We give asymptotic estimations on the area of the sets of points with large Brownian winding, and study the average winding between a planar Brownian motion and a Poisson point process of large intensity on the plane. This allows us to give a new definition of the Lévy area which does not rely on approximations of the Brownian path. It also does not depend on the metric structure on the plane.
{"title":"Lévy area without approximation","authors":"Isao Sauzedde","doi":"10.1214/21-aihp1230","DOIUrl":"https://doi.org/10.1214/21-aihp1230","url":null,"abstract":"We give asymptotic estimations on the area of the sets of points with large Brownian winding, and study the average winding between a planar Brownian motion and a Poisson point process of large intensity on the plane. This allows us to give a new definition of the Lévy area which does not rely on approximations of the Brownian path. It also does not depend on the metric structure on the plane.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"97 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80535605","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 rigorously compute the integrable system for the limiting $(Nrightarrowinfty)$ distribution function of the extreme momentum of $N$ noninteracting fermions when confined to an anharmonic trap $V(q)=q^{2n}$ for $ninmathbb{Z}_{geq 1}$ at positive temperature. More precisely, the edge momentum statistics in the harmonic trap $n=1$ are known to obey the weak asymmetric KPZ crossover law which is realized via the finite temperature Airy kernel determinant or equivalently via a Painlev'e-II integro-differential transcendent, cf. cite{LW,ACQ}. For general $ngeq 2$, a novel higher order finite temperature Airy kernel has recently emerged in physics literature cite{DMS} and we show that the corresponding edge law in momentum space is now governed by a distinguished Painlev'e-II integro-differential hierarchy. Our analysis is based on operator-valued Riemann-Hilbert techniques which produce a Lax pair for an operator-valued Painlev'e-II ODE system that naturally encodes the aforementioned hierarchy. As byproduct, we establish a connection of the integro-differential Painlev'e-II hierarchy to a novel integro-differential mKdV hierarchy.
{"title":"Momenta spacing distributions in anharmonic oscillators and the higher order finite temperature Airy kernel","authors":"Thomas Bothner, M. Cafasso, Sofia Tarricone","doi":"10.1214/21-aihp1211","DOIUrl":"https://doi.org/10.1214/21-aihp1211","url":null,"abstract":"We rigorously compute the integrable system for the limiting $(Nrightarrowinfty)$ distribution function of the extreme momentum of $N$ noninteracting fermions when confined to an anharmonic trap $V(q)=q^{2n}$ for $ninmathbb{Z}_{geq 1}$ at positive temperature. More precisely, the edge momentum statistics in the harmonic trap $n=1$ are known to obey the weak asymmetric KPZ crossover law which is realized via the finite temperature Airy kernel determinant or equivalently via a Painlev'e-II integro-differential transcendent, cf. cite{LW,ACQ}. For general $ngeq 2$, a novel higher order finite temperature Airy kernel has recently emerged in physics literature cite{DMS} and we show that the corresponding edge law in momentum space is now governed by a distinguished Painlev'e-II integro-differential hierarchy. Our analysis is based on operator-valued Riemann-Hilbert techniques which produce a Lax pair for an operator-valued Painlev'e-II ODE system that naturally encodes the aforementioned hierarchy. As byproduct, we establish a connection of the integro-differential Painlev'e-II hierarchy to a novel integro-differential mKdV hierarchy.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"11 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76427404","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}
Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a L'evy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.
{"title":"Pathwise regularization of the stochastic heat equation with multiplicative noise through irregular perturbation","authors":"R. Catellier, Fabian A. Harang","doi":"10.1214/22-aihp1302","DOIUrl":"https://doi.org/10.1214/22-aihp1302","url":null,"abstract":"Existence and uniqueness of solutions to the stochastic heat equation with multiplicative spatial noise is studied. In the spirit of pathwise regularization by noise, we show that a perturbation by a sufficiently irregular continuous path establish wellposedness of such equations, even when the drift and diffusion coefficients are given as generalized functions or distributions. In addition we prove regularity of the averaged field associated to a L'evy fractional stable motion, and use this as an example of a perturbation regularizing the multiplicative stochastic heat equation.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"21 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89241213","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-84828-6_1
Oliver Schlaudt, A. Schmid
{"title":"Cours de Caen 1898–1899","authors":"Oliver Schlaudt, A. Schmid","doi":"10.1007/978-3-030-84828-6_1","DOIUrl":"https://doi.org/10.1007/978-3-030-84828-6_1","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"33 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87989456","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-84828-6_3
Oliver Schlaudt, A. Schmid
{"title":"Manuel de Logistique. Tome I & II","authors":"Oliver Schlaudt, A. Schmid","doi":"10.1007/978-3-030-84828-6_3","DOIUrl":"https://doi.org/10.1007/978-3-030-84828-6_3","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"51 5 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80225858","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}
Pub Date : 2021-01-01DOI: 10.1007/978-3-030-84828-6_2
Oliver Schlaudt, A. Schmid
{"title":"Cours 1905-1906: Histoire de la Logique formelle moderne","authors":"Oliver Schlaudt, A. Schmid","doi":"10.1007/978-3-030-84828-6_2","DOIUrl":"https://doi.org/10.1007/978-3-030-84828-6_2","url":null,"abstract":"","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"207 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83303970","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}
In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and R"uschendorf and Naor and Romik unified these results by establishing a connection between $ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form [B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},] where $phi:mathbb{R}to [0,infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-R"uschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cram'er's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $ell_p^N$ balls. We find that there is a critical parameter $t_{mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{mathrm{crit}}$ the coordinates of random points sampled from $B_{phi,t}^N$ behave like uniform random variables, but for $t leq t_{mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $beta_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -beta_tphi(s)}$.
在《十几个德{菲}内蒂式的结果寻找一个理论》一书中，安。也许吧。Statist. 23(2)(1987)， 397—423]，Diaconis和Freedman研究了来自欧几里得单位球和高维单纯形的随机向量的低维投影，注意到这些随机向量的单个坐标分别看起来像高斯和指数随机变量。在随后的工作中，Rachev和r schendorf以及Naor和Romik通过建立$ell_p^N$球与$p$广义高斯分布之间的联系来统一这些结果。在本文中，我们在一个显著推广和统一的设置中研究类似的问题，观察均匀分布在形式为[B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},]的集合上的随机向量的低维投影，其中$phi:mathbb{R}to [0,infty]$是一个势(包括Orlicz函数的情况)。我们的方法不同于rachev - r schendorf和Naor-Romik，我们的方法基于一个大偏差的视角，以定量版本的克拉姆萨姆定理和吉布斯条件反射原理的形式，提供了一个超越$p$ -广义高斯分布的自然框架，同时揭示了该分布在$ell_p^N$球的几何形状中所起的作用。我们发现存在一个临界参数$t_{mathrm{crit}}$，在该参数处，投影的行为发生相变:对于$t > t_{mathrm{crit}}$，从$B_{phi,t}^N$中采样的随机点的坐标表现得像均匀随机变量，但对于$t leq t_{mathrm{crit}}$，吉布斯条件反射原理开始发挥作用，这里有一个参数$beta_t>0$(逆温度)，使得坐标根据与$e^{ -beta_tphi(s)}$成比例的密度近似分布。
{"title":"A Maxwell principle for generalized Orlicz balls","authors":"S. Johnston, J. Prochno","doi":"10.1214/22-aihp1298","DOIUrl":"https://doi.org/10.1214/22-aihp1298","url":null,"abstract":"In [A dozen de {F}inetti-style results in search of a theory, Ann. Inst. H. Poincar'{e} Probab. Statist. 23(2)(1987), 397--423], Diaconis and Freedman studied low-dimensional projections of random vectors from the Euclidean unit sphere and the simplex in high dimensions, noting that the individual coordinates of these random vectors look like Gaussian and exponential random variables respectively. In subsequent works, Rachev and R\"uschendorf and Naor and Romik unified these results by establishing a connection between $ell_p^N$ balls and a $p$-generalized Gaussian distribution. In this paper, we study similar questions in a significantly generalized and unifying setting, looking at low-dimensional projections of random vectors uniformly distributed on sets of the form [B_{phi,t}^N := Big{(s_1,ldots,s_N)inmathbb{R}^N : sum_{ i =1}^Nphi(s_i)leq t NBig},] where $phi:mathbb{R}to [0,infty]$ is a potential (including the case of Orlicz functions). Our method is different from both Rachev-R\"uschendorf and Naor-Romik, based on a large deviation perspective in the form of quantitative versions of Cram'er's theorem and the Gibbs conditioning principle, providing a natural framework beyond the $p$-generalized Gaussian distribution while simultaneously unraveling the role this distribution plays in relation to the geometry of $ell_p^N$ balls. We find that there is a critical parameter $t_{mathrm{crit}}$ at which there is a phase transition in the behaviour of the projections: for $t > t_{mathrm{crit}}$ the coordinates of random points sampled from $B_{phi,t}^N$ behave like uniform random variables, but for $t leq t_{mathrm{crit}}$ the Gibbs conditioning principle comes into play, and here there is a parameter $beta_t>0$ (the inverse temperature) such that the coordinates are approximately distributed according to a density proportional to $e^{ -beta_tphi(s)}$.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"1 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77180382","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}
Consider a nearest neighbor random walk on the two-dimensional integer lattice, where each vertex is initially labeled either `H' or `V', uniformly and independently. At each discrete time step, the walker resamples the label at its current location (changing `H' to `V' and `V' to `H' with probability $q$). Then, it takes a mean zero horizontal step if the new label is `H', and a mean zero vertical step if the new label is `V'. This model is a randomized version of the deterministic rotor walk, for which its recurrence (i.e., visiting every vertex infinitely often with probability 1) in two dimensions is still an open problem. We answer the analogous question for the the horizontal-vertical walk, by showing that the horizontal-vertical walk is recurrent for $q in (frac{1}{3},1]$.
考虑在二维整数晶格上的最近邻随机游走，其中每个顶点最初被均匀且独立地标记为“H”或“V”。在每个离散的时间步长，行走器在其当前位置重新采样标签(以概率$q$将' H'变为' V'和' V'变为' H')。然后，如果新标签是“H”，它的平均水平步长为零，如果新标签是“V”，它的平均垂直步长为零。该模型是确定性转子行走的随机化版本，其在二维空间中的递归性(即以1的概率无限次访问每个顶点)仍然是一个开放的问题。通过证明水平-垂直行走对于$q in (frac{1}{3}，1]$是循环的，我们回答了水平-垂直行走的类似问题。
{"title":"Recurrence of horizontal–vertical walks","authors":"Swee Hong Chan","doi":"10.1214/22-aihp1277","DOIUrl":"https://doi.org/10.1214/22-aihp1277","url":null,"abstract":"Consider a nearest neighbor random walk on the two-dimensional integer lattice, where each vertex is initially labeled either `H' or `V', uniformly and independently. At each discrete time step, the walker resamples the label at its current location (changing `H' to `V' and `V' to `H' with probability $q$). Then, it takes a mean zero horizontal step if the new label is `H', and a mean zero vertical step if the new label is `V'. This model is a randomized version of the deterministic rotor walk, for which its recurrence (i.e., visiting every vertex infinitely often with probability 1) in two dimensions is still an open problem. We answer the analogous question for the the horizontal-vertical walk, by showing that the horizontal-vertical walk is recurrent for $q in (frac{1}{3},1]$.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"14 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79494304","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 the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of degree d, depending on repulsion strength beta between particles of di?fferent signs and on an activity parameter lambda for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength beta. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d greater or equal 2. In our second and main result, we also show that for large beta and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity increases, assuming that d is greater or equal than 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.
{"title":"Dynamical Gibbs–non-Gibbs transitions in Widom–Rowlinson models on trees","authors":"S. Bergmann, Sascha Kissel, C. Kuelske","doi":"10.1214/22-AIHP1242","DOIUrl":"https://doi.org/10.1214/22-AIHP1242","url":null,"abstract":"We consider the soft-core Widom-Rowlinson model for particles with spins and holes, on a Cayley tree of degree d, depending on repulsion strength beta between particles of di?fferent signs and on an activity parameter lambda for particles. We analyse Gibbsian properties of the time-evolved intermediate Gibbs measure of the static model, under a spin-flip time evolution, in a regime of large repulsion strength beta. We first show that there is a dynamical transition, in which the measure becomes non-Gibbsian at large times, independently of the particle activity, for any d greater or equal 2. In our second and main result, we also show that for large beta and at large times, the measure of the set of bad configurations (discontinuity points) changes from zero to one as the particle activity increases, assuming that d is greater or equal than 4. Our proof relies on a general zero-one law for bad configurations on the tree, and the introduction of a set of uniformly bad configurations given in terms of subtree percolation, which we show to become typical at high particle activity.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"9 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85372205","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}
Inequalities between transportation costs and Fisher information are known to characterize certain concentration properties of Markov processes around their invariant measures. This note provides a new characterization of the quadratic transportation-information inequality $W_2I$ in terms of a dimension-free concentration property for i.i.d. (conditionally on the initial positions) copies of the underlying Markov process. This parallels Gozlan's characterization of the quadratic transportation-entropy inequality $W_2H$. The proof is based on a new Laplace-type principle for the operator norms of Feynman-Kac semigroups, which is of independent interest. Lastly, we illustrate how both our theorem and (a form of) Gozlan's are instances of a general convex-analytic tensorization principle.
已知运输成本和费雪信息之间的不等式表征了马尔可夫过程围绕其不变测度的某些集中性质。本文给出了二次输运信息不等式$W_2I$的一个新的表征，该表征是基于基础马尔可夫过程的i / i / d(有条件地在初始位置上)副本的无维集中性质。这与Gozlan对二次输运-熵不等式的描述相似。本文的证明是基于Feynman-Kac半群算子范数的一个新的拉普拉斯原理，具有独立的研究意义。最后，我们说明了我们的定理和Gozlan的定理(一种形式)是一般凸解析张化原理的实例。
{"title":"A characterization of transportation-information inequalities for Markov processes in terms of dimension-free concentration","authors":"D. Lacker, L. Yeung","doi":"10.1214/22-aihp1249","DOIUrl":"https://doi.org/10.1214/22-aihp1249","url":null,"abstract":"Inequalities between transportation costs and Fisher information are known to characterize certain concentration properties of Markov processes around their invariant measures. This note provides a new characterization of the quadratic transportation-information inequality $W_2I$ in terms of a dimension-free concentration property for i.i.d. (conditionally on the initial positions) copies of the underlying Markov process. This parallels Gozlan's characterization of the quadratic transportation-entropy inequality $W_2H$. The proof is based on a new Laplace-type principle for the operator norms of Feynman-Kac semigroups, which is of independent interest. Lastly, we illustrate how both our theorem and (a form of) Gozlan's are instances of a general convex-analytic tensorization principle.","PeriodicalId":42884,"journal":{"name":"Annales de l Institut Henri Poincare D","volume":"189 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2020-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85848476","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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