>\u0000 </msup>\u0000 <annotation>$mathbb {Z}^m$</annotation>\u0000 </semantics></math>-sequences. Key steps in the proof are an invariant manifold theory for hyperbolic tori and a Lambda Lemma for infinite dimensional coupled map lattices with decaying interaction.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 8","pages":"3333-3426"},"PeriodicalIF":3.0,"publicationDate":"2023-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138770934","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":"Chord measures in integral geometry and their Minkowski problems","authors":"Erwin Lutwak,&nbsp;Dongmeng Xi,&nbsp;Deane Yang,&nbsp;Gaoyong Zhang","doi":"10.1002/cpa.22190","DOIUrl":"10.1002/cpa.22190","url":null,"abstract":"<p>To the families of geometric measures of convex bodies (the area measures of Aleksandrov-Fenchel-Jessen, the curvature measures of Federer, and the recently discovered dual curvature measures) a new family is added. The new family of geometric measures, called chord measures, arises from the study of integral geometric invariants of convex bodies. The Minkowski problems for the new measures and their logarithmic variants are proposed and attacked. When the given ‘data’ is sufficiently regular, these problems are a new type of fully nonlinear partial differential equations involving dual quermassintegrals of functions. Major cases of these Minkowski problems are solved without regularity assumptions.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 7","pages":"3277-3330"},"PeriodicalIF":3.0,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138565628","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":"Overcrowding and separation estimates for the Coulomb gas","authors":"Eric Thoma","doi":"10.1002/cpa.22188","DOIUrl":"10.1002/cpa.22188","url":null,"abstract":"<p>We prove several results for the Coulomb gas in any dimension <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$d ge 2$</annotation>\u0000 </semantics></math> that follow from <i>isotropic averaging</i>, a transport method based on Newton's theorem. First, we prove a high-density Jancovici–Lebowitz–Manificat law, extending the microscopic density bounds of Armstrong and Serfaty and establishing strictly sub-Gaussian tails for charge excess in dimension 2. The existence of microscopic limiting point processes is proved at the edge of the droplet. Next, we prove optimal upper bounds on the <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>-point correlation function for merging points, including a Wegner estimate for the Coulomb gas for <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>k</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <annotation>$k=1$</annotation>\u0000 </semantics></math>. We prove the tightness of the properly rescaled <span></span><math>\u0000 <semantics>\u0000 <mi>k</mi>\u0000 <annotation>$k$</annotation>\u0000 </semantics></math>th minimal particle gap, identifying the correct order in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>=</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 <annotation>$d=2$</annotation>\u0000 </semantics></math> and a three term expansion in <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>d</mi>\u0000 <mo>≥</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$d ge 3$</annotation>\u0000 </semantics></math>, as well as upper and lower tail estimates. In particular, we extend the two-dimensional “perfect-freezing regime” identified by Ameur and Romero to higher dimensions. Finally, we give positive charge discrepancy bounds which are state of the art near the droplet boundary and prove incompressibility of Laughlin states in the fractional quantum Hall effect, starting at large microscopic scales. Using rigidity for fluctuations of smooth linear statistics, we show how to upgrade positive discrepancy bounds to estimates on the absolute discrepancy in certain regions.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 7","pages":"3227-3276"},"PeriodicalIF":3.0,"publicationDate":"2023-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138481467","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":"The anisotropic min-max theory: Existence of anisotropic minimal and CMC surfaces","authors":"Guido De Philippis,&nbsp;Antonio De Rosa","doi":"10.1002/cpa.22189","DOIUrl":"10.1002/cpa.22189","url":null,"abstract":"<p>We prove the existence of nontrivial closed surfaces with constant anisotropic mean curvature with respect to elliptic integrands in closed smooth <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$hskip.001pt 3$</annotation>\u0000 </semantics></math>–dimensional Riemannian manifolds. The constructed min-max surfaces are smooth with at most one singular point. The constant anisotropic mean curvature can be fixed to be any real number. In particular, we partially solve a conjecture of Allard in dimension <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$hskip.001pt 3$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 7","pages":"3184-3226"},"PeriodicalIF":3.0,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.22189","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138473810","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":"Infinite order phase transition in the slow bond TASEP","authors":"Sourav Sarkar,&nbsp;Allan Sly,&nbsp;Lingfu Zhang","doi":"10.1002/cpa.22185","DOIUrl":"10.1002/cpa.22185","url":null,"abstract":"<p>In the slow bond problem the rate of a single edge in the Totally Asymmetric Simple Exclusion Process (TASEP) is reduced from 1 to <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>−</mo>\u0000 <mi>ε</mi>\u0000 </mrow>\u0000 <annotation>$1-varepsilon$</annotation>\u0000 </semantics></math> for some small <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ε</mi>\u0000 <mo>&gt;</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$varepsilon &amp;gt;0$</annotation>\u0000 </semantics></math>. Janowsky and Lebowitz  posed the well-known question of whether such very small perturbations could affect the macroscopic current. Different groups of physicists, using a range of heuristics and numerical simulations reached opposing conclusions on whether the critical value of <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$varepsilon$</annotation>\u0000 </semantics></math> is 0. This was ultimately resolved rigorously in Basu-Sidoravicius-Sly which established that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ε</mi>\u0000 <mi>c</mi>\u0000 </msub>\u0000 <mo>=</mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$varepsilon _c=0$</annotation>\u0000 </semantics></math>.</p><p>Here we study the effect of the current as <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$varepsilon$</annotation>\u0000 </semantics></math> tends to 0 and in doing so explain why it was so challenging to predict on the basis of numerical simulations. In particular we show that the current has an infinite order phase transition at 0, with the effect of the perturbation tending to 0 faster than any polynomial. Our proof focuses on the Last Passage Percolation formulation of TASEP where a slow bond corresponds to reinforcing the diagonal. We give a multiscale analysis to show that when <span></span><math>\u0000 <semantics>\u0000 <mi>ε</mi>\u0000 <annotation>$varepsilon$</annotation>\u0000 </semantics></math> is small the effect of reinforcement remains small compared to the difference between optimal and near optimal geodesics. Since geodesics can be perturbed on many different scales, we inductively bound the tails of the effect of reinforcement by controlling the number of near optimal geodesics and giving new tail estimates for the local time of (near) geodesics along the diagonal.</p>","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 6","pages":"3107-3140"},"PeriodicalIF":3.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138289375","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 sets of solutions of elliptic equations in periodic homogenization","authors":"Fanghua Lin,&nbsp;Zhongwei Shen","doi":"10.1002/cpa.22186","DOIUrl":"10.1002/cpa.22186","url":null,"abstract":"&lt;p&gt;In this paper we study critical sets of solutions &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; of second-order elliptic equations in divergence form with rapidly oscillating and periodic coefficients. Under some condition on the first-order correctors, we show that the &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;d&lt;/mi&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;/mrow&gt;\u0000 &lt;annotation&gt;$(d-2)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;-dimensional Hausdorff measures of the critical sets are bounded uniformly with respect to the period &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;annotation&gt;$varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, provided that doubling indices for solutions are bounded. The key step is an estimate of “turning” of an approximate tangent map, the projection of a non-constant solution &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; onto the subspace of spherical harmonics of order &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;ℓ&lt;/mi&gt;\u0000 &lt;annotation&gt;$ell$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, when the doubling index for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;u&lt;/mi&gt;\u0000 &lt;mi&gt;ε&lt;/mi&gt;\u0000 &lt;/msub&gt;\u0000 &lt;annotation&gt;$u_varepsilon$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; on a sphere &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;∂&lt;/mi&gt;\u0000 &lt;mi&gt;B&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$partial B(0, r)$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; is trapped between &lt;span&gt;&lt;/span&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;mi&gt;δ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$ell -delta$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; and &lt;span&gt;&lt;/span&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;mi&gt;δ&lt;/mi&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;annotation&gt;$ell +delta$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt;, for &lt;span&gt;&lt;/span&gt;&lt;math&gt;\u0000 &lt;semantics&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;annotation&gt;$r$&lt;/annotation&gt;\u0000 &lt;/semantics&gt;&lt;/math&gt; between 1 ","PeriodicalId":10601,"journal":{"name":"Communications on Pure and Applied Mathematics","volume":"77 7","pages":"3143-3183"},"PeriodicalIF":3.0,"publicationDate":"2023-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138297138","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}