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A Riemann Hilbert Approach to the Study of the Generating Function Associated to the Pearcey Process 用黎曼希尔伯特方法研究与皮尔斯过程相关的生成函数
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-04-25 DOI: 10.1007/s11040-023-09455-8
Thomas Chouteau
{"title":"A Riemann Hilbert Approach to the Study of the Generating Function Associated to the Pearcey Process","authors":"Thomas Chouteau","doi":"10.1007/s11040-023-09455-8","DOIUrl":"10.1007/s11040-023-09455-8","url":null,"abstract":"<div><p>Using Riemann–Hilbert methods, we establish a Tracy–Widom like formula for the generating function of the occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order three. We also obtain a non linear coupled heat equation. Combining these two equations we obtain a PDE for the logarithm of the the generating function of the Pearcey process.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4961850","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: 1
Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree Cayley树上具有可计数自旋值集的hc模型的Gibbs测度
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-03-28 DOI: 10.1007/s11040-023-09453-w
R. M. Khakimov, M. T. Makhammadaliev, U. A. Rozikov
{"title":"Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree","authors":"R. M. Khakimov,&nbsp;M. T. Makhammadaliev,&nbsp;U. A. Rozikov","doi":"10.1007/s11040-023-09453-w","DOIUrl":"10.1007/s11040-023-09453-w","url":null,"abstract":"<div><p>In this paper, we study the HC-model with a countable set <span>(mathbb Z)</span> of spin values on a Cayley tree of order <span>(kge 2)</span>. This model is defined by a countable set of parameters (that is, the activity function <span>(lambda _i&gt;0)</span>, <span>(iin mathbb Z)</span>). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:</p><ul>\u0000 <li>\u0000 <p>Let <span>(Lambda =sum _ilambda _i)</span>. For <span>(Lambda =+infty )</span> there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);</p>\u0000 </li>\u0000 <li>\u0000 <p>For <span>(Lambda &lt;+infty )</span>, the uniqueness of TIGM is proved;</p>\u0000 </li>\u0000 <li>\u0000 <p>Let <span>(Lambda _textrm{cr}(k)=frac{k^k}{(k-1)^{k+1}})</span>. If <span>(0&lt;Lambda le Lambda _textrm{cr})</span>, then there is exactly one TPGM that is TIGM;</p>\u0000 </li>\u0000 <li>\u0000 <p>For <span>(Lambda &gt;Lambda _textrm{cr})</span>, there are exactly three TPGMs, one of which is TIGM.</p>\u0000 </li>\u0000 </ul></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09453-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5090994","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: 2
A Comparison of Two Quantum Distances 两个量子距离的比较
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-03-13 DOI: 10.1007/s11040-023-09451-y
Jens Kaad, David Kyed
{"title":"A Comparison of Two Quantum Distances","authors":"Jens Kaad,&nbsp;David Kyed","doi":"10.1007/s11040-023-09451-y","DOIUrl":"10.1007/s11040-023-09451-y","url":null,"abstract":"<div><p>We show that Rieffel’s quantum Gromov–Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov–Hausdorff distance applied to the associated state spaces.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4549296","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: 2
Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree Cayley树上Blume-Emery-Griffiths模型的Gibbs测度
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-03-03 DOI: 10.1007/s11040-023-09448-7
G. Botirov, F. Haydarov, U. Qayumov
{"title":"Gibbs Measures of the Blume–Emery–Griffiths Model on the Cayley Tree","authors":"G. Botirov,&nbsp;F. Haydarov,&nbsp;U. Qayumov","doi":"10.1007/s11040-023-09448-7","DOIUrl":"10.1007/s11040-023-09448-7","url":null,"abstract":"<div><p>In this paper we consider the Blume–Emery–Griffiths model on Cayley trees. We reduce the problem of describing the splitting Gibbs measures of the Blume–Emery–Griffiths model to the description of the solutions of some algebraic equation. Also, we analyse the set of translation-invariant splitting Gibbs measures for a two parametric BEG model on Cayley trees.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09448-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4132891","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
The Near-Critical Two-Point Function and the Torus Plateau for Weakly Self-avoiding Walk in High Dimensions 高维弱自避行走的近临界两点函数和环面平台
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-02-17 DOI: 10.1007/s11040-023-09447-8
Gordon Slade
{"title":"The Near-Critical Two-Point Function and the Torus Plateau for Weakly Self-avoiding Walk in High Dimensions","authors":"Gordon Slade","doi":"10.1007/s11040-023-09447-8","DOIUrl":"10.1007/s11040-023-09447-8","url":null,"abstract":"<div><p>We use the lace expansion to study the long-distance decay of the two-point function of weakly self-avoiding walk on the integer lattice <span>(mathbb {Z}^d)</span> in dimensions <span>(d&gt;4)</span>, in the vicinity of the critical point, and prove an upper bound <span>(|x|^{-(d-2)}exp [-c|x|/xi ])</span>, where the correlation length <span>(xi )</span> has a square root divergence at the critical point. As an application, we prove that the two-point function for weakly self-avoiding walk on a discrete torus in dimensions <span>(d{&gt;}4)</span> has a “plateau.” We also discuss the significance and consequences of the plateau for the analysis of critical behaviour on the torus.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09447-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4678192","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: 7
On the Integrability of a Four-Prototype Rössler System 关于四原型Rössler系统的可积性
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-02-16 DOI: 10.1007/s11040-023-09449-6
Jaume Llibre, Claudia Valls
{"title":"On the Integrability of a Four-Prototype Rössler System","authors":"Jaume Llibre,&nbsp;Claudia Valls","doi":"10.1007/s11040-023-09449-6","DOIUrl":"10.1007/s11040-023-09449-6","url":null,"abstract":"<div><p>We consider a four-prototype Rossler system introduced by Otto Rössler among others as prototypes of the simplest autonomous differential equations (in the sense of minimal dimension, minimal number of parameters, minimal number of nonlinear terms) having chaotic behavior. We contribute towards the understanding of its chaotic behavior by studying its integrability from different points of view. We show that it is neither Darboux integrable, nor <span>(C^1)</span>-integrable.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4934495","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
Mean-field behavior of Nearest-Neighbor Oriented Percolation on the BCC Lattice Above 8 + 1 Dimensions 8 + 1维以上BCC格上最近邻定向渗流的平均场行为
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-02-14 DOI: 10.1007/s11040-022-09441-6
Lung-Chi Chen, Satoshi Handa, Yoshinori Kamijima
{"title":"Mean-field behavior of Nearest-Neighbor Oriented Percolation on the BCC Lattice Above 8 + 1 Dimensions","authors":"Lung-Chi Chen,&nbsp;Satoshi Handa,&nbsp;Yoshinori Kamijima","doi":"10.1007/s11040-022-09441-6","DOIUrl":"10.1007/s11040-022-09441-6","url":null,"abstract":"<div><p>In this paper, we consider nearest-neighbor oriented percolation with independent Bernoulli bond-occupation probability on the <i>d</i>-dimensional body-centered cubic (BCC) lattice <span>({mathbb {L}^d})</span> and the set of non-negative integers <span>({{mathbb {Z}}_+})</span>. Thanks to the orderly structure of the BCC lattice, we prove that the infrared bound holds on <span>({mathbb {L}^d} times {{mathbb {Z}}_+})</span> in all dimensions <span>(dge 9)</span>. As opposed to ordinary percolation, we have to deal with complex numbers due to asymmetry induced by time-orientation, which makes it hard to bound the bootstrap functions in the lace-expansion analysis. By investigating the Fourier–Laplace transform of the random-walk Green function and the two-point function, we derive the key properties to obtain the upper bounds and resolve a problematic issue in Nguyen and Yang’s bound. The issue is caused by the fact that the Fourier transform of the random-walk transition probability can take the value <span>(-1)</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4569654","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
Long Time Asymptotic Behavior for the Nonlocal mKdV Equation in Solitonic Space–Time Regions 孤子时空区域中非局部mKdV方程的长时间渐近行为
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-01-28 DOI: 10.1007/s11040-023-09445-w
Xuan Zhou, Engui Fan
{"title":"Long Time Asymptotic Behavior for the Nonlocal mKdV Equation in Solitonic Space–Time Regions","authors":"Xuan Zhou,&nbsp;Engui Fan","doi":"10.1007/s11040-023-09445-w","DOIUrl":"10.1007/s11040-023-09445-w","url":null,"abstract":"<div><p>We study the long time asymptotic behavior for the Cauchy problem of an integrable real nonlocal mKdV equation with nonzero initial data in the solitonic regions </p><div><div><span>$$begin{aligned}&amp;q_t(x,t)-6sigma q(x,t)q(-x,-t)q_{x}(x,t)+q_{xxx}(x,t)=0, &amp;quad q(x,0)=q_{0}(x), lim _{xrightarrow pm infty } q_{0}(x)=q_{pm }, end{aligned}$$</span></div></div><p>where <span>(|q_{pm }|=1)</span> and <span>(q_{+}=delta q_{-})</span>, <span>(sigma delta =-1)</span>. In our previous article, we have obtained long time asymptotics for the nonlocal mKdV equation in the solitonic region <span>(-6&lt;xi &lt;6)</span> with <span>(xi =frac{x}{t})</span>. In this paper, we give the asymptotic expansion of the solution <i>q</i>(<i>x</i>, <i>t</i>) for other solitonic regions <span>(xi &lt;-6)</span> and <span>(xi &gt;6)</span>. Based on the Riemann–Hilbert formulation of the Cauchy problem, further using the <span>({bar{partial }})</span> steepest descent method, we derive different long time asymptotic expansions of the solution <i>q</i>(<i>x</i>, <i>t</i>) in above two different space-time solitonic regions. In the region <span>(xi &lt;-6)</span>, phase function <span>(theta (z))</span> has four stationary phase points on the <span>({mathbb {R}})</span>. Correspondingly, <i>q</i>(<i>x</i>, <i>t</i>) can be characterized with an <span>({mathcal {N}}(Lambda ))</span>-soliton on discrete spectrum, the leading order term on continuous spectrum and an residual error term, which are affected by a function <span>(textrm{Im}nu (zeta _i))</span>. In the region <span>(xi &gt;6)</span>, phase function <span>(theta (z))</span> has four stationary phase points on <span>(i{mathbb {R}})</span>, the corresponding asymptotic approximations can be characterized with an <span>({mathcal {N}}(Lambda ))</span>-soliton with diverse residual error order <span>({mathcal {O}}(t^{-1}))</span>.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09445-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5082235","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: 1
Lusztig Factorization Dynamics of the Full Kostant–Toda Lattices 全Kostant-Toda格的Lusztig分解动力学
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-01-17 DOI: 10.1007/s11040-022-09444-3
Nicholas M. Ercolani, Jonathan Ramalheira-Tsu
{"title":"Lusztig Factorization Dynamics of the Full Kostant–Toda Lattices","authors":"Nicholas M. Ercolani,&nbsp;Jonathan Ramalheira-Tsu","doi":"10.1007/s11040-022-09444-3","DOIUrl":"10.1007/s11040-022-09444-3","url":null,"abstract":"<div><p>We study extensions of the classical Toda lattices at several different space–time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant–Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel–Lie algebras. Our study brings into play factorizations of Loewner–Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space–time scales. Along the way we derive a novel, minimal box–ball system for the Full Kostant–Toda Lattice that does not involve any capacities or colorings, and which has a natural interpretation in terms of the Robinson–Schensted–Knuth algorithm. We provide as well an extension of O’Connell’s ordinary differential equations to the Full Kostant–Toda Lattice.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4680574","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: 1
Heisenberg Dynamics for Non Self-Adjoint Hamiltonians: Symmetries and Derivations 非自伴随哈密顿量的Heisenberg动力学:对称性和推导
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2022-12-19 DOI: 10.1007/s11040-022-09443-4
F. Bagarello
{"title":"Heisenberg Dynamics for Non Self-Adjoint Hamiltonians: Symmetries and Derivations","authors":"F. Bagarello","doi":"10.1007/s11040-022-09443-4","DOIUrl":"10.1007/s11040-022-09443-4","url":null,"abstract":"<div><p>In some recent literature the role of non self-adjoint Hamiltonians, <span>(Hne H^dagger )</span>, is often considered in connection with gain-loss systems. The dynamics for these systems is, most of the times, given in terms of a Schrödinger equation. In this paper we rather focus on the Heisenberg-like picture of quantum mechanics, stressing the (few) similarities and the (many) differences with respected to the standard Heisenberg picture for systems driven by self-adjoint Hamiltonians. In particular, the role of the symmetries, *-derivations and integrals of motion is discussed.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2022-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-022-09443-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5042872","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: 2
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