We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of several models of random constrained permutations. This family is parameterized by two real parameters. For a specific choice of the parameters, the skew Brownian permuton coincides with the Baxter permuton, that is, the permuton limit of Baxter permutations. We prove that for another specific choice of the parameters, the skew Brownian permuton coincides with the biased Brownian separable permuton, a one‐parameter family of permutons previously studied in the literature as the limit of uniform permutations in substitution‐closed classes. This brings two different limiting objects under the same roof, identifying a new larger universality class. The skew Brownian permuton is constructed in terms of flows of solutions of certain stochastic differential equations (SDEs) driven by two‐dimensional correlated Brownian excursions in the nonnegative quadrant. We call these SDEs skew perturbed Tanaka equations because they are a mixture of the perturbed Tanaka equations and the equations encoding skew Brownian motions. We prove existence and uniqueness of (strong) solutions for these new SDEs. In addition, we show that some natural permutons arising from Liouville quantum gravity (LQG) spheres decorated with two Schramm–Loewner evolution (SLE) curves are skew Brownian permutons and such permutons cover almost the whole range of possible parameters. Some connections between constrained permutations and decorated planar maps have been investigated in the literature at the discrete level; this paper establishes this connection directly at the continuum level. Proving the latter result, we also give an SDE interpretation of some quantities related to SLE‐decorated LQG spheres.
{"title":"The skew Brownian permuton: A new universality class for random constrained permutations","authors":"J. Borga","doi":"10.1112/plms.12519","DOIUrl":"https://doi.org/10.1112/plms.12519","url":null,"abstract":"We construct a new family of random permutons, called skew Brownian permuton, which describes the limits of several models of random constrained permutations. This family is parameterized by two real parameters. For a specific choice of the parameters, the skew Brownian permuton coincides with the Baxter permuton, that is, the permuton limit of Baxter permutations. We prove that for another specific choice of the parameters, the skew Brownian permuton coincides with the biased Brownian separable permuton, a one‐parameter family of permutons previously studied in the literature as the limit of uniform permutations in substitution‐closed classes. This brings two different limiting objects under the same roof, identifying a new larger universality class. The skew Brownian permuton is constructed in terms of flows of solutions of certain stochastic differential equations (SDEs) driven by two‐dimensional correlated Brownian excursions in the nonnegative quadrant. We call these SDEs skew perturbed Tanaka equations because they are a mixture of the perturbed Tanaka equations and the equations encoding skew Brownian motions. We prove existence and uniqueness of (strong) solutions for these new SDEs. In addition, we show that some natural permutons arising from Liouville quantum gravity (LQG) spheres decorated with two Schramm–Loewner evolution (SLE) curves are skew Brownian permutons and such permutons cover almost the whole range of possible parameters. Some connections between constrained permutations and decorated planar maps have been investigated in the literature at the discrete level; this paper establishes this connection directly at the continuum level. Proving the latter result, we also give an SDE interpretation of some quantities related to SLE‐decorated LQG spheres.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41688555","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 class of ℓ$ell$ ‐adic local systems on A1$mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y′′(z)=zy(z)$y^{prime prime }(z)=zy(z)$ . We employ the geometric Langlands correspondence to construct the sought‐after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GLn$mathrm{GL}_n$ , we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞$infty$ . These conjectures, in particular, imply cohomological rigidity of Airy sheaves.
{"title":"Airy sheaves for reductive groups","authors":"Konstantin Jakob, Masoud Kamgarpour, Lingfei Yi","doi":"10.1112/plms.12494","DOIUrl":"https://doi.org/10.1112/plms.12494","url":null,"abstract":"We construct a class of ℓ$ell$ ‐adic local systems on A1$mathbb {A}^1$ that generalizes the Airy sheaves defined by N. Katz to reductive groups. These sheaves are finite field analogues of generalizations of the classical Airy equation y′′(z)=zy(z)$y^{prime prime }(z)=zy(z)$ . We employ the geometric Langlands correspondence to construct the sought‐after local systems as eigenvalues of certain rigid Hecke eigensheaves, following the methods developed by Heinloth, Ngô, and Yun. The construction is motivated by a special case of Adler and Yu's construction of tame supercuspidal representations. The representations that we consider can be viewed as deeper analogues of simple supercuspidals. For GLn$mathrm{GL}_n$ , we compute the Frobenius trace of the local systems in question and show that they agree with Katz's Airy sheaves. We make precise conjectures about the ramification behavior of the local systems at ∞$infty$ . These conjectures, in particular, imply cohomological rigidity of Airy sheaves.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48415682","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12360","DOIUrl":"https://doi.org/10.1112/plms.12360","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43199106","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 L$L$ be a second‐order elliptic operator with analytic coefficients defined in B1⊆Rn$B_1subseteq mathbb {R}^n$ . We construct explicitly and canonically a fundamental solution for the operator, that is, a function u:Br0→R$u:B_{r_0}rightarrow mathbb {R}$ such that Lu=δ0$Lu=delta _0$ . As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of |x|$vert {x}vert$ , plus homogeneous polynomials multiplied by log(|x|)$log (vert {x}vert )$ if the dimension n$n$ is even) which improves the classical result of [6]. The control we have on the complexity of each homogeneous term is optimal and in particular, when L$L$ is the Laplace–Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to Kodaira [8]. The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by Z2$mathbb {Z}^2$ .
{"title":"Expansion of the fundamental solution of a second‐order elliptic operator with analytic coefficients","authors":"Federico Franceschini, Federico Glaudo","doi":"10.1112/plms.12556","DOIUrl":"https://doi.org/10.1112/plms.12556","url":null,"abstract":"Let L$L$ be a second‐order elliptic operator with analytic coefficients defined in B1⊆Rn$B_1subseteq mathbb {R}^n$ . We construct explicitly and canonically a fundamental solution for the operator, that is, a function u:Br0→R$u:B_{r_0}rightarrow mathbb {R}$ such that Lu=δ0$Lu=delta _0$ . As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of |x|$vert {x}vert$ , plus homogeneous polynomials multiplied by log(|x|)$log (vert {x}vert )$ if the dimension n$n$ is even) which improves the classical result of [6]. The control we have on the complexity of each homogeneous term is optimal and in particular, when L$L$ is the Laplace–Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to Kodaira [8]. The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by Z2$mathbb {Z}^2$ .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48102219","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 study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.
{"title":"On sufficient conditions for spanning structures in dense graphs","authors":"R. Lang, Nicolás Sanhueza-Matamala","doi":"10.1112/plms.12552","DOIUrl":"https://doi.org/10.1112/plms.12552","url":null,"abstract":"We study structural conditions in dense graphs that guarantee the existence of vertex‐spanning substructures such as Hamilton cycles. It is easy to see that every Hamiltonian graph is connected, has a perfect fractional matching and, excluding the bipartite case, contains an odd cycle. A simple consequence of the Robust Expander Theorem of Kühn, Osthus and Treglown tells us that any large enough graph that robustly satisfies these properties must already be Hamiltonian. Our main result generalises this phenomenon to powers of cycles and graphs of sublinear bandwidth subject to natural generalisations of connectivity, matchings and odd cycles. This answers a question of Ebsen, Maesaka, Reiher, Schacht and Schülke and solves the embedding problem that underlies multiple lines of research on sufficient conditions for spanning structures in dense graphs. As applications, we recover and establish Bandwidth Theorems in a variety of settings including Ore‐type degree conditions, Pósa‐type degree conditions, deficiency‐type conditions, locally dense and inseparable graphs, multipartite graphs as well as robust expanders.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49425832","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}
Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group VA[Γ]${rm VA}[Gamma ]$ of a Coxeter graph Γ$Gamma$ mixing the standard presentation of the Artin group A[Γ]$A[Gamma ]$ with the standard presentation of the Coxeter group W[Γ]$W[Gamma ]$ and some mixed relations that mimic the action of W[Γ]$W[Gamma ]$ on its root system. By definition, we have two epimorphisms πK:VA[Γ]→W[Γ]$pi _K:{rm VA}[Gamma ]rightarrow W[Gamma ]$ and πP:VA[Γ]→W[Γ]$pi _P:{rm VA}[Gamma ]rightarrow W[Gamma ]$ whose kernels are denoted by KVA[Γ]${rm KVA}[Gamma ]$ and PVA[Γ]${rm PVA}[Gamma ]$ , respectively. We calculate presentations for these two subgroups. In particular, KVA[Γ]${rm KVA}[Gamma ]$ is an Artin group. We prove that the center of any virtual Artin group is trivial. In the case where Γ$Gamma$ is of spherical type or of affine type, we show that each free of infinity parabolic subgroup of KVA[Γ]${rm KVA}[Gamma ]$ is also of spherical type or of affine type, and we show that VA[Γ]${rm VA}[Gamma ]$ has a solution to the word problem. In the case where Γ$Gamma$ is of spherical type we show that KVA[Γ]${rm KVA}[Gamma ]$ satisfies the K(π,1)$K(pi ,1)$ conjecture and we infer the cohomological dimension of KVA[Γ]${rm KVA}[Gamma ]$ and the virtual cohomological dimension of VA[Γ]${rm VA}[Gamma ]$ . In the case where Γ$Gamma$ is of affine type we determine upper bounds for the cohomological dimension of KVA[Γ]${rm KVA}[Gamma ]$ and for the virtual cohomological dimension of VA[Γ]${rm VA}[Gamma ]$ .
{"title":"Virtual Artin groups","authors":"P. Bellingeri, L. Paris, A. Thiel","doi":"10.1112/plms.12491","DOIUrl":"https://doi.org/10.1112/plms.12491","url":null,"abstract":"Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group VA[Γ]${rm VA}[Gamma ]$ of a Coxeter graph Γ$Gamma$ mixing the standard presentation of the Artin group A[Γ]$A[Gamma ]$ with the standard presentation of the Coxeter group W[Γ]$W[Gamma ]$ and some mixed relations that mimic the action of W[Γ]$W[Gamma ]$ on its root system. By definition, we have two epimorphisms πK:VA[Γ]→W[Γ]$pi _K:{rm VA}[Gamma ]rightarrow W[Gamma ]$ and πP:VA[Γ]→W[Γ]$pi _P:{rm VA}[Gamma ]rightarrow W[Gamma ]$ whose kernels are denoted by KVA[Γ]${rm KVA}[Gamma ]$ and PVA[Γ]${rm PVA}[Gamma ]$ , respectively. We calculate presentations for these two subgroups. In particular, KVA[Γ]${rm KVA}[Gamma ]$ is an Artin group. We prove that the center of any virtual Artin group is trivial. In the case where Γ$Gamma$ is of spherical type or of affine type, we show that each free of infinity parabolic subgroup of KVA[Γ]${rm KVA}[Gamma ]$ is also of spherical type or of affine type, and we show that VA[Γ]${rm VA}[Gamma ]$ has a solution to the word problem. In the case where Γ$Gamma$ is of spherical type we show that KVA[Γ]${rm KVA}[Gamma ]$ satisfies the K(π,1)$K(pi ,1)$ conjecture and we infer the cohomological dimension of KVA[Γ]${rm KVA}[Gamma ]$ and the virtual cohomological dimension of VA[Γ]${rm VA}[Gamma ]$ . In the case where Γ$Gamma$ is of affine type we determine upper bounds for the cohomological dimension of KVA[Γ]${rm KVA}[Gamma ]$ and for the virtual cohomological dimension of VA[Γ]${rm VA}[Gamma ]$ .","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44872128","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 consider the full three‐dimensional Ginzburg–Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the ‘first critical field’ Hc1$H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter ε$varepsilon$ . This onset of vorticity is directly related to an ‘isoflux problem’ on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [22] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below Hc1+Clog|logε|${H_{c_1}}+ C log {|log varepsilon |}$ , the total vorticity remains bounded independently of ε$varepsilon$ , with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three‐dimensional setting a two‐dimensional result of [28]. We finish by showing an improved estimate on the value of Hc1${H_{c_1}}$ in some specific simple geometries.
{"title":"Bounded vorticity for the 3D Ginzburg–Landau model and an isoflux problem","authors":"Carlos Rom'an, E. Sandier, S. Serfaty","doi":"10.1112/plms.12505","DOIUrl":"https://doi.org/10.1112/plms.12505","url":null,"abstract":"We consider the full three‐dimensional Ginzburg–Landau model of superconductivity with applied magnetic field, in the regime where the intensity of the applied field is close to the ‘first critical field’ Hc1$H_{c_1}$ at which vortex filaments appear, and in the asymptotics of a small inverse Ginzburg–Landau parameter ε$varepsilon$ . This onset of vorticity is directly related to an ‘isoflux problem’ on curves (finding a curve that maximizes the ratio of a magnetic flux by its length), whose study was initiated in [22] and which we continue here. By assuming a nondegeneracy condition for this isoflux problem, which we show holds at least for instance in the case of a ball, we prove that if the intensity of the applied field remains below Hc1+Clog|logε|${H_{c_1}}+ C log {|log varepsilon |}$ , the total vorticity remains bounded independently of ε$varepsilon$ , with vortex lines concentrating near the maximizer of the isoflux problem, thus extending to the three‐dimensional setting a two‐dimensional result of [28]. We finish by showing an improved estimate on the value of Hc1${H_{c_1}}$ in some specific simple geometries.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"126 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41478879","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 prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov–Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.
{"title":"Collapsed ancient solutions of the Ricci flow on compact homogeneous spaces","authors":"Francesco Pediconi, Sammy Sbiti","doi":"10.1112/plms.12478","DOIUrl":"https://doi.org/10.1112/plms.12478","url":null,"abstract":"We prove a general existence theorem for collapsed ancient solutions to the Ricci flow on compact homogeneous spaces and we show that they converge in the Gromov–Hausdorff topology, under a suitable rescaling, to an Einstein metric on the base of a torus fibration. This construction generalizes all previous known examples in the literature.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43813158","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 show that for each cM∈[1,25)${mathbf {c}}_{mathrm{M}} in [1,25)$ , there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge cM${mathbf {c}}_{mathrm{M}}$ . An earlier series of works by Ding–Dubédat–Dunlap–Falconet, Gwynne–Miller, and others showed that such a metric exists and is unique in the subcritical case cM∈(−∞,1)${mathbf {c}}_{mathrm{M}} in (-infty ,1)$ , which corresponds to coupling constant γ∈(0,2)$gamma in (0,2)$ . The critical case cM=1${mathbf {c}}_{mathrm{M}} = 1$ corresponds to γ=2$gamma =2$ and the supercritical case cM∈(1,25)${mathbf {c}}_{mathrm{M}} in (1,25)$ corresponds to γ∈C$gamma in mathbb {C}$ with |γ|=2$|gamma | = 2$ . Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for cM∈[1,25)$mathbf {c}_{mathrm{M}} in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for cM∈(−∞,1)$mathbf {c}_{mathrm{M}} in (-infty ,1)$ to the full parameter range cM∈(−∞,25)$mathbf {c}_{mathrm{M}} in (-infty ,25)$ . Our argument is substantially different from the proof of the characterization of the LQG metric for cM∈(−∞,1)$mathbf {c}_{mathrm{M}} in (-infty ,1)$ . In particular, the core part of the argument is simpler and does not use confluence of geodesics.
我们证明，对于每个cM∈[1,25)${mathbf {c}}_{mathrm{M}} in [1,25)$，存在一个与具有物质中心电荷cM ${mathbf {c}}_{mathrm{M}}$的刘维尔量子引力(LQG)相关的唯一度量。ding - dubsamdat - dunlap - falconet, Gwynne-Miller等人早期的一系列研究表明，在次临界情况cM∈(−∞，1)${mathbf {c}}_{mathrm{M}} in (-infty ,1)$下，这种度量存在并且是唯一的，它对应于耦合常数γ∈(0,2)$gamma in (0,2)$。临界情况cM=1 ${mathbf {c}}_{mathrm{M}} = 1$对应γ=2 $gamma =2$，超临界情况cM∈(1,25)${mathbf {c}}_{mathrm{M}} in (1,25)$对应γ∈C $gamma in mathbb {C}$，其中|γ|=2 $|gamma | = 2$。我们的度量被构建为称为Liouville第一通道渗透的近似过程的极限，Ding和Gwynne(2020)先前证明cM∈[1,25)$mathbf {c}_{mathrm{M}} in [1,25)$是紧的。在本文中，我们证明了子序列极限是由一个自然公理表唯一表征的。这将Gwynne和Miller(2019)对cM∈(−∞，1)$mathbf {c}_{mathrm{M}} in (-infty ,1)$证明的LQG度量的表征扩展到cM∈(−∞，25)$mathbf {c}_{mathrm{M}} in (-infty ,25)$的全参数范围。我们的论证与cM∈(−∞，1)$mathbf {c}_{mathrm{M}} in (-infty ,1)$的LQG度量的表征的证明有本质的不同。特别是，论证的核心部分更简单，并且没有使用测地线的合流。
{"title":"Uniqueness of the critical and supercritical Liouville quantum gravity metrics","authors":"Jian Ding, Ewain Gwynne","doi":"10.1112/plms.12492","DOIUrl":"https://doi.org/10.1112/plms.12492","url":null,"abstract":"We show that for each cM∈[1,25)${mathbf {c}}_{mathrm{M}} in [1,25)$ , there is a unique metric associated with Liouville quantum gravity (LQG) with matter central charge cM${mathbf {c}}_{mathrm{M}}$ . An earlier series of works by Ding–Dubédat–Dunlap–Falconet, Gwynne–Miller, and others showed that such a metric exists and is unique in the subcritical case cM∈(−∞,1)${mathbf {c}}_{mathrm{M}} in (-infty ,1)$ , which corresponds to coupling constant γ∈(0,2)$gamma in (0,2)$ . The critical case cM=1${mathbf {c}}_{mathrm{M}} = 1$ corresponds to γ=2$gamma =2$ and the supercritical case cM∈(1,25)${mathbf {c}}_{mathrm{M}} in (1,25)$ corresponds to γ∈C$gamma in mathbb {C}$ with |γ|=2$|gamma | = 2$ . Our metric is constructed as the limit of an approximation procedure called Liouville first passage percolation, which was previously shown to be tight for cM∈[1,25)$mathbf {c}_{mathrm{M}} in [1,25)$ by Ding and Gwynne (2020). In this paper, we show that the subsequential limit is uniquely characterized by a natural list of axioms. This extends the characterization of the LQG metric proven by Gwynne and Miller (2019) for cM∈(−∞,1)$mathbf {c}_{mathrm{M}} in (-infty ,1)$ to the full parameter range cM∈(−∞,25)$mathbf {c}_{mathrm{M}} in (-infty ,25)$ . Our argument is substantially different from the proof of the characterization of the LQG metric for cM∈(−∞,1)$mathbf {c}_{mathrm{M}} in (-infty ,1)$ . In particular, the core part of the argument is simpler and does not use confluence of geodesics.","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":" ","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44928588","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}
{"title":"Issue Information","authors":"","doi":"10.1112/plms.12359","DOIUrl":"https://doi.org/10.1112/plms.12359","url":null,"abstract":"","PeriodicalId":49667,"journal":{"name":"Proceedings of the London Mathematical Society","volume":"123 1","pages":""},"PeriodicalIF":1.8,"publicationDate":"2021-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41957421","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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