We study fluctuations of the current at the boundary for the half-space asymmetric simple exclusion process (ASEP) and the height function of the half-space six-vertex model at the boundary at large times. We establish a phase transition depending on the effective density of particles at the boundary, with Gaussian symplectic ensemble (GSE) and Gaussian orthogonal ensemble (GOE) limits as well as the Baik–Rains crossover distribution near the critical point. This was previously known for half-space last-passage percolation, and recently established for the half-space log-gamma polymer and Kardar–Parisi–Zhang equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half-space six-vertex model and a half-space Hall–Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half-space ASEP and six-vertex model, indicating a hidden free fermionic structure.
{"title":"Boundary current fluctuations for the half-space ASEP and six-vertex model","authors":"Jimmy He","doi":"10.1112/plms.12585","DOIUrl":"https://doi.org/10.1112/plms.12585","url":null,"abstract":"We study fluctuations of the current at the boundary for the half-space asymmetric simple exclusion process (ASEP) and the height function of the half-space six-vertex model at the boundary at large times. We establish a phase transition depending on the effective density of particles at the boundary, with Gaussian symplectic ensemble (GSE) and Gaussian orthogonal ensemble (GOE) limits as well as the Baik–Rains crossover distribution near the critical point. This was previously known for half-space last-passage percolation, and recently established for the half-space log-gamma polymer and Kardar–Parisi–Zhang equation in the groundbreaking work of Imamura, Mucciconi, and Sasamoto. The proof uses the underlying algebraic structure of these models in a crucial way to obtain exact formulas. In particular, we show a relationship between the half-space six-vertex model and a half-space Hall–Littlewood measure with two boundary parameters, which is then matched to a free boundary Schur process via a new identity of symmetric functions. Fredholm Pfaffian formulas are established for the half-space ASEP and six-vertex model, indicating a hidden free fermionic structure.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"27 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139920108","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 locally flatly embedded 2-sphere in a compact 4-manifold <mjx-container aria-label="upper X" ctxtmenu_counter="0" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true"><mjx-semantics><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic- data-semantic-role="latinletter" data-semantic-speech="upper X" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden="true" display="inline" unselectable="on"><math altimg="/cms/asset/0849428e-8bdc-41aa-8881-92e8cadc8a45/plms12583-math-0001.png" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mi data-semantic-="" data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic-role="latinletter" data-semantic-speech="upper X" data-semantic-type="identifier">X</mi>$X$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is called a spine if the inclusion map is a homotopy equivalence. A spine is called simple if the complement of the 2-sphere has abelian fundamental group. We prove that if two simple spines represent the same generator of <mjx-container aria-label="upper H 2 left parenthesis upper X right parenthesis" ctxtmenu_counter="1" ctxtmenu_oldtabindex="1" jax="CHTML" role="application" sre-explorer- style="font-size: 103%; position: relative;" tabindex="0"><mjx-math aria-hidden="true"><mjx-semantics><mjx-mrow data-semantic-annotation="clearspeak:simple" data-semantic-children="2,6" data-semantic-content="7,0" data-semantic- data-semantic-role="simple function" data-semantic-speech="upper H 2 left parenthesis upper X right parenthesis" data-semantic-type="appl"><mjx-msub data-semantic-children="0,1" data-semantic- data-semantic-parent="8" data-semantic-role="simple function" data-semantic-type="subscript"><mjx-mi data-semantic-annotation="clearspeak:simple" data-semantic-font="italic" data-semantic- data-semantic-operator="appl" data-semantic-parent="2" data-semantic-role="simple function" data-semantic-type="identifier"><mjx-c></mjx-c></mjx-mi><mjx-script style="vertical-align: -0.15em; margin-left: -0.057em;"><mjx-mn data-semantic-annotation="clearspeak:simple" data-semantic-font="normal" data-semantic- data-semantic-parent="2" data-semantic-role="integer" data-semantic-type="number" size="s"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added="true" data-semantic- data-semantic-operator="appl" data-semantic-parent="8" data-semantic-role="application" data-semantic-type="punctuation" style="margin-left: 0.056em; margin-right: 0.056em;"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children="4" data-semantic-content="3,5" data-semantic- data-semantic-parent="8" data-semantic-role="leftright" data-semantic-type="fenced"><mjx-mo data-semantic- data-semantic-operator="fenced" data-semantic-parent="6" data-semantic-role="open" data-semant
{"title":"Simple spines of homotopy 2-spheres are unique","authors":"Patrick Orson, Mark Powell","doi":"10.1112/plms.12583","DOIUrl":"https://doi.org/10.1112/plms.12583","url":null,"abstract":"A locally flatly embedded 2-sphere in a compact 4-manifold <mjx-container aria-label=\"upper X\" ctxtmenu_counter=\"0\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi></mjx-semantics></mjx-math><mjx-assistive-mml aria-hidden=\"true\" display=\"inline\" unselectable=\"on\"><math altimg=\"/cms/asset/0849428e-8bdc-41aa-8881-92e8cadc8a45/plms12583-math-0001.png\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><semantics><mi data-semantic-=\"\" data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic-role=\"latinletter\" data-semantic-speech=\"upper X\" data-semantic-type=\"identifier\">X</mi>$X$</annotation></semantics></math></mjx-assistive-mml></mjx-container> is called a spine if the inclusion map is a homotopy equivalence. A spine is called simple if the complement of the 2-sphere has abelian fundamental group. We prove that if two simple spines represent the same generator of <mjx-container aria-label=\"upper H 2 left parenthesis upper X right parenthesis\" ctxtmenu_counter=\"1\" ctxtmenu_oldtabindex=\"1\" jax=\"CHTML\" role=\"application\" sre-explorer- style=\"font-size: 103%; position: relative;\" tabindex=\"0\"><mjx-math aria-hidden=\"true\"><mjx-semantics><mjx-mrow data-semantic-annotation=\"clearspeak:simple\" data-semantic-children=\"2,6\" data-semantic-content=\"7,0\" data-semantic- data-semantic-role=\"simple function\" data-semantic-speech=\"upper H 2 left parenthesis upper X right parenthesis\" data-semantic-type=\"appl\"><mjx-msub data-semantic-children=\"0,1\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"simple function\" data-semantic-type=\"subscript\"><mjx-mi data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"italic\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"2\" data-semantic-role=\"simple function\" data-semantic-type=\"identifier\"><mjx-c></mjx-c></mjx-mi><mjx-script style=\"vertical-align: -0.15em; margin-left: -0.057em;\"><mjx-mn data-semantic-annotation=\"clearspeak:simple\" data-semantic-font=\"normal\" data-semantic- data-semantic-parent=\"2\" data-semantic-role=\"integer\" data-semantic-type=\"number\" size=\"s\"><mjx-c></mjx-c></mjx-mn></mjx-script></mjx-msub><mjx-mo data-semantic-added=\"true\" data-semantic- data-semantic-operator=\"appl\" data-semantic-parent=\"8\" data-semantic-role=\"application\" data-semantic-type=\"punctuation\" style=\"margin-left: 0.056em; margin-right: 0.056em;\"><mjx-c></mjx-c></mjx-mo><mjx-mrow data-semantic-children=\"4\" data-semantic-content=\"3,5\" data-semantic- data-semantic-parent=\"8\" data-semantic-role=\"leftright\" data-semantic-type=\"fenced\"><mjx-mo data-semantic- data-semantic-operator=\"fenced\" data-semantic-parent=\"6\" data-semantic-role=\"open\" data-semant","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"89 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139759190","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 this paper, we prove quantitative results about geodesic approximations to submanifolds in negatively curved spaces. Among the main tools is a new and general Jarník–Besicovitch type theorem in Diophantine approximation. Our framework allows manifolds of variable negative curvature, a variety of geometric targets, and logarithm laws as well as spiraling phenomena in both measure and dimension aspect. Several of the results are new also for manifolds of constant negative sectional curvature. We further establish a large intersection property of Falconer in this context.
{"title":"Diophantine approximation, large intersections and geodesics in negative curvature","authors":"Anish Ghosh, Debanjan Nandi","doi":"10.1112/plms.12581","DOIUrl":"https://doi.org/10.1112/plms.12581","url":null,"abstract":"In this paper, we prove quantitative results about geodesic approximations to submanifolds in negatively curved spaces. Among the main tools is a new and general Jarník–Besicovitch type theorem in Diophantine approximation. Our framework allows manifolds of variable negative curvature, a variety of geometric targets, and logarithm laws as well as spiraling phenomena in both measure and dimension aspect. Several of the results are new also for manifolds of constant negative sectional curvature. We further establish a large intersection property of Falconer in this context.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"1 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139759187","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 apply the machinery of Bridgeland stability conditions on derived categories of coherent sheaves to describe the geometry of classical moduli spaces associated with canonical genus 4 space curves via an effective control over its wall-crossing. This article provides the first description of a moduli space of Pandharipande–Thomas stable pairs that is used as an intermediate step toward the description of the associated Hilbert scheme, which in turn is the first example where the components of a classical moduli space were completely determined via wall-crossing. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme. There are several long standing open problems regarding classical sheaf theoretic moduli spaces, and the present work will shed light on further studies of such moduli spaces such as Hilbert schemes of curves and moduli of stable pairs that are very hard to tackle without the wall-crossing techniques.
{"title":"Geometry of canonical genus 4 curves","authors":"Fatemeh Rezaee","doi":"10.1112/plms.12577","DOIUrl":"https://doi.org/10.1112/plms.12577","url":null,"abstract":"We apply the machinery of Bridgeland stability conditions on derived categories of coherent sheaves to describe the geometry of classical moduli spaces associated with canonical genus 4 space curves via an effective control over its wall-crossing. This article provides the first description of a moduli space of Pandharipande–Thomas stable pairs that is used as an intermediate step toward the description of the associated Hilbert scheme, which in turn is the first example where the components of a classical moduli space were completely determined via wall-crossing. We give a full list of irreducible components of the space of stable pairs, along with a birational description of each component, and a partial list for the Hilbert scheme. There are several long standing open problems regarding classical sheaf theoretic moduli spaces, and the present work will shed light on further studies of such moduli spaces such as Hilbert schemes of curves and moduli of stable pairs that are very hard to tackle without the wall-crossing techniques.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"13 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2024-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139474829","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 this paper, we study the non-Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky (Selecta Math. (N.S.) 4 (1998) 279–320).
{"title":"On the existence of harmonic metrics on non-Hermitian Yang–Mills bundles","authors":"Changpeng Pan, Zhenghan Shen, Xi Zhang","doi":"10.1112/plms.12580","DOIUrl":"https://doi.org/10.1112/plms.12580","url":null,"abstract":"In this paper, we study the non-Hermitian Yang–Mills (NHYM) bundles over compact Kähler manifolds. We show that the existence of harmonic metrics is equivalent to the semisimplicity of NHYM bundles, which confirms the Conjecture 8.7 in Kaledin and Verbitsky (<i>Selecta Math</i>. (N.S.) 4 (1998) 279–320).","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"80 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139035722","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 twisted commutative algebra is (for us) a commutative -algebra equipped with an action of the infinite general linear group. In such algebras, the “-prime” ideals assume the duties fulfilled by prime ideals in ordinary commutative algebra, and so it is crucial to understand them. Unfortunately, distinct -primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish -primes. This yields an effective method for analyzing -primes.
{"title":"The spectrum of a twisted commutative algebra","authors":"Andrew Snowden","doi":"10.1112/plms.12576","DOIUrl":"https://doi.org/10.1112/plms.12576","url":null,"abstract":"A twisted commutative algebra is (for us) a commutative <math altimg=\"urn:x-wiley:00246115:media:plms12576:plms12576-math-0001\" display=\"inline\" location=\"graphic/plms12576-math-0001.png\">\u0000<semantics>\u0000<mi mathvariant=\"bold\">Q</mi>\u0000$mathbf {Q}$</annotation>\u0000</semantics></math>-algebra equipped with an action of the infinite general linear group. In such algebras, the “<math altimg=\"urn:x-wiley:00246115:media:plms12576:plms12576-math-0002\" display=\"inline\" location=\"graphic/plms12576-math-0002.png\">\u0000<semantics>\u0000<mi mathvariant=\"bold\">GL</mi>\u0000$mathbf {GL}$</annotation>\u0000</semantics></math>-prime” ideals assume the duties fulfilled by prime ideals in ordinary commutative algebra, and so it is crucial to understand them. Unfortunately, distinct <math altimg=\"urn:x-wiley:00246115:media:plms12576:plms12576-math-0003\" display=\"inline\" location=\"graphic/plms12576-math-0003.png\">\u0000<semantics>\u0000<mi mathvariant=\"bold\">GL</mi>\u0000$mathbf {GL}$</annotation>\u0000</semantics></math>-primes can have the same radical, which obstructs one from studying them geometrically. We show that this problem can be eliminated by working with super vector spaces: doing so provides enough geometry to distinguish <math altimg=\"urn:x-wiley:00246115:media:plms12576:plms12576-math-0004\" display=\"inline\" location=\"graphic/plms12576-math-0004.png\">\u0000<semantics>\u0000<mi mathvariant=\"bold\">GL</mi>\u0000$mathbf {GL}$</annotation>\u0000</semantics></math>-primes. This yields an effective method for analyzing <math altimg=\"urn:x-wiley:00246115:media:plms12576:plms12576-math-0005\" display=\"inline\" location=\"graphic/plms12576-math-0005.png\">\u0000<semantics>\u0000<mi mathvariant=\"bold\">GL</mi>\u0000$mathbf {GL}$</annotation>\u0000</semantics></math>-primes.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"33 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139035805","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 a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super-polynomial rate.
{"title":"Rates of mixing for the measure of maximal entropy of dispersing billiard maps","authors":"Mark F. Demers, Alexey Korepanov","doi":"10.1112/plms.12578","DOIUrl":"https://doi.org/10.1112/plms.12578","url":null,"abstract":"In a recent work, Baladi and Demers constructed a measure of maximal entropy for finite horizon dispersing billiard maps and proved that it is unique, mixing and moreover Bernoulli. We show that this measure enjoys natural probabilistic properties for Hölder continuous observables, such as at least polynomial decay of correlations and the Central Limit Theorem. The results of Baladi and Demers are subject to a condition of sparse recurrence to singularities. We use a similar and slightly stronger condition, and it has a direct effect on our rate of decay of correlations. For billiard tables with bounded complexity (a property conjectured to be generic), we show that the sparse recurrence condition is always satisfied and the correlations decay at a super-polynomial rate.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"40 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826593","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 this paper, we study rational Collet–Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, these maps are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree .
{"title":"Slowly recurrent Collet–Eckmann maps with non-empty Fatou set","authors":"Magnus Aspenberg, Mats Bylund, Weiwei Cui","doi":"10.1112/plms.12574","DOIUrl":"https://doi.org/10.1112/plms.12574","url":null,"abstract":"In this paper, we study rational Collet–Eckmann maps for which the Julia set is not the whole sphere and for which the critical points are recurrent at a slow rate. In families where the orders of the critical points are fixed, we prove that such maps are Lebesgue density points of hyperbolic maps. In particular, if all critical points are simple, these maps are Lebesgue density points of hyperbolic maps in the full space of rational maps of any degree <math altimg=\"urn:x-wiley:00246115:media:plms12574:plms12574-math-0001\" display=\"inline\" location=\"graphic/plms12574-math-0001.png\">\u0000<semantics>\u0000<mrow>\u0000<mi>d</mi>\u0000<mo>⩾</mo>\u0000<mn>2</mn>\u0000</mrow>\u0000$d geqslant 2$</annotation>\u0000</semantics></math>.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"53 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2023-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138826651","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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