Mathematical Physics, Analysis and Geometry最新文献

{"title":"Iterated Foldings of Discrete Spaces and Their Limits: Candidates for the Role of Brownian Map in Higher Dimensions","authors":"Luca Lionni,&nbsp;Jean-François Marckert","doi":"10.1007/s11040-021-09410-5","DOIUrl":"10.1007/s11040-021-09410-5","url":null,"abstract":"<div><p>In this last decade, an important stochastic model emerged: the Brownian map. It is the limit of various models of random combinatorial maps after rescaling: it is a random metric space with Hausdorff dimension 4, almost surely homeomorphic to the 2-sphere, and possesses some deep connections with Liouville quantum gravity in 2D. In this paper, we present a sequence of random objects that we call <span>(Dhbox {th})</span>-random feuilletages (denoted by <span>(mathbf{r}[{D}])</span>), indexed by a parameter <span>(Dge 0)</span> and which are candidate to play the role of the Brownian map in dimension <i>D</i>. The construction relies on some objects that we name iterated Brownian snakes, which are branching analogues of iterated Brownian motions, and which are moreover limits of iterated discrete snakes. In the planar <span>(D=2)</span> case, the family of discrete snakes considered coincides with some family of (random) labeled trees known to encode planar quadrangulations. Iterating snakes provides a sequence of random trees <span>((mathbf{t}^{(j)}, jge 1))</span>. The <span>(Dhbox {th})</span>-random feuilletage <span>(mathbf{r}[{D}])</span> is built using <span>((mathbf{t}^{(1)},ldots ,mathbf{t}^{(D)}))</span>: <span>(mathbf{r}[{0}])</span> is a deterministic circle, <span>(mathbf{r}[{1}])</span> is Aldous’ continuum random tree, <span>(mathbf{r}[{2}])</span> is the Brownian map, and somehow, <span>(mathbf{r}[{D}])</span> is obtained by quotienting <span>(mathbf{t}^{(D)})</span> by <span>(mathbf{r}[{D-1}])</span>. A discrete counterpart to <span>(mathbf{r}[{D}])</span> is introduced and called the <span>(D)</span>th random discrete feuilletage with <span>(n+D)</span> nodes (<span>(mathbf{R}_{n}[D])</span>). The proof of the convergence of <span>(mathbf{R}_{n}[D])</span> to <span>(mathbf{r}[{D}])</span> after appropriate rescaling in some functional space is provided (however, the convergence obtained is too weak to imply the Gromov-Hausdorff convergence). An upper bound on the diameter of <span>(mathbf{R}_{n}[D])</span> is <span>(n^{1/2^{D}})</span>. Some elements allowing to conjecture that the Hausdorff dimension of <span>(mathbf{r}[{D}])</span> is <span>(2^D)</span> are given.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09410-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5057628","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}

{"title":"A Lax Formulation of a Generalized q-Garnier System","authors":"Takao Suzuki","doi":"10.1007/s11040-021-09412-3","DOIUrl":"10.1007/s11040-021-09412-3","url":null,"abstract":"<div><p>Recently, a birational representation of an extended affine Weyl group of type <span>(A_{mn-1}^{(1)}times A_{m-1}^{(1)}times A_{m-1}^{(1)})</span> was proposed with the aid of a cluster mutation. In this article we formulate this representation in a framework of a system of <i>q</i>-difference equations with <span>(mntimes mn)</span> matrices. This formulation is called a Lax form and is used to derive a generalization of the <i>q</i>-Garnier system.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09412-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5060879","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}

{"title":"Constant Mean Curvature Surfaces Based on Fundamental Quadrilaterals","authors":"Alexander I. Bobenko,&nbsp;Sebastian Heller,&nbsp;Nick Schmitt","doi":"10.1007/s11040-021-09397-z","DOIUrl":"10.1007/s11040-021-09397-z","url":null,"abstract":"<div><p>We describe the construction of CMC surfaces with symmetries in <span>(mathbb {S}^{3})</span> and <span>(mathbb {R}^{3})</span> using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the space. The fundamental piece is constructed by the generalized Weierstrass representation using a geometric flow on the space of potentials.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09397-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4276441","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}

{"title":"Sharpness of the Phase Transition for the Orthant Model","authors":"Thomas Beekenkamp","doi":"10.1007/s11040-021-09408-z","DOIUrl":"10.1007/s11040-021-09408-z","url":null,"abstract":"<div><p>The orthant model is a directed percolation model on <span>(mathbb {Z}^{d})</span>, in which all clusters are infinite. We prove a sharp threshold result for this model: if <i>p</i> is larger than the critical value above which the cluster of 0 is contained in a cone, then the shift from 0 that is required to contain the cluster of 0 in that cone is exponentially small. As a consequence, above this critical threshold, a shape theorem holds for the cluster of 0, as well as ballisticity of the random walk on this cluster.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09408-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4918467","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}

{"title":"Kasteleyn Theorem, Geometric Signatures and KP-II Divisors on Planar Bipartite Networks in the Disk","authors":"Simonetta Abenda","doi":"10.1007/s11040-021-09405-2","DOIUrl":"10.1007/s11040-021-09405-2","url":null,"abstract":"<div><p>Maximal minors of Kasteleyn sign matrices on planar bipartite graphs in the disk count dimer configurations with prescribed boundary conditions, and the weighted version of such matrices provides a natural parametrization of the totally non–negative part of real Grassmannians (Postnikov et al. J. Algebr. Combin. <b>30</b>(2), 173–191, 2009; Lam J. Lond. Math. Soc. (2) <b>92</b>(3), 633–656, 2015; Lam 2016; Speyer 2016; Affolter et al. 2019). In this paper we provide a geometric interpretation of such variant of Kasteleyn theorem: a signature is Kasteleyn if and only if it is geometric in the sense of Abenda and Grinevich (2019). We apply this geometric characterization to explicitly solve the associated system of relations and provide a new proof that the parametrization of positroid cells induced by Kasteleyn weighted matrices coincides with that of Postnikov boundary measurement map. Finally we use Kasteleyn system of relations to associate algebraic geometric data to KP multi-soliton solutions. Indeed the KP wave function solves such system of relations at the nodes of the spectral curve if the dual graph of the latter represents the soliton data. Therefore the construction of the divisor is automatically invariant, and finally it coincides with that in Abenda and Grinevich (Sel. Math. New Ser. <b>25</b>(3), 43, 2019; Abenda and Grinevich 2020) for the present class of graphs.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09405-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4556725","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}

{"title":"Sums of Two-Parameter Deformations of Multiple Polylogarithms","authors":"Masaki Kato","doi":"10.1007/s11040-021-09407-0","DOIUrl":"10.1007/s11040-021-09407-0","url":null,"abstract":"<div><p>In this paper, we introduce a generating function of sums of two-parameter deformations of multiple polylogarithms, denoted by Φ₂(<i>a</i>;<i>p</i>,<i>q</i>), and study a <i>q</i>-difference equation satisfied by it. We show that this <i>q</i>-difference equation can be solved by expanding Φ₂(<i>a</i>;<i>p</i>,<i>q</i>) into power series of the parameter <i>p</i> and then using the method of variation of constants. By letting <span>(p rightarrow 0)</span> in the main theorem, we find that the generating function of sums of <i>q</i>-interpolated multiple zeta values can be written in terms of the <i>q</i>-hypergeometric function ₃<i>ϕ</i>₂, which is due to Li-Wakabayashi.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09407-0.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4363914","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}

{"title":"From Auto-Bäcklund Transformations to Auto-Bäcklund Transformations, and Torqued ABS Equations","authors":"Dan-da Zhang,&nbsp;Da-jun Zhang,&nbsp;Peter H. van der Kamp","doi":"10.1007/s11040-021-09406-1","DOIUrl":"10.1007/s11040-021-09406-1","url":null,"abstract":"<div><p>We provide a method which from a given auto-Bäcklund transformation (auto-BT) produces another auto-BT for a different equation. We apply the method to the natural auto-BTs for the ABS quad equations, which gives rise to torqued versions of ABS equations and explains the origin of each auto-BT listed in Atkinson (J. Phys. A: Math. Theor. <b>41</b>(8pp), 135202, 2008). The method is also applied to non-natural auto-BTs for ABS equations, which yields 3D consistent cubes which have not been found in Boll (J. Nonl. Math. Phys. <b>18</b>, 337–365, 2011), and to a multi-quadratic ABS* equation giving rise to a multi-quartic equation.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4919320","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}

{"title":"Solitons for the Modified Camassa-Holm Equation and their Interactions Via Dressing Method","authors":"Hui Mao,&nbsp;Yonghui Kuang","doi":"10.1007/s11040-021-09395-1","DOIUrl":"10.1007/s11040-021-09395-1","url":null,"abstract":"<div><p>In this paper, we develop the dressing method to study the modified Camassa-Holm equation with the help of reciprocal transformation and the associated modified Camassa- Holm equation. Based on this method, some different soliton solutions, in particular dark solitons to the modified Camassa-Holm equation are presented and their interactions are investigated.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09395-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5142250","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}

{"title":"New Condition on Uniqueness of Gibbs Measure for Models with Uncountable Set of Spin Values on a Cayley Tree","authors":"F. H. Haydarov","doi":"10.1007/s11040-021-09404-3","DOIUrl":"10.1007/s11040-021-09404-3","url":null,"abstract":"<div><p>In this paper we consider a model with nearest-neighbor interactions with spin space [0, 1] on Cayley trees of order <i>k</i> ⩾ 2. In Yu et al. (2013), a sufficient condition of uniqueness for the splitting Gibbs measure of the model is given. We investigate the sufficient condition of uniqueness and obtain better estimates.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-021-09404-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4809844","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}

{"title":"Density of States and Lifshitz Tails for Discrete 1D Random Dirac Operators","authors":"Roberto A. Prado,&nbsp;César R. de Oliveira,&nbsp;Edmundo C. de Oliveira","doi":"10.1007/s11040-021-09403-4","DOIUrl":"10.1007/s11040-021-09403-4","url":null,"abstract":"<div><p>We study the density of states and Lifshitz tails for a family of random Dirac operators on the one-dimensional lattice <span>(mathbb {Z})</span>. These operators consist of the sum of a discrete free Dirac operator with a random potential. The potential is a diagonal matrix formed by two different scalar potentials, which are sequences of independent and identically distributed random variables according to a Borel probability measure of compact support in <span>(mathbb {R})</span>. The existence of the density of state measure for these Dirac operators is obtained through two approaches by finite-volume quantities. By using one of these approaches, we show that the distribution function of the density of states decays exponentially for energies near the spectral band edges, i.e., we establish Lifshitz tails for these operators. Lifshitz tails are established first for Dirac operators restricted to appropriate subspaces of energies and, using this, extended to the full operators, including the occurrence of internal tails in the case of spectral gap.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"24 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2021-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4590762","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}