For a given dimension d $ge$ 2 and a finite measure $nu$ on (0, +$infty$), we consider $xi$ a Poisson point process on R d x (0, +$infty$) with intensity measure dc $otimes$ $nu$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Sigma$ = $cup$ (c,r)$in$$xi$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Sigma$ and at infinite speed inside $Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $mu$ x when x goes to infinity, where $mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $mu$ as a function of the measure $nu$ associated with the underlying Boolean model.
{"title":"Continuity of the time constant in a continuous model of first passage percolation","authors":"J.-B. Gouéré, Marie Théret","doi":"10.1214/21-aihp1222","DOIUrl":"https://doi.org/10.1214/21-aihp1222","url":null,"abstract":"For a given dimension d $ge$ 2 and a finite measure $nu$ on (0, +$infty$), we consider $xi$ a Poisson point process on R d x (0, +$infty$) with intensity measure dc $otimes$ $nu$ where dc denotes the Lebesgue measure on R d. We consider the Boolean model $Sigma$ = $cup$ (c,r)$in$$xi$ B(c, r) where B(c, r) denotes the open ball centered at c with radius r. For every x, y $in$ R d we define T (x, y) as the minimum time needed to travel from x to y by a traveler that walks at speed 1 outside $Sigma$ and at infinite speed inside $Sigma$. By a standard application of Kingman sub-additive theorem, one easily shows that T (0, x) behaves like $mu$ x when x goes to infinity, where $mu$ is a constant named the time constant in classical first passage percolation. In this paper we investigate the regularity of $mu$ as a function of the measure $nu$ associated with the underlying Boolean model.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"59 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84307803","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $mu$ and $nu$ are supercritical offspring distributions with means $bar mu < bar nu$ then every space-time set that is recurrent with respect to the offspring distribution $mu$ is also recurrent with respect to the offspring distribution $nu$ and similarly that every space-time set that is transient with respect to the offspring distribution $nu$ is also transient with respect to the offspring distribution $mu$. To prove this, we introduce a new order on probability measures that we call the germ order and prove more generally that the same result holds whenever $mu$ is smaller than $nu$ in the germ order. Our work is inspired by the work of Johnson and Junge (AIHP 2018), who used related stochastic orders to study the frog model.
研究了图上分支随机漫步的时空集的递归性和暂态性与子代分布的关系。在这里，我们说一个时空集$A$是循环的，如果它在分支随机漫步永远存在的情况下几乎可以肯定地被无限次访问，如果它几乎可以肯定地被无限次访问，那么$A$是短暂的。我们证明，如果$mu$和$nu$是均值为$bar mu < bar nu$的超临界后代分布，那么每一个关于后代分布循环的时空集$mu$对于后代分布也是循环的$nu$同样，每一个关于后代分布瞬态的时空集$nu$对于后代分布也是瞬态的$mu$。为了证明这一点，我们在概率测度上引入了一个新的阶数，我们称之为胚芽阶数，并更一般地证明，当胚芽阶中$mu$小于$nu$时，同样的结果成立。我们的工作灵感来自Johnson和Junge (AIHP 2018)的工作，他们使用相关的随机顺序来研究青蛙模型。
{"title":"Transience and recurrence of sets for branching random walk via non-standard stochastic orders","authors":"Tom Hutchcroft","doi":"10.1214/21-aihp1186","DOIUrl":"https://doi.org/10.1214/21-aihp1186","url":null,"abstract":"We study how the recurrence and transience of space-time sets for a branching random walk on a graph depends on the offspring distribution. Here, we say that a space-time set $A$ is recurrent if it is visited infinitely often almost surely on the event that the branching random walk survives forever, and say that $A$ is transient if it is visited at most finitely often almost surely. We prove that if $mu$ and $nu$ are supercritical offspring distributions with means $bar mu < bar nu$ then every space-time set that is recurrent with respect to the offspring distribution $mu$ is also recurrent with respect to the offspring distribution $nu$ and similarly that every space-time set that is transient with respect to the offspring distribution $nu$ is also transient with respect to the offspring distribution $mu$. To prove this, we introduce a new order on probability measures that we call the germ order and prove more generally that the same result holds whenever $mu$ is smaller than $nu$ in the germ order. Our work is inspired by the work of Johnson and Junge (AIHP 2018), who used related stochastic orders to study the frog model.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"30 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73351567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct a two-parameter family of diffusions on the set of open subsets of $(0,1)$ that arise as diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains.
我们在$(0,1)$的开子集集合上构造了作为双参数有序中餐馆过程上下链的扩散极限的双参数扩散族。
{"title":"Diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains","authors":"Kelvin Rivera-Lopez, Douglas Rizzolo","doi":"10.1214/22-aihp1256","DOIUrl":"https://doi.org/10.1214/22-aihp1256","url":null,"abstract":"We construct a two-parameter family of diffusions on the set of open subsets of $(0,1)$ that arise as diffusive limits of two-parameter ordered Chinese Restaurant Process up-down chains.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88822866","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a one dimension Kac model with conservation of energy and an exclusion rule: Fix a number of particles $n$, and an energy $E>0$. Let each of the particles have an energy $x_j geq 0$, with $sum_{j=1}^n x_j = E$. For some $epsilon$, the allowed configurations $(x_1,dots,x_n)$ are those that satisfy $|x_i - x_j| geq epsilon$ for all $ineq j$. At each step of the process, a pair $(i,j)$ of particles is selected uniformly at random, and then they "collide", and there is a repartition of their total energy $x_i + x_j$ between them producing new energies $x^*_i$ and $x^*_j$ with $x^*_i + x^*_j = x_i + x_j$, but with the restriction that exclusion rule is still observed for the new pair of energies. This process bears some resemblance to Kac models for Fermions in which the exclusion represents the effects of the Pauli exclusion principle. However, the "non-quantized" exclusion rule here, with only a lower bound on the gaps, introduces interesting novel features, and a strong notion of Kac's chaos is required to derive an evolution equation for the evolution of rescaled empirical measures for the process, as we show here.
{"title":"A Kac model with exclusion","authors":"E. Carlen, B. Wennberg","doi":"10.1214/22-aihp1276","DOIUrl":"https://doi.org/10.1214/22-aihp1276","url":null,"abstract":"We consider a one dimension Kac model with conservation of energy and an exclusion rule: Fix a number of particles $n$, and an energy $E>0$. Let each of the particles have an energy $x_j geq 0$, with $sum_{j=1}^n x_j = E$. For some $epsilon$, the allowed configurations $(x_1,dots,x_n)$ are those that satisfy $|x_i - x_j| geq epsilon$ for all $ineq j$. At each step of the process, a pair $(i,j)$ of particles is selected uniformly at random, and then they \"collide\", and there is a repartition of their total energy $x_i + x_j$ between them producing new energies $x^*_i$ and $x^*_j$ with $x^*_i + x^*_j = x_i + x_j$, but with the restriction that exclusion rule is still observed for the new pair of energies. This process bears some resemblance to Kac models for Fermions in which the exclusion represents the effects of the Pauli exclusion principle. However, the \"non-quantized\" exclusion rule here, with only a lower bound on the gaps, introduces interesting novel features, and a strong notion of Kac's chaos is required to derive an evolution equation for the evolution of rescaled empirical measures for the process, as we show here.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"38 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77677825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
R. Sanchis, Diogo C. dos Santos, Roger W. C. Silva
We consider the Constrained-degree percolation model with random constraints on the square lattice and prove a non-trivial phase transition. In this model, each vertex has an independently distributed random constraint $jin {0,1,2,3}$ with probability $rho_j$. Each edge $e$ tries to open at a random uniform time $U_e$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when $rho_3$ is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, together with a coarse-graining argument.
考虑具有随机约束的约束度渗流模型，证明了其非平凡相变。在该模型中，每个顶点都有一个独立分布的随机约束$j In {0,1,2,3}$，概率$rho_j$。每条边$e$尝试在一个随机的统一时间$U_e$打开，独立于所有其他边。如果在时间$U_e$，它的两个端点的度数都严格小于它们各自附加的约束，它就会成功。我们证明，当$rho_3$足够大时，该模型经历了一个非平凡的相变。该证明由解耦不等式、局部事件概率的连续性以及粗粒度论证组成。
{"title":"Constrained-degree percolation in random environment","authors":"R. Sanchis, Diogo C. dos Santos, Roger W. C. Silva","doi":"10.1214/21-aihp1231","DOIUrl":"https://doi.org/10.1214/21-aihp1231","url":null,"abstract":"We consider the Constrained-degree percolation model with random constraints on the square lattice and prove a non-trivial phase transition. In this model, each vertex has an independently distributed random constraint $jin {0,1,2,3}$ with probability $rho_j$. Each edge $e$ tries to open at a random uniform time $U_e$, independently of all other edges. It succeeds if at time $U_e$ both its end-vertices have degrees strictly smaller than their respectively attached constraints. We show that this model undergoes a non-trivial phase transition when $rho_3$ is sufficiently large. The proof consists of a decoupling inequality, the continuity of the probability for local events, together with a coarse-graining argument.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"3 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74487519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the dimer and Ising models on a planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $Gamma$ in the Klein bottle $K$. Let $Gamma_{mn}$ denote the graph obtained by pasting $m$ rows and $n$ columns of copies of $Gamma$, which embeds in $K$ for $n$ odd and in the torus $mathbb{T}^2$ for $n$ even. We compute the dimer partition function $Z_{mn}$ of $Gamma_{mn}$ for $n$ odd, in terms of the well-known characteristic polynomial $P$ of $Gamma_{12}subsetmathbb{T}^2$ together with a new characteristic polynomial $R$ of $Gammasubset K$. Using this result together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in the bipartite case, this partition function has the asymptotic expansion $log Z_{mn}=mn f_0/2 +mathrm{fsc}+o(1)$, where $f_0$ is the bulk free energy for $Gamma_{12}subsetmathbb{T}^2$ and $mathrm{fsc}$ an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph $Gamma$, but only on the zeros of $P$ on the unit torus and on an explicit (purely imaginary) shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of $P$. We then show that this asymptotic expansion holds for the Ising partition function as well, with $mathrm{fsc}$ taking a particularly simple form: it vanishes in the subcritical regime, is equal to $log(2)$ in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blote, Cardy and Nightingale.
{"title":"The dimer and Ising models on Klein bottles","authors":"David Cimasoni","doi":"10.4171/aihpd/166","DOIUrl":"https://doi.org/10.4171/aihpd/166","url":null,"abstract":"We study the dimer and Ising models on a planar weighted graph with periodic-antiperiodic boundary conditions, i.e. a graph $Gamma$ in the Klein bottle $K$. Let $Gamma_{mn}$ denote the graph obtained by pasting $m$ rows and $n$ columns of copies of $Gamma$, which embeds in $K$ for $n$ odd and in the torus $mathbb{T}^2$ for $n$ even. We compute the dimer partition function $Z_{mn}$ of $Gamma_{mn}$ for $n$ odd, in terms of the well-known characteristic polynomial $P$ of $Gamma_{12}subsetmathbb{T}^2$ together with a new characteristic polynomial $R$ of $Gammasubset K$. Using this result together with work of Kenyon, Sun and Wilson [arXiv:1310.2603], we show that in the bipartite case, this partition function has the asymptotic expansion $log Z_{mn}=mn f_0/2 +mathrm{fsc}+o(1)$, where $f_0$ is the bulk free energy for $Gamma_{12}subsetmathbb{T}^2$ and $mathrm{fsc}$ an explicit finite-size correction term. The remarkable feature of this later term is its universality: it does not depend on the graph $Gamma$, but only on the zeros of $P$ on the unit torus and on an explicit (purely imaginary) shape parameter. A similar expansion is also obtained in the non-bipartite case, assuming a conjectural condition on the zeros of $P$. We then show that this asymptotic expansion holds for the Ising partition function as well, with $mathrm{fsc}$ taking a particularly simple form: it vanishes in the subcritical regime, is equal to $log(2)$ in the supercritical regime, and to an explicit function of the shape parameter at criticality. These results are in full agreement with the conformal field theory predictions of Blote, Cardy and Nightingale.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"14 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81413791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Levy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuous-time process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a uniform large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation estimate for the quenched fluctuations and the asymptotics of quenched moments, showing that, unexpectedly, the asymptotically stable diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion. Although the Levy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings lead to the conclusion that superdiffusion is a metastable behavior, which develops into normal diffusion on long timescales.
{"title":"Large fluctuations and transport properties of the Lévy–Lorentz gas","authors":"M. Zamparo","doi":"10.1214/22-AIHP1283","DOIUrl":"https://doi.org/10.1214/22-AIHP1283","url":null,"abstract":"The Levy-Lorentz gas describes the motion of a particle on the real line in the presence of a random array of scattering points, whose distances between neighboring points are heavy-tailed i.i.d. random variables with finite mean. The motion is a continuous-time, constant-speed interpolation of the simple symmetric random walk on the marked points. In this paper we study the large fluctuations of the continuous-time process and the resulting transport properties of the model, both annealed and quenched, confirming and extending previous work by physicists that pertain to the annealed framework. Specifically, focusing on the particle displacement, and under the assumption that the tail distribution of the interdistances between scatterers is regularly varying at infinity, we prove a uniform large deviation principle for the annealed fluctuations and present the asymptotics of annealed moments, demonstrating annealed superdiffusion. Then, we provide an upper large deviation estimate for the quenched fluctuations and the asymptotics of quenched moments, showing that, unexpectedly, the asymptotically stable diffusive regime conditional on a typical arrangement of the scatterers is normal diffusion. Although the Levy-Lorentz gas seems to be accepted as a model for anomalous diffusion, our findings lead to the conclusion that superdiffusion is a metastable behavior, which develops into normal diffusion on long timescales.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"27 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87658293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider vector spin glasses whose energy function is a Gaussian random field with covariance given in terms of the matrix of scalar products. For essentially any model in this class, we give an upper bound for the limit free energy, which is expected to be sharp. The bound is expressed in terms of an infinite-dimensional Hamilton-Jacobi equation.
{"title":"Free energy upper bound for mean-field vector spin glasses","authors":"J. Mourrat","doi":"10.1214/22-aihp1292","DOIUrl":"https://doi.org/10.1214/22-aihp1292","url":null,"abstract":"We consider vector spin glasses whose energy function is a Gaussian random field with covariance given in terms of the matrix of scalar products. For essentially any model in this class, we give an upper bound for the limit free energy, which is expected to be sharp. The bound is expressed in terms of an infinite-dimensional Hamilton-Jacobi equation.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81467196","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We describe the translation invariant stationary states (TIS) of the one-dimensional facilitated asymmetric exclusion process in continuous time, in which a particle at site $iinmathbb{Z}$ jumps to site $i+1$ (respectively $i-1$) with rate $p$ (resp. $1-p$), provided that site $i-1$ (resp. $i+1$) is occupied and site $i+1$ (resp. $i-1$) is empty. All TIS states with density $rhole1/2$ are supported on trapped configurations in which no two adjacent sites are occupied; we prove that if in this case the initial state is Bernoulli then the final state is independent of $p$. This independence also holds for the system on a finite ring. For $rho>1/2$ there is only one TIS. It is the infinite volume limit of the probability distribution that gives uniform weight to all configurations in which no two holes are adjacent, and is isomorphic to the Gibbs measure for hard core particles with nearest neighbor exclusion.
{"title":"Stationary states of the one-dimensional facilitated asymmetric exclusion process","authors":"Arvind Ayyer, S. Goldstein, J. Lebowitz, E. Speer","doi":"10.1214/22-AIHP1264","DOIUrl":"https://doi.org/10.1214/22-AIHP1264","url":null,"abstract":"We describe the translation invariant stationary states (TIS) of the one-dimensional facilitated asymmetric exclusion process in continuous time, in which a particle at site $iinmathbb{Z}$ jumps to site $i+1$ (respectively $i-1$) with rate $p$ (resp. $1-p$), provided that site $i-1$ (resp. $i+1$) is occupied and site $i+1$ (resp. $i-1$) is empty. All TIS states with density $rhole1/2$ are supported on trapped configurations in which no two adjacent sites are occupied; we prove that if in this case the initial state is Bernoulli then the final state is independent of $p$. This independence also holds for the system on a finite ring. For $rho>1/2$ there is only one TIS. It is the infinite volume limit of the probability distribution that gives uniform weight to all configurations in which no two holes are adjacent, and is isomorphic to the Gibbs measure for hard core particles with nearest neighbor exclusion.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"143 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79620788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We give elements towards the classification of quantum Airy structures based on the $W(mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of $mathfrak{gl}_r$ for twists by arbitrary elements of the Weyl group $mathfrak{S}_{r}$. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion a la Chekhov--Eynard--Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard--Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves and give precise conjectures for application in open $r$-spin intersection theory.
{"title":"Higher Airy structures and topological recursion for singular spectral curves","authors":"G. Borot, Reinier Kramer, Yannik Schuler","doi":"10.4171/aihpd/168","DOIUrl":"https://doi.org/10.4171/aihpd/168","url":null,"abstract":"We give elements towards the classification of quantum Airy structures based on the $W(mathfrak{gl}_r)$-algebras at self-dual level based on twisted modules of the Heisenberg VOA of $mathfrak{gl}_r$ for twists by arbitrary elements of the Weyl group $mathfrak{S}_{r}$. In particular, we construct a large class of such quantum Airy structures. We show that the system of linear ODEs forming the quantum Airy structure and determining uniquely its partition function is equivalent to a topological recursion a la Chekhov--Eynard--Orantin on singular spectral curves. In particular, our work extends the definition of the Bouchard--Eynard topological recursion (valid for smooth curves) to a large class of singular curves, and indicates impossibilities to extend naively the definition to other types of singularities. We also discuss relations to intersection theory on moduli spaces of curves and give precise conjectures for application in open $r$-spin intersection theory.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"9 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73940384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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