Abstract We prove that if the set of unordered pairs of real numbers is coloured by finitely many colours, there is a set of reals homeomorphic to the rationals whose pairs have at most two colours. Our proof uses large cardinals and verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.
{"title":"Proof of a conjecture of Galvin","authors":"Dilip Raghavan, S. Todorcevic","doi":"10.1017/fmp.2020.12","DOIUrl":"https://doi.org/10.1017/fmp.2020.12","url":null,"abstract":"Abstract We prove that if the set of unordered pairs of real numbers is coloured by finitely many colours, there is a set of reals homeomorphic to the rationals whose pairs have at most two colours. Our proof uses large cardinals and verifies a conjecture of Galvin from the 1970s. We extend this result to an essentially optimal class of topological spaces in place of the reals.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"8 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.12","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"56602373","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}
Abstract We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in $ell $ -adic cohomology is Galois invariant for all $ell $ .
摘要证明了四元数Shimura变上同调自同构形式的Jacquet-Langlands对应是由一个Hodge类实现的。在酉相似群上的Shimura变的Kottwitz猜想的条件下，我们还证明了该类在$ well $ -进上同调中的象对所有$ well $都是伽罗瓦不变的。
{"title":"Hodge classes and the Jacquet–Langlands correspondence","authors":"Atsushi Ichino, Kartik Prasanna","doi":"10.1017/fmp.2023.20","DOIUrl":"https://doi.org/10.1017/fmp.2023.20","url":null,"abstract":"Abstract We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in \u0000$ell $\u0000 -adic cohomology is Galois invariant for all \u0000$ell $\u0000 .","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"11 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44408880","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}
Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.
麦克唐纳过程是利用麦克唐纳对称函数的柯西和恒等式建立的整数分区序列的度量。这些度量是揭示许多概率系统的可积性的有用工具，包括kardar - paris - zhang (KPZ)方程及其普适类中的许多其他模型。在本文中，我们发展了这些模型的半空间变体和相应的半空间麦克唐纳过程的结构理论。这些过程是用利特尔伍德和恒等式而不是柯西恒等式建立的，它们的分析比它们的全空间对应物要困难得多。我们计算了一般半空间麦克唐纳测度的可观测量的矩和拉普拉斯变换。引入新的动力学来保持这类测度，我们将它们与各种随机过程联系起来，特别是半象限中的log-gamma聚合物(它们也与半象限中的随机六顶点模型和半空间ASEP有关)。对于聚合物模型，我们给出了配分函数拉普拉斯变换的显式积分公式。非严格鞍点渐近导致定向聚合物自由能收敛于Tracy-Widom(与高斯正交或辛系综相关)或高斯分布，这取决于边界上权值的平均大小。
{"title":"HALF-SPACE MACDONALD PROCESSES","authors":"Guillaume Barraquand, A. Borodin, Ivan Corwin","doi":"10.1017/FMP.2020.3","DOIUrl":"https://doi.org/10.1017/FMP.2020.3","url":null,"abstract":"Macdonald processes are measures on sequences of integer partitions built using the Cauchy summation identity for Macdonald symmetric functions. These measures are a useful tool to uncover the integrability of many probabilistic systems, including the Kardar–Parisi–Zhang (KPZ) equation and a number of other models in its universality class. In this paper, we develop the structural theory behind half-space variants of these models and the corresponding half-space Macdonald processes. These processes are built using a Littlewood summation identity instead of the Cauchy identity, and their analysis is considerably harder than their full-space counterparts. We compute moments and Laplace transforms of observables for general half-space Macdonald measures. Introducing new dynamics preserving this class of measures, we relate them to various stochastic processes, in particular the log-gamma polymer in a half-quadrant (they are also related to the stochastic six-vertex model in a half-quadrant and the half-space ASEP). For the polymer model, we provide explicit integral formulas for the Laplace transform of the partition function. Nonrigorous saddle-point asymptotics yield convergence of the directed polymer free energy to either the Tracy–Widom (associated to the Gaussian orthogonal or symplectic ensemble) or the Gaussian distribution depending on the average size of weights on the boundary.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"8 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/FMP.2020.3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45210114","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}
For each $tin mathbb{R}$, we define the entire function $$begin{eqnarray}H_{t}(z):=int _{0}^{infty }e^{tu^{2}}unicode[STIX]{x1D6F7}(u)cos (zu),du,end{eqnarray}$$ where $unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$begin{eqnarray}unicode[STIX]{x1D6F7}(u):=mathop{sum }_{n=1}^{infty }(2unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3unicode[STIX]{x1D70B}n^{2}e^{5u})exp (-unicode[STIX]{x1D70B}n^{2}e^{4u}).end{eqnarray}$$ Newman showed that there exists a finite constant $unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant) such that the zeros of $H_{t}$ are all real precisely when $tgeqslant unicode[STIX]{x1D6EC}$. The Riemann hypothesis is equivalent to the assertion $unicode[STIX]{x1D6EC}leqslant 0$, and Newman conjectured the complementary bound $unicode[STIX]{x1D6EC}geqslant 0$. In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $unicode[STIX]{x1D6EC}
{"title":"THE DE BRUIJN–NEWMAN CONSTANT IS NON-NEGATIVE","authors":"B. Rodgers, T. Tao","doi":"10.1017/fmp.2020.6","DOIUrl":"https://doi.org/10.1017/fmp.2020.6","url":null,"abstract":"For each $tin mathbb{R}$, we define the entire function $$begin{eqnarray}H_{t}(z):=int _{0}^{infty }e^{tu^{2}}unicode[STIX]{x1D6F7}(u)cos (zu),du,end{eqnarray}$$ where $unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$begin{eqnarray}unicode[STIX]{x1D6F7}(u):=mathop{sum }_{n=1}^{infty }(2unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3unicode[STIX]{x1D70B}n^{2}e^{5u})exp (-unicode[STIX]{x1D70B}n^{2}e^{4u}).end{eqnarray}$$ Newman showed that there exists a finite constant $unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant) such that the zeros of $H_{t}$ are all real precisely when $tgeqslant unicode[STIX]{x1D6EC}$. The Riemann hypothesis is equivalent to the assertion $unicode[STIX]{x1D6EC}leqslant 0$, and Newman conjectured the complementary bound $unicode[STIX]{x1D6EC}geqslant 0$. In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $unicode[STIX]{x1D6EC}<tleqslant 0$, until one establishes that the zeros of $H_{0}$ are in local equilibrium, in the sense that they locally behave (on average) as if they were equally spaced in an arithmetic progression, with gaps staying close to the global average gap size. But this latter claim is inconsistent with the known results about the local distribution of zeros of the Riemann zeta function, such as the pair correlation estimates of Montgomery.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2018-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.6","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48796125","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}
What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.
{"title":"THE EXACT MINIMUM NUMBER OF TRIANGLES IN GRAPHS WITH GIVEN ORDER AND SIZE","authors":"Hong Liu, O. Pikhurko, Katherine Staden","doi":"10.1017/fmp.2020.7","DOIUrl":"https://doi.org/10.1017/fmp.2020.7","url":null,"abstract":"What is the minimum number of triangles in a graph of given order and size? Motivated by earlier results of Mantel and Turán, Rademacher solved the first nontrivial case of this problem in 1941. The problem was revived by Erdős in 1955; it is now known as the Erdős–Rademacher problem. After attracting much attention, it was solved asymptotically in a major breakthrough by Razborov in 2008. In this paper, we provide an exact solution for all large graphs whose edge density is bounded away from $1$, which in this range confirms a conjecture of Lovász and Simonovits from 1975. Furthermore, we give a description of the extremal graphs.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.7","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48554034","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}
Abstract Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of $operatorname {mathrm {GL}}_n$ , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.
{"title":"On the Kottwitz conjecture for local shtuka spaces","authors":"D. Hansen, Tasho Kaletha, Jared Weinstein","doi":"10.1017/fmp.2022.7","DOIUrl":"https://doi.org/10.1017/fmp.2022.7","url":null,"abstract":"Abstract Kottwitz’s conjecture describes the contribution of a supercuspidal representation to the cohomology of a local Shimura variety in terms of the local Langlands correspondence. A natural extension of this conjecture concerns Scholze’s more general spaces of local shtukas. Using a new Lefschetz–Verdier trace formula for v-stacks, we prove the extended conjecture, disregarding the action of the Weil group, and modulo a virtual representation whose character vanishes on the locus of elliptic elements. As an application, we show that, for an irreducible smooth representation of an inner form of \u0000$operatorname {mathrm {GL}}_n$\u0000 , the L-parameter constructed by Fargues–Scholze agrees with the usual semisimplified parameter arising from local Langlands.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"10 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48036646","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}
In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0)in mathbb{R}^{2}$ , and the other is varied horizontally, over $(z,1)$ , $zin mathbb{R}$ , the polymer weight profile as a function of $zin mathbb{R}$ is locally Brownian; indeed, by Hammond [‘Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation’, Preprint (2016), arXiv:1609.02971, Theorem 2.11 and Proposition 2.5], the law of the profile is known to enjoy a very strong comparison to Brownian bridge on a given compact interval, with a Radon–Nikodym derivative in every $L^{p}$ space for $pin (1,infty )$ , uniformly in the scaling parameter, provided that an affine adjustment is made to the weight profile before the comparison is made. In this article, we generalize this narrow wedge case and study polymer weight profiles begun from a very general initial condition. We prove that the profiles on a compact interval resemble Brownian bridge in a uniform sense: splitting the compact interval into a random but controlled number of patches, the profile in each patch after affine adjustment has a Radon–Nikodym derivative that lies in every $L^{p}$ space for $pin (1,3)$ . This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in Hammond [‘Exponents governing the rarity of disjoint polymers in Brownian last passage percolation’, Preprint (2017a), arXiv:1709.04110] using techniques from Hammond (2016) and [‘Modulus of continuity of polymer weight profiles in Brownian last passage percolation’, Preprint (2017b), arXiv:1709.04115].
{"title":"A PATCHWORK QUILT SEWN FROM BROWNIAN FABRIC: REGULARITY OF POLYMER WEIGHT PROFILES IN BROWNIAN LAST PASSAGE PERCOLATION","authors":"A. Hammond","doi":"10.1017/fmp.2019.2","DOIUrl":"https://doi.org/10.1017/fmp.2019.2","url":null,"abstract":"In last passage percolation models lying in the Kardar–Parisi–Zhang (KPZ) universality class, the energy of long energy-maximizing paths may be studied as a function of the paths’ pair of endpoint locations. Scaled coordinates may be introduced, so that these maximizing paths, or polymers, now cross unit distances with unit-order fluctuations, and have scaled energy, or weight, of unit order. In this article, we consider Brownian last passage percolation in these scaled coordinates. In the narrow wedge case, when one endpoint of such polymers is fixed, say at $(0,0)in mathbb{R}^{2}$ , and the other is varied horizontally, over $(z,1)$ , $zin mathbb{R}$ , the polymer weight profile as a function of $zin mathbb{R}$ is locally Brownian; indeed, by Hammond [‘Brownian regularity for the Airy line ensemble, and multi-polymer watermelons in Brownian last passage percolation’, Preprint (2016), arXiv:1609.02971, Theorem 2.11 and Proposition 2.5], the law of the profile is known to enjoy a very strong comparison to Brownian bridge on a given compact interval, with a Radon–Nikodym derivative in every $L^{p}$ space for $pin (1,infty )$ , uniformly in the scaling parameter, provided that an affine adjustment is made to the weight profile before the comparison is made. In this article, we generalize this narrow wedge case and study polymer weight profiles begun from a very general initial condition. We prove that the profiles on a compact interval resemble Brownian bridge in a uniform sense: splitting the compact interval into a random but controlled number of patches, the profile in each patch after affine adjustment has a Radon–Nikodym derivative that lies in every $L^{p}$ space for $pin (1,3)$ . This result is proved by harnessing an understanding of the uniform coalescence structure in the field of polymers developed in Hammond [‘Exponents governing the rarity of disjoint polymers in Brownian last passage percolation’, Preprint (2017a), arXiv:1709.04110] using techniques from Hammond (2016) and [‘Modulus of continuity of polymer weight profiles in Brownian last passage percolation’, Preprint (2017b), arXiv:1709.04115].","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"7 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2019.2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43016269","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}
We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.
我们发展了有限群、一般群或特殊群、线性群和酉群的复不可约特征的特征级概念。我们用Lusztig标记和度来刻画一个字符的级别。然后，我们证明了具有不太大中心化子的元素的特征值的显式上界，并导出了与这些共轭类相对应的随机游动的覆盖数和混合时间的上界。我们还用某些对偶对刻画了特征的级别，并证明了特征值的显式指数特征界，前提是级别不太大。还提供了几个进一步的应用。其他有限经典群的相关结果在续集[Guralnick et al.‘有限经典群中的字符级别和字符边界’，Preprint，2019，arXiv:1904.08070]中通过不同的方法获得。
{"title":"CHARACTER LEVELS AND CHARACTER BOUNDS","authors":"R. Guralnick, M. Larsen, P. Tiep","doi":"10.1017/fmp.2019.9","DOIUrl":"https://doi.org/10.1017/fmp.2019.9","url":null,"abstract":"We develop the concept of character level for the complex irreducible characters of finite, general or special, linear and unitary groups. We give characterizations of the level of a character in terms of its Lusztig label and in terms of its degree. Then we prove explicit upper bounds for character values at elements with not-too-large centralizers and derive upper bounds on the covering number and mixing time of random walks corresponding to these conjugacy classes. We also characterize the level of the character in terms of certain dual pairs and prove explicit exponential character bounds for the character values, provided that the level is not too large. Several further applications are also provided. Related results for other finite classical groups are obtained in the sequel [Guralnick et al. ‘Character levels and character bounds for finite classical groups’, Preprint, 2019, arXiv:1904.08070] by different methods.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":"8 1","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2019.9","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42841124","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}
A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.
{"title":"ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES","authors":"V. Delecroix, É. Goujard, P. Zograf, A. Zorich","doi":"10.1017/fmp.2020.2","DOIUrl":"https://doi.org/10.1017/fmp.2020.2","url":null,"abstract":"A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2020.2","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48129260","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}
We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.
{"title":"THE SET OF QUANTUM CORRELATIONS IS NOT CLOSED","authors":"William Slofstra","doi":"10.1017/fmp.2018.3","DOIUrl":"https://doi.org/10.1017/fmp.2018.3","url":null,"abstract":"We construct a linear system nonlocal game which can be played perfectly using a limit of finite-dimensional quantum strategies, but which cannot be played perfectly on any finite-dimensional Hilbert space, or even with any tensor-product strategy. In particular, this shows that the set of (tensor-product) quantum correlations is not closed. The constructed nonlocal game provides another counterexample to the ‘middle’ Tsirelson problem, with a shorter proof than our previous paper (though at the loss of the universal embedding theorem). We also show that it is undecidable to determine if a linear system game can be played perfectly with a finite-dimensional strategy, or a limit of finite-dimensional quantum strategies.","PeriodicalId":56024,"journal":{"name":"Forum of Mathematics Pi","volume":" ","pages":""},"PeriodicalIF":2.3,"publicationDate":"2017-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1017/fmp.2018.3","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44700996","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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