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A New Regularized Siegel-Weil Type Formula. Part I 一种新的正规化西格尔-韦尔公式。第一部分
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2024-01-22 DOI: 10.1007/s00039-024-00657-y
David Ginzburg, David Soudry
{"title":"A New Regularized Siegel-Weil Type Formula. Part I","authors":"David Ginzburg, David Soudry","doi":"10.1007/s00039-024-00657-y","DOIUrl":"https://doi.org/10.1007/s00039-024-00657-y","url":null,"abstract":"<p>In this paper, we prove a formula, realizing certain residual Eisenstein series on symplectic groups as regularized kernel integrals. These Eisenstein series, as well as the kernel integrals, are attached to Speh representations. This forms an initial step to a new type of a regularized Siegel-Weil formula that we propose. This new formula bears the same relation to the generalized doubling integrals of Cai, Friedberg, Ginzburg and Kaplan, as does the regularized Siegel-Weil formula to the doubling integrals of Piatetski-Shapiro and Rallis.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2024-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510830","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}
引用次数: 0
Relations between scaling exponents in unimodular random graphs 单模随机图中标度指数之间的关系
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-11-09 DOI: 10.1007/s00039-023-00654-7
James R. Lee
{"title":"Relations between scaling exponents in unimodular random graphs","authors":"James R. Lee","doi":"10.1007/s00039-023-00654-7","DOIUrl":"https://doi.org/10.1007/s00039-023-00654-7","url":null,"abstract":"<p>We investigate the validity of the “Einstein relations” in the general setting of unimodular random networks. These are equalities relating scaling exponents: </p><span> $$begin{aligned} d_{w} &amp;= d_{f} + tilde{zeta }, d_{s} &amp;= 2 d_{f}/d_{w}, end{aligned}$$ </span><p> where <i>d</i><sub><i>w</i></sub> is the walk dimension, <i>d</i><sub><i>f</i></sub> is the fractal dimension, <i>d</i><sub><i>s</i></sub> is the spectral dimension, and <span>(tilde{zeta })</span> is the resistance exponent. Roughly speaking, this relates the mean displacement and return probability of a random walker to the density and conductivity of the underlying medium. We show that if <i>d</i><sub><i>f</i></sub> and <span>(tilde{zeta } geqslant 0)</span> exist, then <i>d</i><sub><i>w</i></sub> and <i>d</i><sub><i>s</i></sub> exist, and the aforementioned equalities hold. Moreover, our primary new estimate <span>(d_{w} geqslant d_{f} + tilde{zeta })</span> is established for all <span>(tilde{zeta } in mathbb{R})</span>.</p><p>For the uniform infinite planar triangulation (UIPT), this yields the consequence <i>d</i><sub><i>w</i></sub>=4 using <i>d</i><sub><i>f</i></sub>=4 (Angel in Geom. Funct. Anal. 13(5):935–974, 2003) and <span>(tilde{zeta }=0)</span> (established here as a consequence of the Liouville Quantum Gravity theory, following Gwynne-Miller 2020 and (Ding and Gwynne in Commun. Math. Phys. 374(3):1877–1934, 2020)). The conclusion <i>d</i><sub><i>w</i></sub>=4 had been previously established by Gwynne and Hutchcroft (2018) using more elaborate methods. A new consequence is that <i>d</i><sub><i>w</i></sub>=<i>d</i><sub><i>f</i></sub> for the uniform infinite Schnyder-wood decorated triangulation, implying that the simple random walk is subdiffusive, since <i>d</i><sub><i>f</i></sub>&gt;2.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72364946","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}
引用次数: 4
GOE statistics on the moduli space of surfaces of large genus 大亏格曲面模空间的GOE统计
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-11-02 DOI: 10.1007/s00039-023-00655-6
Zeév Rudnick
{"title":"GOE statistics on the moduli space of surfaces of large genus","authors":"Zeév Rudnick","doi":"10.1007/s00039-023-00655-6","DOIUrl":"https://doi.org/10.1007/s00039-023-00655-6","url":null,"abstract":"<p>For a compact hyperbolic surface, we define a smooth linear statistic, mimicking the number of Laplace eigenvalues in a short energy window. We study the variance of this statistic, when averaged over the moduli space <span>(mathcal{M}_{g})</span> of all genus <i>g</i> surfaces with respect to the Weil-Petersson measure. We show that in the double limit, first taking the large genus limit and then the short window limit, we recover GOE statistics for the variance. The proof makes essential use of Mirzakhani’s integration formula.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509201","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}
引用次数: 11
Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case 经典波动方法和现代规范变换：一维情况下的谱渐近性
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-10-31 DOI: 10.1007/s00039-023-00650-x
Jeffrey Galkowski, Leonid Parnovski, Roman Shterenberg
{"title":"Classical wave methods and modern gauge transforms: spectral asymptotics in the one dimensional case","authors":"Jeffrey Galkowski, Leonid Parnovski, Roman Shterenberg","doi":"10.1007/s00039-023-00650-x","DOIUrl":"https://doi.org/10.1007/s00039-023-00650-x","url":null,"abstract":"<p>In this article, we consider the asymptotic behaviour of the spectral function of Schrödinger operators on the real line. Let <span>(H: L^{2}(mathbb{R})to L^{2}(mathbb{R}))</span> have the form </p><span>$$ H:=-frac{d^{2}}{dx^{2}}+Q, $$</span><p> where <i>Q</i> is a formally self-adjoint first order differential operator with smooth coefficients, bounded with all derivatives. We show that the kernel of the spectral projector, <span>({1}_{(-infty ,rho ^{2}]}(H))</span>, has a complete asymptotic expansion in powers of <i>ρ</i>. This settles the 1-dimensional case of a conjecture made by the last two authors.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509199","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}
引用次数: 0
Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties Bers的同时一致化与Poincaréholonomy变种的交集
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-10-31 DOI: 10.1007/s00039-023-00653-8
Shinpei Baba
{"title":"Bers’ simultaneous uniformization and the intersection of Poincaré holonomy varieties","authors":"Shinpei Baba","doi":"10.1007/s00039-023-00653-8","DOIUrl":"https://doi.org/10.1007/s00039-023-00653-8","url":null,"abstract":"<p>We consider the space of ordered pairs of distinct <span>({mathbb{C}{mathrm{P}}}^{1})</span>-structures on Riemann surfaces (of any orientations) which have identical holonomy, so that the quasi-Fuchsian space is identified with a connected component of this space. This space holomorphically maps to the product of the Teichmüller spaces minus its diagonal.</p><p>In this paper, we prove that this mapping is a complete local branched covering map. As a corollary, we reprove Bers’ simultaneous uniformization theorem without any quasi-conformal deformation theory. Our main theorem is that the intersection of arbitrary two Poincaré holonomy varieties (<span>(operatorname{SL}_{2}mathbb{C})</span>-opers) is a non-empty discrete set, which is closely related to the mapping.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509200","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}
引用次数: 0
Contribution of n-cylinder square-tiled surfaces to Masur–Veech volume of $mathcal{H}(2g-2)$ n柱方形瓷砖表面对Masur–Veech体积的贡献$mathcal｛H｝（2g-2）$
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-10-12 DOI: 10.1007/s00039-023-00652-9
Ivan Yakovlev
{"title":"Contribution of n-cylinder square-tiled surfaces to Masur–Veech volume of $mathcal{H}(2g-2)$","authors":"Ivan Yakovlev","doi":"10.1007/s00039-023-00652-9","DOIUrl":"https://doi.org/10.1007/s00039-023-00652-9","url":null,"abstract":"<p>We find the generating function for the contributions of <i>n</i>-cylinder square-tiled surfaces to the Masur–Veech volume of <span>(mathcal{H}(2g-2))</span>. It is a bivariate generalization of the generating function for the total volumes obtained by Sauvaget via intersection theory. Our approach is, however, purely combinatorial. It relies on the study of counting functions for certain families of metric ribbon graphs. Their top-degree terms are polynomials, whose (normalized) coefficients are cardinalities of certain families of metric plane trees. These polynomials are analogues of Kontsevich polynomials that appear as part of his proof of Witten’s conjecture.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509213","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}
引用次数: 0
Concentration of invariant means and dynamics of chain stabilizers in continuous geometries 连续几何中不变均值的集中与链稳定器的动力学
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-10-12 DOI: 10.1007/s00039-023-00651-w
Friedrich Martin Schneider
{"title":"Concentration of invariant means and dynamics of chain stabilizers in continuous geometries","authors":"Friedrich Martin Schneider","doi":"10.1007/s00039-023-00651-w","DOIUrl":"https://doi.org/10.1007/s00039-023-00651-w","url":null,"abstract":"<p>We prove a concentration inequality for invariant means on topological groups, namely for such adapted to a chain of amenable topological subgroups. The result is based on an application of Azuma’s martingale inequality and provides a method for establishing extreme amenability. Building on this technique, we exhibit new examples of extremely amenable groups arising from von Neumann’s continuous geometries. Along the way, we also answer a question by Pestov on dynamical concentration in direct products of amenable topological groups.</p>","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71509198","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}
引用次数: 1
Isometric group actions with vanishing rate of escape on $$textrm{CAT}(0)$$ spaces $$textrm{CAT}(0)$$空间上具有逃逸消失率的等长群作用
1区 数学
Geometric and Functional Analysis Pub Date : 2023-01-24 DOI: 10.1007/s00039-023-00628-9
Hiroyasu Izeki
{"title":"Isometric group actions with vanishing rate of escape on $$textrm{CAT}(0)$$ spaces","authors":"Hiroyasu Izeki","doi":"10.1007/s00039-023-00628-9","DOIUrl":"https://doi.org/10.1007/s00039-023-00628-9","url":null,"abstract":"Let $Gamma$ be a finitely generated group equipped with a symmetric and nondegenerate probability measure $mu$ with finite second moment, and $Y$ a CAT(0) space which is either proper or of finite telescopic dimension. We show that if an isometric action of $Gamma$ on $Y$ has vanishing rate of escape with respect to $mu$ and does not fix a point in the boundary at infinity of $Y$, then there exists a flat subspace in $Y$ which is left invariant under the action of $Gamma$. In the proof of this result, an equivariant $mu$-harmonic map from $Gamma$ into $Y$ plays an important role.","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136198725","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}
引用次数: 1
Tori Approximation of Families of Diagonally Invariant Measures 对角不变测度族的Tori逼近
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2023-01-02 DOI: 10.1007/s00039-023-00646-7
O. Solan, Y. Yifrach
{"title":"Tori Approximation of Families of Diagonally Invariant Measures","authors":"O. Solan, Y. Yifrach","doi":"10.1007/s00039-023-00646-7","DOIUrl":"https://doi.org/10.1007/s00039-023-00646-7","url":null,"abstract":"","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47938375","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}
引用次数: 0
Publisher Correction to: The generalized doubling method: local theory 出版商更正：广义加倍方法：局部理论
IF 2.2 1区 数学
Geometric and Functional Analysis Pub Date : 2022-11-29 DOI: 10.1007/s00039-022-00622-7
Yuanqing Cai, S. Friedberg, Eyal Kaplan
{"title":"Publisher Correction to: The generalized doubling method: local theory","authors":"Yuanqing Cai, S. Friedberg, Eyal Kaplan","doi":"10.1007/s00039-022-00622-7","DOIUrl":"https://doi.org/10.1007/s00039-022-00622-7","url":null,"abstract":"","PeriodicalId":12478,"journal":{"name":"Geometric and Functional Analysis","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46602211","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}
引用次数: 0
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