We consider the ABC model on a ring in a strongly asymmetric regime. The main result asserts that the particles almost always form three pure domains (one of each species) and that this segregated shape evolves, in a proper time scale, as a Brownian motion on the circle, which may have a drift. This is, to our knowledge, the first proof of a zero-temperature limit for a non-reversible dynamics whose invariant measure is not explicitly known.
{"title":"Evolution of the ABC model among the segregated configurations in the zero-temperature limit","authors":"R. Misturini","doi":"10.1214/14-AIHP648","DOIUrl":"https://doi.org/10.1214/14-AIHP648","url":null,"abstract":"We consider the ABC model on a ring in a strongly asymmetric regime. The main result asserts that the particles almost always form three pure domains (one of each species) and that this segregated shape evolves, in a proper time scale, as a Brownian motion on the circle, which may have a drift. This is, to our knowledge, the first proof of a zero-temperature limit for a non-reversible dynamics whose invariant measure is not explicitly known.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"127 1","pages":"669-702"},"PeriodicalIF":1.5,"publicationDate":"2014-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74145082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the empirical spectral distribution (ESD) in the limit where n goes to infinity of a fixed n by n matrix M_n plus small random noise of the form f(n)X_n, where X_n has iid mean 0, variance 1/n entries and f(n) goes to 0 as n goes to infinity. It is known for certain M_n, in the case where X_n is iid complex Gaussian, that the limiting distribution of the ESD of M_n+f(n)X_n can be dramatically different from that for M_n. We prove a general universality result showing, with some conditions on M_n and f(n), that the limiting distribution of the ESD does not depend on the type of distribution used for the random entries of X_n. We use the universality result to exactly compute the limiting ESD for two families where it was not previously known. The proof of the main result incorporates the Tao-Vu replacement principle and a version of the Lindeberg replacement strategy, along with the newly-defined notion of stability of sets of rows of a matrix.
{"title":"Universality of the ESD for a fixed matrix plus small random noise: A stability approach","authors":"Philip Matchett Wood","doi":"10.1214/15-AIHP702","DOIUrl":"https://doi.org/10.1214/15-AIHP702","url":null,"abstract":"We study the empirical spectral distribution (ESD) in the limit where n goes to infinity of a fixed n by n matrix M_n plus small random noise of the form f(n)X_n, where X_n has iid mean 0, variance 1/n entries and f(n) goes to 0 as n goes to infinity. It is known for certain M_n, in the case where X_n is iid complex Gaussian, that the limiting distribution of the ESD of M_n+f(n)X_n can be dramatically different from that for M_n. We prove a general universality result showing, with some conditions on M_n and f(n), that the limiting distribution of the ESD does not depend on the type of distribution used for the random entries of X_n. We use the universality result to exactly compute the limiting ESD for two families where it was not previously known. The proof of the main result incorporates the Tao-Vu replacement principle and a version of the Lindeberg replacement strategy, along with the newly-defined notion of stability of sets of rows of a matrix.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"45 1","pages":"1877-1896"},"PeriodicalIF":1.5,"publicationDate":"2014-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85485984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Suppose that (X;Y;Z) is a random walk in Z 3 that moves in the following way: on the rst visit to a vertex only Z changes by 1 equally likely, while on later visits to the same vertex (X;Y ) performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.
{"title":"Martingale defocusing and transience of a self-interacting random walk","authors":"Y. Peres, Bruno Schapira, Perla Sousi","doi":"10.1214/14-AIHP667","DOIUrl":"https://doi.org/10.1214/14-AIHP667","url":null,"abstract":"Suppose that (X;Y;Z) is a random walk in Z 3 that moves in the following way: on the rst visit to a vertex only Z changes by 1 equally likely, while on later visits to the same vertex (X;Y ) performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"6 1","pages":"1009-1022"},"PeriodicalIF":1.5,"publicationDate":"2014-03-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75149642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hormander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.
{"title":"A parametrix approach for some degenerate stable driven SDEs","authors":"Lorick Huang, S. Menozzi","doi":"10.1214/15-AIHP704","DOIUrl":"https://doi.org/10.1214/15-AIHP704","url":null,"abstract":"We consider a stable driven degenerate stochastic differential equation, whose coefficients satisfy a kind of weak Hormander condition. Under mild smoothness assumptions we prove the uniqueness of the martingale problem for the associated generator under some dimension constraints. Also, when the driving noise is scalar and tempered, we establish density bounds reflecting the multi-scale behavior of the process.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"87 1","pages":"1925-1975"},"PeriodicalIF":1.5,"publicationDate":"2014-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81343791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pour chaque entier k≥2, on introduit une suite d’arbres discrets k-aires construite recursivement en choisissant a chaque etape une arete uniformement parmi les aretes de l’arbre pre-existant et greffant sur son « milieu » k−1 nouvelles aretes. Lorsque k=2, cette procedure correspond a un algorithme introduit par Remy. Pour chaque entier k≥2, nous decrivons la limite d’echelle de ces arbres lorsque le nombre d’etapes n tend vers l’infini : ils grandissent a la vitesse n1/k vers un arbre reel aleatoire k-aire qui appartient a la famille des arbres de fragmentation auto-similaires. Cette convergence a lieu en probabilite, pour la topologie de Gromov–Hausdorff–Prokhorov. Nous etudions egalement l’emboitement des arbres limites quand k varie.
{"title":"Scaling limits of k-ary growing trees","authors":"Bénédicte Haas, R. Stephenson","doi":"10.1214/14-AIHP622","DOIUrl":"https://doi.org/10.1214/14-AIHP622","url":null,"abstract":"Pour chaque entier k≥2, on introduit une suite d’arbres discrets k-aires construite recursivement en choisissant a chaque etape une arete uniformement parmi les aretes de l’arbre pre-existant et greffant sur son « milieu » k−1 nouvelles aretes. Lorsque k=2, cette procedure correspond a un algorithme introduit par Remy. Pour chaque entier k≥2, nous decrivons la limite d’echelle de ces arbres lorsque le nombre d’etapes n tend vers l’infini : ils grandissent a la vitesse n1/k vers un arbre reel aleatoire k-aire qui appartient a la famille des arbres de fragmentation auto-similaires. Cette convergence a lieu en probabilite, pour la topologie de Gromov–Hausdorff–Prokhorov. Nous etudions egalement l’emboitement des arbres limites quand k varie.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"58 1","pages":"1314-1341"},"PeriodicalIF":1.5,"publicationDate":"2014-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83952474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived as a corollary, and it is in fact extended to a wide class of radially symmetric measures. Résumé. En utilisant des arguments géométriques élémentaires, on démontre des inégalités de corrélation pour des mesures de probabilité à symétrie radiale. Plus précisément on montre que, parmi la famille des ensembles width-decreasing , le ratio de corrélation est minimisé par des bandes. Comme les ouverts convexes symétriques appartiennent à cette famille, on retrouve comme corollaire le résultat de Pitt sur la validité de la conjecture de corrélation gaussiennne en dimension 2, qui est étendue dans ce papier à une large classe de mesures à symétrie radiale.
。利用初等几何论证，证明了平面上径向对称概率测度的相关不等式。准确地说，在无限条的类别中，宽度递减集对的相关比是最小的。由于相对于原点对称的开凸集是宽度递减集，皮特的高斯相关不等式(长期存在的高斯相关猜想的二维情况)被作为一个推论推导出来，它实际上被推广到一个广泛的类径向对称测度。的简历。所有适用的论点都是关于或更大的范围的，例如:或更大的范围的，例如:或更大范围的。加上在montre que上的prassicimement, parmi la famille des ensembles的宽度递减，le ratio de conacimation和minimisispardes bands的宽度递减。Comme les ouvers convents symsamtriques appartient conette famille，在返回Comme corcolllaire le consamsult de Pitt sur la validitest de consamicationgaussienne维2上，Comme corcolllaire le consamicsult de Pitt sur la validitest de consamicationne维2上，Comme cocollaire de consamicationdne维2上，Comme cocollaire de consamicationdne维2上，Comme cocollaire de consamicationes在consamictritrie radiale上的大类测度。
{"title":"A geometric approach to correlation inequalities in the plane","authors":"A. Figalli, F. Maggi, A. Pratelli","doi":"10.1214/12-AIHP494","DOIUrl":"https://doi.org/10.1214/12-AIHP494","url":null,"abstract":". By elementary geometric arguments, correlation inequalities for radially symmetric probability measures are proved in the plane. Precisely, it is shown that the correlation ratio for pairs of width-decreasing sets is minimized within the class of infinite strips. Since open convex sets which are symmetric with respect to the origin turn out to be width-decreasing sets, Pitt’s Gaussian correlation inequality (the two-dimensional case of the long-standing Gaussian correlation conjecture) is derived as a corollary, and it is in fact extended to a wide class of radially symmetric measures. Résumé. En utilisant des arguments géométriques élémentaires, on démontre des inégalités de corrélation pour des mesures de probabilité à symétrie radiale. Plus précisément on montre que, parmi la famille des ensembles width-decreasing , le ratio de corrélation est minimisé par des bandes. Comme les ouverts convexes symétriques appartiennent à cette famille, on retrouve comme corollaire le résultat de Pitt sur la validité de la conjecture de corrélation gaussiennne en dimension 2, qui est étendue dans ce papier à une large classe de mesures à symétrie radiale.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"53 6 1","pages":"1-14"},"PeriodicalIF":1.5,"publicationDate":"2014-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83228171","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use a different approach based on two ingredients: we first study typical geodesics and then all geodesics by an ''entrapment'' strategy. Our results give geometrical characterizations of some subsets of interest, in terms of geodesics, boundary points and concatenations of geodesics that are not homotopic to 0.
{"title":"Geodesics in Brownian surfaces (Brownian maps)","authors":"Jérémie Bettinelli","doi":"10.1214/14-AIHP666","DOIUrl":"https://doi.org/10.1214/14-AIHP666","url":null,"abstract":"We define a class a metric spaces we call Brownian surfaces, arising as the scaling limits of random maps on surfaces with a boundary and we study the geodesics from a uniformly chosen random point. These spaces generalize the well-known Brownian map and our results generalize the properties shown by Le Gall on geodesics in the latter space. We use a different approach based on two ingredients: we first study typical geodesics and then all geodesics by an ''entrapment'' strategy. Our results give geometrical characterizations of some subsets of interest, in terms of geodesics, boundary points and concatenations of geodesics that are not homotopic to 0.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"7 1","pages":"612-646"},"PeriodicalIF":1.5,"publicationDate":"2014-01-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83898202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a fixed point. We define two different procedures of estimation, the first one using kernel rules, the second one inspired from projection methods. Both adapted estimators are tuned by using the Goldenshluger and Lepski methodology. After deriving lower bounds, we show that these procedures satisfy oracle inequalities and are optimal from the minimax point of view on anisotropic Holder balls. Furthermore, our results allow us to measure precisely the influence of $mathrm{f}_X(x)$ on rates of convergence, where $mathrm{f}_X$ is the density of $X$. Finally, some simulations illustrate the good behavior of our tuned estimates in practice.
本文考虑了在多元环境下，用独立样本分布的$(X,Y)$估计$f$，即给定$X$的$Y$的条件密度问题。我们考虑f(x，.)$的估计，其中$x$是一个不动点。我们定义了两种不同的估计过程，第一种是使用核规则，第二种是受投影方法的启发。两个适应的估计量都是通过使用Goldenshluger和Lepski方法来调整的。在推导出下界之后，我们证明了这些方法满足oracle不等式，并且从极大极小的角度来看，对于各向异性的Holder球是最优的。此外，我们的结果允许我们精确地测量$mathrm{f}_X(x)$对收敛速率的影响，其中$mathrm{f}_X$是$ x $的密度。最后，一些仿真说明了我们的调优估计在实践中的良好行为。
{"title":"Adaptive pointwise estimation of conditional density function","authors":"K. Bertin, C. Lacour, V. Rivoirard","doi":"10.1214/14-AIHP665","DOIUrl":"https://doi.org/10.1214/14-AIHP665","url":null,"abstract":"In this paper we consider the problem of estimating $f$, the conditional density of $Y$ given $X$, by using an independent sample distributed as $(X,Y)$ in the multivariate setting. We consider the estimation of $f(x,.)$ where $x$ is a fixed point. We define two different procedures of estimation, the first one using kernel rules, the second one inspired from projection methods. Both adapted estimators are tuned by using the Goldenshluger and Lepski methodology. After deriving lower bounds, we show that these procedures satisfy oracle inequalities and are optimal from the minimax point of view on anisotropic Holder balls. Furthermore, our results allow us to measure precisely the influence of $mathrm{f}_X(x)$ on rates of convergence, where $mathrm{f}_X$ is the density of $X$. Finally, some simulations illustrate the good behavior of our tuned estimates in practice.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"40 1","pages":"939-980"},"PeriodicalIF":1.5,"publicationDate":"2013-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81136973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.
{"title":"PATHWISE SOLVABILITY OF STOCHASTIC INTEGRAL EQUATIONS WITH GENERALIZED DRIFT AND NON-SMOOTH DISPERSION FUNCTIONS","authors":"I. Karatzas, J. Ruf","doi":"10.1214/14-AIHP660","DOIUrl":"https://doi.org/10.1214/14-AIHP660","url":null,"abstract":"We study one-dimensional stochastic integral equations with non-smooth dispersion coefficients, and with drift components that are not restricted to be absolutely continuous with respect to Lebesgue measure. In the spirit of Lamperti, Doss and Sussmann, we relate solutions of such equations to solutions of certain ordinary integral equations, indexed by a generic element of the underlying probability space. This relation allows us to solve the stochastic integral equations in a pathwise sense.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"113 1","pages":"915-938"},"PeriodicalIF":1.5,"publicationDate":"2013-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76070873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A necessary and sufficient condition is obtained for the existence of strong stationary times for ergodic one-dimensional diffusions, whatever the initial distribution. The strong stationary times are constructed through intertwinings with dual processes, in the Diaconis-Fill sense, taking values in the set of segments of the extended line $mathbb{R}sqcup{-infty,+infty}$. They can be seen as natural $h$-transforms of the extensions to the diffusion framework of the evolving sets of Morris-Peres. Starting from a singleton set, the dual process begins by evolving into true segments in the same way a Bessel process of dimension 3 escapes from 0. The strong stationary time corresponds to the first time the full segment $[-infty,+infty]$ is reached. The benchmark Ornstein-Uhlenbeck process cannot be treated in this way, it will nevertheless be seen how to use other strong times to recover its optimal exponential rate of convergence in the total variation sense.
{"title":"Strong stationary times for one-dimensional diffusions","authors":"L. Miclo","doi":"10.1214/16-aihp745","DOIUrl":"https://doi.org/10.1214/16-aihp745","url":null,"abstract":"A necessary and sufficient condition is obtained for the existence of strong stationary times for ergodic one-dimensional diffusions, whatever the initial distribution. The strong stationary times are constructed through intertwinings with dual processes, in the Diaconis-Fill sense, taking values in the set of segments of the extended line $mathbb{R}sqcup{-infty,+infty}$. They can be seen as natural $h$-transforms of the extensions to the diffusion framework of the evolving sets of Morris-Peres. Starting from a singleton set, the dual process begins by evolving into true segments in the same way a Bessel process of dimension 3 escapes from 0. The strong stationary time corresponds to the first time the full segment $[-infty,+infty]$ is reached. The benchmark Ornstein-Uhlenbeck process cannot be treated in this way, it will nevertheless be seen how to use other strong times to recover its optimal exponential rate of convergence in the total variation sense.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"1 1","pages":"957-996"},"PeriodicalIF":1.5,"publicationDate":"2013-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79807977","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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