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Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices 修正奇异高斯单元集合的离散、连续和渐近及其大汉克尔矩阵的最小特征值
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-03-02 DOI: 10.1007/s11040-024-09477-w
Dan Wang, Mengkun Zhu
{"title":"Discrete, Continuous and Asymptotic for a Modified Singularly Gaussian Unitary Ensemble and the Smallest Eigenvalue of Its Large Hankel Matrices","authors":"Dan Wang,&nbsp;Mengkun Zhu","doi":"10.1007/s11040-024-09477-w","DOIUrl":"10.1007/s11040-024-09477-w","url":null,"abstract":"<div><p>This paper focuses on the characteristics of the Hankel determinant generated by a modified singularly Gaussian weight. The weight function is defined as: </p><div><div><span>$$begin{aligned} w(z;t)=|z|^{alpha }textrm{e}^{-frac{1}{z^2}-tleft( z^2-frac{1}{z^2}right) }, ~zin {mathbb {R}}, end{aligned}$$</span></div></div><p>where <span>(alpha &gt;1)</span> and <span>(tin (0,1))</span> are parameters. Using ladder operator techniques, we derive a series of difference formulas for the auxiliary quantities and recurrence coefficients. We present the difference equations for the recurrence coefficients and the logarithmic derivative of the Hankel determinant. We then use the “t-dependence\" to obtain the differential identities satisfied by the auxiliary quantities and the logarithmic derivative of the Hankel determinant. To obtain the large <i>n</i> asymptotic expressions of the recurrence coefficients, we use the Coulomb fluid method together with the related difference equations, which depend on <i>n</i> either being odd or even. We also obtain the reduction forms of the second-order differential equations satisfied by the orthogonal polynomials generated by this weight. Two special cases coincide with the bi-confluent Heun equation and the double confluent Heun equation, respectively. Finally, we calculate the asymptotic behavior of the smallest eigenvalue of large Hankel matrices generated by this weight. Our result not only covers the classical result of Szegö (Trans Am Math Soc 40:450–461, 1936) but also determines our next research direction.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140019413","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
On the KPZ Scaling and the KPZ Fixed Point for TASEP 关于 KPZ 比例和 TASEP 的 KPZ 固定点
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-01-29 DOI: 10.1007/s11040-024-09475-y
Yuta Arai
{"title":"On the KPZ Scaling and the KPZ Fixed Point for TASEP","authors":"Yuta Arai","doi":"10.1007/s11040-024-09475-y","DOIUrl":"10.1007/s11040-024-09475-y","url":null,"abstract":"<div><p>We consider all totally asymmetric simple exclusion processes (TASEPs) whose transition probabilities are given by the Schütz-type formulas and which jump with homogeneous rates. We show that the multi-point distribution of particle positions and the KPZ scaling are described using the probability generating function of the rightmost particle’s jump. For all TASEPs satisfying certain assumptions, we also prove the pointwise convergence of the kernels appearing in the joint distribution of particle positions to those appearing in the KPZ fixed point formula. Our result generalizes the result of Matetski, Quastel, and Remenik [18] in the sense that we provide the KPZ fixed point formulation for a class of TASEPs, instead of for one specific TASEP.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139646134","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
On the Integrable Structure of Deformed Sine Kernel Determinants 论变形正弦核决定因素的积分结构
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-01-27 DOI: 10.1007/s11040-024-09476-x
Tom Claeys, Sofia Tarricone
{"title":"On the Integrable Structure of Deformed Sine Kernel Determinants","authors":"Tom Claeys,&nbsp;Sofia Tarricone","doi":"10.1007/s11040-024-09476-x","DOIUrl":"10.1007/s11040-024-09476-x","url":null,"abstract":"<div><p>We study a family of Fredholm determinants associated to deformations of the sine kernel, parametrized by a weight function <i>w</i>. For a specific choice of <i>w</i>, this kernel describes bulk statistics of finite temperature free fermions. We establish a connection between these determinants and a system of integro-differential equations generalizing the fifth Painlevé equation, and we show that they allow us to solve an integrable PDE explicitly for a large class of initial data.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-01-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139582011","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
Multicritical Schur Measures and Higher-Order Analogues of the Tracy–Widom Distribution 多临界舒尔量和特雷西-维多姆分布的高阶类似物
IF 0.9 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-01-25 DOI: 10.1007/s11040-023-09472-7
Dan Betea, Jérémie Bouttier, Harriet Walsh
{"title":"Multicritical Schur Measures and Higher-Order Analogues of the Tracy–Widom Distribution","authors":"Dan Betea,&nbsp;Jérémie Bouttier,&nbsp;Harriet Walsh","doi":"10.1007/s11040-023-09472-7","DOIUrl":"10.1007/s11040-023-09472-7","url":null,"abstract":"<div><p>We introduce multicritical Schur measures, which are probability laws on integer partitions which give rise to non-generic fluctuations at their edge. They are in the same universality classes as one-dimensional momentum-space models of free fermions in flat confining potentials, studied by Le Doussal, Majumdar and Schehr. These universality classes involve critical exponents of the form <span>(1/(2m+1))</span>, with <i>m</i> a positive integer, and asymptotic distributions given by Fredholm determinants constructed from higher order Airy kernels, extending the generic Tracy–Widom GUE distribution recovered for <span>(m=1)</span>. We also compute limit shapes for the multicritical Schur measures, discuss the finite temperature setting, and exhibit an exact mapping to the multicritical unitary matrix models previously encountered by Periwal and Shevitz.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 1","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-01-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139581744","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
Tau-Function of the Multi-component CKP Hierarchy 多组分 CKP 层次结构的 Tau 功能
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2024-01-02 DOI: 10.1007/s11040-023-09473-6
A. Zabrodin
{"title":"Tau-Function of the Multi-component CKP Hierarchy","authors":"A. Zabrodin","doi":"10.1007/s11040-023-09473-6","DOIUrl":"10.1007/s11040-023-09473-6","url":null,"abstract":"<div><p>We consider multi-component Kadomtsev-Petviashvili hierarchy of type C (the multi-component CKP hierarchy) originally defined with the help of matrix pseudo-differential operators via the Lax-Sato formalism. Starting from the bilinear relation for the wave functions, we prove existence of the tau-function for the multi-component CKP hierarchy and provide a formula which expresses the wave functions through the tau-function. We also find how this tau-function is related to the tau-function of the multi-component Kadomtsev-Petviashvili hierarchy. The tau-function of the multi-component CKP hierarchy satisfies an integral relation which, unlike the integral relation for the latter tau-function, is no longer bilinear but has a more complicated form.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139084228","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
Complex Creation Operator and Planar Automorphic Functions 复生成算子与平面自同构函数
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-11-01 DOI: 10.1007/s11040-023-09471-8
Ghanmi Allal, Imlal Lahcen
{"title":"Complex Creation Operator and Planar Automorphic Functions","authors":"Ghanmi Allal,&nbsp;Imlal Lahcen","doi":"10.1007/s11040-023-09471-8","DOIUrl":"10.1007/s11040-023-09471-8","url":null,"abstract":"<div><p>We provide a concrete characterization of the poly-analytic planar automorphic functions, a special class of non analytic planar automorphic functions with respect to the Appell–Humbert automorphy factor, arising as images of the holomorphic ones by means of the creation differential operator. This is closely connected to the spectral theory of the magnetic Laplacian on the complex plane.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134795352","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
Surgery Transformations and Spectral Estimates of (delta ) Beam Operators (delta )光束算子的手术变换和谱估计
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-10-17 DOI: 10.1007/s11040-023-09470-9
Aftab Ali, Muhammad Usman
{"title":"Surgery Transformations and Spectral Estimates of (delta ) Beam Operators","authors":"Aftab Ali,&nbsp;Muhammad Usman","doi":"10.1007/s11040-023-09470-9","DOIUrl":"10.1007/s11040-023-09470-9","url":null,"abstract":"<div><p>We introduce <span>(delta )</span> type vertex conditions for beam operators, the fourth-order differential operator, on finite, compact and connected metric graphs. Our study the effect of certain geometrical alterations (graph surgery) of the graph on their spectra. Results are obtained for a class of vertex conditions which can be seen as an analogue of <span>(delta )</span>-conditions for graphs Laplacian. There are a number of possible candidates of <span>(delta )</span> type conditions for beam operators. We develop surgery principles and record the monotonicity properties of their spectrum, keeping in view the possibility that vertex conditions may change within the same class after certain graph alterations. We also demonstrate the applications of surgery principles by obtaining several lower and upper estimates on the eigenvalues.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134797016","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
Cohomology of Lie Algebra Morphism Triples and Some Applications 李代数态射三元组的上同调及其一些应用
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-09-25 DOI: 10.1007/s11040-023-09468-3
Apurba Das
{"title":"Cohomology of Lie Algebra Morphism Triples and Some Applications","authors":"Apurba Das","doi":"10.1007/s11040-023-09468-3","DOIUrl":"10.1007/s11040-023-09468-3","url":null,"abstract":"<div><p>A Lie algebra morphism triple is a triple <span>((mathfrak {g}, mathfrak {h}, phi ))</span> consisting of two Lie algebras <span>(mathfrak {g}, mathfrak {h})</span> and a Lie algebra homomorphism <span>(phi : mathfrak {g} rightarrow mathfrak {h})</span>. We define representations and cohomology of Lie algebra morphism triples. As applications of our cohomology, we study some aspects of deformations, abelian extensions of Lie algebra morphism triples and classify skeletal sh Lie algebra morphism triples. Finally, we consider the cohomology of Lie group morphism triples and find a relation with the cohomology of Lie algebra morphism triples.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134797413","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
The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve (y=x^{n}) 用代数曲线分离的不连续分段线性微分系统的第16 Hilbert问题 (y=x^{n})
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-09-19 DOI: 10.1007/s11040-023-09467-4
Jaume Llibre, Claudia Valls
{"title":"The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve (y=x^{n})","authors":"Jaume Llibre,&nbsp;Claudia Valls","doi":"10.1007/s11040-023-09467-4","DOIUrl":"10.1007/s11040-023-09467-4","url":null,"abstract":"<div><p>We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve <span>(y=x^n)</span> with <span>(n ge 2)</span>. We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of <i>n</i>, proving the extended 16th Hilbert problem in this case. In particular, we show that for <span>(n=2)</span> this bound can be reached.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09467-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"7183673","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 Inverse Spectral Map for Dimers 二聚体的逆光谱图
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-09-12 DOI: 10.1007/s11040-023-09466-5
T. George, A. B. Goncharov, R. Kenyon
{"title":"The Inverse Spectral Map for Dimers","authors":"T. George,&nbsp;A. B. Goncharov,&nbsp;R. Kenyon","doi":"10.1007/s11040-023-09466-5","DOIUrl":"10.1007/s11040-023-09466-5","url":null,"abstract":"<div><p>In 2015, Vladimir Fock proved that the spectral transform, associating to an element of a dimer cluster integrable system its spectral data, is birational by constructing an inverse map using theta functions on Jacobians of spectral curves. We provide an alternate construction of the inverse map that involves only rational functions in the spectral data.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134795864","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: 3
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