Pub Date : 2020-11-17DOI: 10.1215/00127094-2022-0077
Jian Ding, Mateo Wirth
For the two-dimensional random field Ising model where the random field is given by i.i.d. mean zero Gaussian variables with variance $epsilon^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as $epsilon to 0$, at zero temperature the correlation length scales as $e^{epsilon^{-4/3+o(1)}}$ (and our upper bound applies for all positive temperatures). As a proof ingredient, we establish a growth rate for the two-dimensional greedy lattice animal normalized by its boundary size, which may be of independent interest.
{"title":"Correlation length of the two-dimensional random field Ising model via greedy lattice animal","authors":"Jian Ding, Mateo Wirth","doi":"10.1215/00127094-2022-0077","DOIUrl":"https://doi.org/10.1215/00127094-2022-0077","url":null,"abstract":"For the two-dimensional random field Ising model where the random field is given by i.i.d. mean zero Gaussian variables with variance $epsilon^2$, we study (one natural notion of) the correlation length, which is the critical size of a box at which the influences of the random field and of the boundary condition on the spin magnetization are comparable. We show that as $epsilon to 0$, at zero temperature the correlation length scales as $e^{epsilon^{-4/3+o(1)}}$ (and our upper bound applies for all positive temperatures). As a proof ingredient, we establish a growth rate for the two-dimensional greedy lattice animal normalized by its boundary size, which may be of independent interest.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42377842","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}
Let $S$ be a closed surface and $text{Diff}_{text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k in mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every periodic group of bounded exponent $G subset text{Diff}_{text{Vol}}(S)$ is a finite group.
{"title":"The Burnside problem for Diffω(S2)","authors":"Sebastián Hurtado, Alejandro Kocsard, Federico Rodríguez-Hertz","doi":"10.1215/00127094-2020-0028","DOIUrl":"https://doi.org/10.1215/00127094-2020-0028","url":null,"abstract":"Let $S$ be a closed surface and $text{Diff}_{text{Vol}}(S)$ be the group of volume preserving diffeomorphisms of $S$. A finitely generated group $G$ is periodic of bounded exponent if there exists $k in mathbb{N}$ such that every element of $G$ has order at most $k$. We show that every periodic group of bounded exponent $G subset text{Diff}_{text{Vol}}(S)$ is a finite group.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84054124","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}
Pub Date : 2020-11-09DOI: 10.1215/00127094-2022-0015
Laurin Kohler-Schindler, V. Tassion
We prove a general Russo-Seymour-Welsh result valid for any invariant planar percolation process satisfying positive association. This means that the probability of crossing a rectangle in the long direction is related by a homeomorphism to the probability of crossing it in the short direction. This homeomorphism is universal in the sense that it depends only on the aspect ratio of the rectangle, and is uniform in the scale and the considered model.
{"title":"Crossing probabilities for planar percolation","authors":"Laurin Kohler-Schindler, V. Tassion","doi":"10.1215/00127094-2022-0015","DOIUrl":"https://doi.org/10.1215/00127094-2022-0015","url":null,"abstract":"We prove a general Russo-Seymour-Welsh result valid for any invariant planar percolation process satisfying positive association. This means that the probability of crossing a rectangle in the long direction is related by a homeomorphism to the probability of crossing it in the short direction. This homeomorphism is universal in the sense that it depends only on the aspect ratio of the rectangle, and is uniform in the scale and the considered model.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46918048","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}
Pub Date : 2020-11-01DOI: 10.1215/00127094-2020-0035
H. Darmon, Jan Vonk
A rigid meromorphic cocycle is a class in the rst cohomology of the discrete group Γ := SL2(Z[1/p]) with values in the multiplicative group of non-zero rigid meromorphic functions on the p-adic upper half plane Hp := P1(Cp) − P1(Qp). Such a class can be evaluated at the real quadratic irrationalities in Hp, which are referred to as “RM points”. Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lis: their zeroes and poles are contained in a nite union of Γ-orbits of RM points, and their RM values are conjectured to lie in ring class elds of real quadratic elds. ese RM values enjoy striking parallels with the CM values of modular functions on SL2(Z)H: in particular they seem to factor just like the dierences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high p-adic accuracy leads to convincing numerical evidence for the algebraicity and factorisation of the resulting singular moduli for real quadratic elds.
{"title":"Singular moduli for real quadratic fields: A rigid analytic approach","authors":"H. Darmon, Jan Vonk","doi":"10.1215/00127094-2020-0035","DOIUrl":"https://doi.org/10.1215/00127094-2020-0035","url":null,"abstract":"A rigid meromorphic cocycle is a class in the rst cohomology of the discrete group Γ := SL2(Z[1/p]) with values in the multiplicative group of non-zero rigid meromorphic functions on the p-adic upper half plane Hp := P1(Cp) − P1(Qp). Such a class can be evaluated at the real quadratic irrationalities in Hp, which are referred to as “RM points”. Rigid meromorphic cocycles can be envisaged as the real quadratic counterparts of Borcherds’ singular theta lis: their zeroes and poles are contained in a nite union of Γ-orbits of RM points, and their RM values are conjectured to lie in ring class elds of real quadratic elds. ese RM values enjoy striking parallels with the CM values of modular functions on SL2(Z)H: in particular they seem to factor just like the dierences of classical singular moduli, as described by Gross and Zagier. A fast algorithm for computing rigid meromorphic cocycles to high p-adic accuracy leads to convincing numerical evidence for the algebraicity and factorisation of the resulting singular moduli for real quadratic elds.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45131514","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}
Pub Date : 2020-11-01DOI: 10.1215/00127094-2020-0065
Marco Antei, M. Emsalem, C. Gasbarri
{"title":"Erratum for “Heights of vector bundles and the fundamental group scheme of a curve”","authors":"Marco Antei, M. Emsalem, C. Gasbarri","doi":"10.1215/00127094-2020-0065","DOIUrl":"https://doi.org/10.1215/00127094-2020-0065","url":null,"abstract":"","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45595554","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}
Pub Date : 2020-10-07DOI: 10.1215/00127094-2022-0066
Francisco Arana-Herrera
We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichm"uller space that lie within a Teichm"uller metric ball of given center and large radius. Estimates of the same kind are also proved for sector and bisector counts. These estimates effectivize asymptotic counting results of Athreya, Bufetov, Eskin, and Mirzakhani.
{"title":"Effective mapping class group dynamics, I: Counting lattice points in Teichmüller space","authors":"Francisco Arana-Herrera","doi":"10.1215/00127094-2022-0066","DOIUrl":"https://doi.org/10.1215/00127094-2022-0066","url":null,"abstract":"We prove a quantitative estimate with a power saving error term for the number of points in a mapping class group orbit of Teichm\"uller space that lie within a Teichm\"uller metric ball of given center and large radius. Estimates of the same kind are also proved for sector and bisector counts. These estimates effectivize asymptotic counting results of Athreya, Bufetov, Eskin, and Mirzakhani.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46729931","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}
Pub Date : 2020-09-29DOI: 10.1215/00127094-2022-0041
Paul Balmer, Martin Gallauer
We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.
{"title":"Finite permutation resolutions","authors":"Paul Balmer, Martin Gallauer","doi":"10.1215/00127094-2022-0041","DOIUrl":"https://doi.org/10.1215/00127094-2022-0041","url":null,"abstract":"We prove that every finite dimensional representation of a finite group over a field of characteristic p admits a finite resolution by p-permutation modules. The proof involves a reformulation in terms of derived categories.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43816454","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}
Pub Date : 2020-09-11DOI: 10.1215/00127094-2022-0013
Gal Binyamini, R. Cluckers, D. Novikov
We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${mathbb{C}}(!(t)!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical Pila-Wilkie counting theorem in the archimean situation. Some non-archimedean analogs of this type have been developed in the work of Cluckers-Comte-Loeser for the field ${mathbb{Q}}_p$, but the situation in ${mathbb{C}}(!(t)!)$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree $r$ on the transcendental part of a subanalytic set over ${mathbb{C}}(!(t)!)$ is automatically finite, but give examples showing that their number may grow arbitrarily quickly even for analytic sets. Thus no analog of the Pila-Wilkie theorem can be expected to hold for general analytic sets. On the other hand we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in $r$, thus showing that the analog of Wilkie's conjecture does hold in this context.
我们考虑域$｛mathbb｛C｝｝（！（t）！）$上解析或可定义子集上的多项式曲线的计数问题，作为度数$r$的函数。这种类型的结果可以通过与经典的Pila-Wilkie计数定理在阿基米德情形下的类比来预期。在Cluckers Comte Loeser的工作中，已经为字段$｛mathbb｛Q｝｝_p$开发了一些这种类型的非阿基米德类似物，但在$｛ mathbb｛C｝（！（t）！）$中的情况似乎明显不同。我们证明了$｛mathbb｛C｝｝（！（t）！）$上的子分析集超越部分上的固定次数$r$的多项式曲线集是自动有限的，但给出的例子表明，即使对于分析集，它们的数量也可以任意快速增长。因此，Pila-Wilkie定理的任何类似物都不可能适用于一般分析集。另一方面，我们证明，如果限制由Pfafian或Noetherian函数定义的变种，那么这个数字最多以$r$的形式多项式增长，从而表明Wilkie猜想的类似物在这种情况下确实成立。
{"title":"Point counting and Wilkie’s conjecture for non-Archimedean Pfaffian and Noetherian functions","authors":"Gal Binyamini, R. Cluckers, D. Novikov","doi":"10.1215/00127094-2022-0013","DOIUrl":"https://doi.org/10.1215/00127094-2022-0013","url":null,"abstract":"We consider the problem of counting polynomial curves on analytic or definable subsets over the field ${mathbb{C}}(!(t)!)$, as a function of the degree $r$. A result of this type could be expected by analogy with the classical Pila-Wilkie counting theorem in the archimean situation. \u0000Some non-archimedean analogs of this type have been developed in the work of Cluckers-Comte-Loeser for the field ${mathbb{Q}}_p$, but the situation in ${mathbb{C}}(!(t)!)$ appears to be significantly different. We prove that the set of polynomial curves of a fixed degree $r$ on the transcendental part of a subanalytic set over ${mathbb{C}}(!(t)!)$ is automatically finite, but give examples showing that their number may grow arbitrarily quickly even for analytic sets. Thus no analog of the Pila-Wilkie theorem can be expected to hold for general analytic sets. On the other hand we show that if one restricts to varieties defined by Pfaffian or Noetherian functions, then the number grows at most polynomially in $r$, thus showing that the analog of Wilkie's conjecture does hold in this context.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47553123","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}
Pub Date : 2020-09-11DOI: 10.1215/00127094-2022-0059
Jeongseok Oh, Richard P. Thomas
Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for $SO(r,mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.
{"title":"Counting sheaves on Calabi–Yau 4-folds, I","authors":"Jeongseok Oh, Richard P. Thomas","doi":"10.1215/00127094-2022-0059","DOIUrl":"https://doi.org/10.1215/00127094-2022-0059","url":null,"abstract":"Borisov-Joyce constructed a real virtual cycle on compact moduli spaces of stable sheaves on Calabi-Yau 4-folds, using derived differential geometry. We construct an algebraic virtual cycle. A key step is a localisation of Edidin-Graham's square root Euler class for $SO(r,mathbb C)$ bundles to the zero locus of an isotropic section, or to the support of an isotropic cone. We prove a torus localisation formula, making the invariants computable and extending them to the noncompact case when the fixed locus is compact. We give a $K$-theoretic refinement by defining $K$-theoretic square root Euler classes and their localised versions. In a sequel we prove our invariants reproduce those of Borisov-Joyce.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46681636","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}
Pub Date : 2020-08-21DOI: 10.1215/00127094-2022-0042
Lin Chen
Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(!(t)!)$-equivariant and $U^-(!(t)!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and arXiv:1607.00586. We also generalize our results to the Beilinson-Drinfeld Grassmannian $Gr_{G,X^I}$ and to the affine flag variety $Fl_G$.
{"title":"Nearby cycles on Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and long intertwining functor","authors":"Lin Chen","doi":"10.1215/00127094-2022-0042","DOIUrl":"https://doi.org/10.1215/00127094-2022-0042","url":null,"abstract":"Let $G$ be a reductive group and $U,U^-$ be the unipotent radicals of a pair of opposite parabolic subgroups $P,P^-$. We prove that the DG-categories of $U(!(t)!)$-equivariant and $U^-(!(t)!)$-equivariant D-modules on the affine Grassmannian $Gr_G$ are canonically dual to each other. We show that the unit object witnessing this duality is given by nearby cycles on the Drinfeld-Gaitsgory-Vinberg interpolation Grassmannian defined in arXiv:1805.07721. We study various properties of the mentioned nearby cycles, in particular compare them with the nearby cycles studied in arXiv:1411.4206 and arXiv:1607.00586. We also generalize our results to the Beilinson-Drinfeld Grassmannian $Gr_{G,X^I}$ and to the affine flag variety $Fl_G$.","PeriodicalId":11447,"journal":{"name":"Duke Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":2.5,"publicationDate":"2020-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47423413","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}
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