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Propagation of Chaos and Residual Dependence in Gibbs Measures on Finite Sets
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2025-03-15 DOI: 10.1007/s11040-025-09503-5
Jonas Jalowy, Zakhar Kabluchko, Matthias Löwe
{"title":"Propagation of Chaos and Residual Dependence in Gibbs Measures on Finite Sets","authors":"Jonas Jalowy,&nbsp;Zakhar Kabluchko,&nbsp;Matthias Löwe","doi":"10.1007/s11040-025-09503-5","DOIUrl":"10.1007/s11040-025-09503-5","url":null,"abstract":"<div><p>We compare a mean-field Gibbs distribution on a finite state space on <i>N</i> spins to that of an explicit simple mixture of product measures. This illustrates the situation beyond the so-called <i>increasing propagation of chaos</i> introduced by Ben Arous and Zeitouni [3], where marginal distributions of size <span>(k=o(N))</span> are compared to product measures.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-025-09503-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143629691","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Recurrence Relations for the Generalized Laguerre and Charlier Orthogonal Polynomials and Discrete Painlevé Equations on the (D_{6}^{(1)}) Sakai Surface
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2025-03-15 DOI: 10.1007/s11040-025-09502-6
Xing Li, Anton Dzhamay, Galina Filipuk, Da-jun Zhang
{"title":"Recurrence Relations for the Generalized Laguerre and Charlier Orthogonal Polynomials and Discrete Painlevé Equations on the (D_{6}^{(1)}) Sakai Surface","authors":"Xing Li,&nbsp;Anton Dzhamay,&nbsp;Galina Filipuk,&nbsp;Da-jun Zhang","doi":"10.1007/s11040-025-09502-6","DOIUrl":"10.1007/s11040-025-09502-6","url":null,"abstract":"<div><p>This paper concerns the discrete version of the <i>Painlevé identification problem</i>, i.e., how to recognize a certain recurrence relation as a discrete Painlevé equation. Often some clues can be seen from the setting of the problem, e.g., when the recurrence is connected with some differential Painlevé equation, or from the geometry of the configuration of indeterminate points of the equation. The main message of our paper is that, in fact, this only allows us to identify the <i>configuration space</i> of the dynamic system, but not the dynamics themselves. The <i>refined version</i> of the identification problem lies in determining, up to the conjugation, the translation direction of the dynamics, which in turn requires the full power of the geometric theory of Painlevé equations. To illustrate this point, in this paper we consider two examples of such recurrences that appear in the theory of orthogonal polynomials. We choose these examples because they get regularized on the same family of Sakai surfaces, but at the same time are not equivalent, since they result in non-equivalent translation directions. In addition, we show the effectiveness of a recently proposed identification procedure for discrete Painlevé equations using Sakai’s geometric approach for answering such questions. In particular, this approach requires no a priori knowledge of a possible type of the equation.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143621992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Matrix Solutions of the Cubic Szegő Equation on the Real Line
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2025-03-09 DOI: 10.1007/s11040-025-09500-8
Ruoci Sun
{"title":"Matrix Solutions of the Cubic Szegő Equation on the Real Line","authors":"Ruoci Sun","doi":"10.1007/s11040-025-09500-8","DOIUrl":"10.1007/s11040-025-09500-8","url":null,"abstract":"<div><p>This paper is dedicated to studying matrix solutions of the cubic Szegő equation on the real line, which is introduced in Pocovnicu [Anal PDE 4(3):379–404, 2011; Dyn Syst A 31(3):607–649, 2011] and Gérard–Pushnitski (Commun Math Phys 405:167, 2024), leading to the following cubic matrix Szegő equation on <span>({mathbb {R}})</span>, </p><div><div><span>$$begin{aligned} i partial _t U = Pi _{ge 0} left( U U ^* U right) , quad widehat{left( Pi _{ge 0} Uright) }(xi )= {textbf{1}}_{xi ge 0}{hat{U}}(xi )in {mathbb {C}}^{M times N}. end{aligned}$$</span></div></div><p>Inspired by the space-periodic case in Sun (The matrix Szegő equation, arXiv:2309.12136), we establish its Lax pair structure via double Hankel operators and Toeplitz operators. Then the explicit formula in Gérard–Pushnitski (Commun Math Phys 405:167, 2024) can be extended to two equivalent formulas in the matrix equation case, which both express every solution explicitly in terms of its initial datum and the time variable.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143581173","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
A Novel Discrete Integrable System Related to Hyper-Elliptic Curves of Genus Two
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2025-02-25 DOI: 10.1007/s11040-025-09501-7
Jing-Rui Wu, Xing-Biao Hu
{"title":"A Novel Discrete Integrable System Related to Hyper-Elliptic Curves of Genus Two","authors":"Jing-Rui Wu,&nbsp;Xing-Biao Hu","doi":"10.1007/s11040-025-09501-7","DOIUrl":"10.1007/s11040-025-09501-7","url":null,"abstract":"<div><p>Motivated by the discrete-time Toda (HADT) equation and quotient-quotient-difference (QQD) scheme together with their hungry forms (hHADT equation and hQQD scheme), we derive a new class of discrete integrable systems by considering the determinant structures of bivariate orthogonal polynomials associated with the genus-two hyper-elliptic curves. The corresponding Lax pairs are expressed through the recurrence relations of this class of bivariate orthogonal polynomials. Our study emphasizes the richer structures of genus-two hyper-elliptic curves, in contrast to the genus-one curve considered in the HADT and QQD cases, as well as in the hHADT and hQQD cases.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143489469","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Umbilicity and the First Stability Eigenvalue of a Subclass of CMC Hypersurfaces Immersed in Certain Einstein Manifolds
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2025-02-03 DOI: 10.1007/s11040-025-09499-y
Henrique F. de Lima, Ary V. F. Leite, Marco Antonio L. Velásquez
{"title":"Umbilicity and the First Stability Eigenvalue of a Subclass of CMC Hypersurfaces Immersed in Certain Einstein Manifolds","authors":"Henrique F. de Lima,&nbsp;Ary V. F. Leite,&nbsp;Marco Antonio L. Velásquez","doi":"10.1007/s11040-025-09499-y","DOIUrl":"10.1007/s11040-025-09499-y","url":null,"abstract":"<div><p>We study the umbilicity of constant mean curvature (CMC) complete hypersurfaces immersed in an Einstein manifold satisfying appropriate curvature constraints. In this setting, we obtain new characterization results for totally umbilical hypersurfaces via suitable maximum principles which deal with the notions of convergence to zero at infinity and polynomial volume growth. Afterwards, we establish optimal estimates for the first eigenvalue of the stability operator of CMC compact hypersurfaces in such an Einstein manifold. In particular, we derive a nonexistence result concerning strongly stable CMC hypersurfaces.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143107870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Lipschitz-Type Estimate for the Frog Model with Bernoulli Initial Configuration 具有Bernoulli初始构型的Frog模型的lipschitz型估计
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2025-01-02 DOI: 10.1007/s11040-024-09497-6
Van Hao Can, Naoki Kubota, Shuta Nakajima
{"title":"Lipschitz-Type Estimate for the Frog Model with Bernoulli Initial Configuration","authors":"Van Hao Can,&nbsp;Naoki Kubota,&nbsp;Shuta Nakajima","doi":"10.1007/s11040-024-09497-6","DOIUrl":"10.1007/s11040-024-09497-6","url":null,"abstract":"<div><p>We consider the frog model with Bernoulli initial configuration, which is an interacting particle system on the multidimensional lattice consisting of two states of particles: active and sleeping. Active particles perform independent simple random walks. On the other hand, although sleeping particles do not move at first, they become active and can move around when touched by active particles. Initially, only the origin has one active particle, and the other sites have sleeping particles according to a Bernoulli distribution. Then, starting from the original active particle, active ones are gradually generated and propagate across the lattice, with time. It is of interest to know how the propagation of active particles behaves as the parameter of the Bernoulli distribution varies. In this paper, we treat the so-called time constant describing the speed of propagation, and prove that the absolute difference between the time constants for parameters <span>(p,q in (0,1])</span> is bounded from above and below by multiples of <span>(|p-q|)</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"28 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2025-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142912939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Trees and Superintegrable Lotka–Volterra Families 树与超积分洛特卡-伏特拉家族
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-11-26 DOI: 10.1007/s11040-024-09496-7
Peter H. van der Kamp, G. R. W. Quispel, David I. McLaren
{"title":"Trees and Superintegrable Lotka–Volterra Families","authors":"Peter H. van der Kamp,&nbsp;G. R. W. Quispel,&nbsp;David I. McLaren","doi":"10.1007/s11040-024-09496-7","DOIUrl":"10.1007/s11040-024-09496-7","url":null,"abstract":"<div><p>To any tree on <i>n</i> vertices we associate an <i>n</i>-dimensional Lotka–Volterra system with <span>(3n-2)</span> parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits <span>(n-1)</span> functionally independent integrals. We also show how each system can be reduced to an (<span>(n-1)</span>)-dimensional system which is superintegrable and solvable by quadratures.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 4","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Equality of Magnetization and Edge Current for Interacting Lattice Fermions at Positive Temperature 正温下相互作用晶格费米子的磁化和边缘电流相等
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-11-25 DOI: 10.1007/s11040-024-09495-8
Jonas Lampart, Massimo Moscolari, Stefan Teufel, Tom Wessel
{"title":"Equality of Magnetization and Edge Current for Interacting Lattice Fermions at Positive Temperature","authors":"Jonas Lampart,&nbsp;Massimo Moscolari,&nbsp;Stefan Teufel,&nbsp;Tom Wessel","doi":"10.1007/s11040-024-09495-8","DOIUrl":"10.1007/s11040-024-09495-8","url":null,"abstract":"<div><p>We prove that the magnetization is equal to the edge current in the thermodynamic limit for a large class of models of lattice fermions with finite-range interactions satisfying local indistinguishability of the Gibbs state, a condition known to hold for sufficiently high temperatures. Our result implies that edge currents in such systems are determined by bulk properties and are therefore stable against large perturbations near the boundaries. Moreover, the equality persists also after taking the derivative with respect to the chemical potential. We show that this form of bulk-edge correspondence is essentially a consequence of homogeneity in the bulk and locality of the Gibbs state. An important intermediate result is a new version of Bloch’s theorem for two-dimensional systems, stating that persistent currents vanish in the bulk.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 4","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-024-09495-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694728","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Braided Hopf algebras and gauge transformations 编织霍普夫数组和规整变换
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-11-20 DOI: 10.1007/s11040-024-09492-x
Paolo Aschieri, Giovanni Landi, Chiara Pagani
{"title":"Braided Hopf algebras and gauge transformations","authors":"Paolo Aschieri,&nbsp;Giovanni Landi,&nbsp;Chiara Pagani","doi":"10.1007/s11040-024-09492-x","DOIUrl":"10.1007/s11040-024-09492-x","url":null,"abstract":"<div><p>We study infinitesimal gauge transformations of <i>K</i>-equivariant noncommutative principal bundles, for <i>K</i> a triangular Hopf algebra. They form a Lie algebra of derivations in the category of <i>K</i>-modules. We study Drinfeld twist deformations of these infinitesimal gauge transformations. We give several examples from abelian and Jordanian twist deformations. These include the quantum Lie algebra of gauge transformations of the instanton bundle and of the orthogonal bundle on the quantum sphere <span>(S^4_theta )</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 4","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679594","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
Index of Bipolar Surfaces to Otsuki Tori 双极表面到大月鸟迹的索引
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-11-13 DOI: 10.1007/s11040-024-09494-9
Egor Morozov
{"title":"Index of Bipolar Surfaces to Otsuki Tori","authors":"Egor Morozov","doi":"10.1007/s11040-024-09494-9","DOIUrl":"10.1007/s11040-024-09494-9","url":null,"abstract":"<div><p>For each rational number <span>(p/qin (1/2,sqrt{2}/2))</span> one can construct an <span>(mathbb {S}^1)</span>-equivariant minimal torus in <span>(mathbb {S}^3)</span> called Otsuki torus and denoted by <span>(O_{p/q})</span>. The Lawson’s bipolar surface construction applied to <span>(O_{p/q})</span> gives a minimal torus <span>(widetilde{O}_{p/q})</span> in <span>(mathbb {S}^4)</span>. In this paper we give upper and lower bounds on the Morse index and the nullity of these tori for <i>p</i>/<i>q</i> close to <span>(sqrt{2}/2)</span>. We also state a numerically assisted conjecture concerning the general case.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 4","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142600583","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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: 0
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