Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size n, rescaled by the factor n−1 lnn, converges in probability as n → ∞ in the sense of GromovHausdorff-Prokhorov, to the unit interval endowed with the usual distance and Lebesgue measure. This enables us to explain and extend some recent results of Kuba and Panholzer [15] on multiple isolation of nodes in random recursive trees.
{"title":"The cut-tree of large recursive trees","authors":"J. Bertoin","doi":"10.1214/13-AIHP597","DOIUrl":"https://doi.org/10.1214/13-AIHP597","url":null,"abstract":"Imagine a graph which is progressively destroyed by cutting its edges one after the other in a uniform random order. The so-called cut-tree records key steps of this destruction process. It can be viewed as a random metric space equipped with a natural probability mass. In this work, we show that the cut-tree of a random recursive tree of size n, rescaled by the factor n−1 lnn, converges in probability as n → ∞ in the sense of GromovHausdorff-Prokhorov, to the unit interval endowed with the usual distance and Lebesgue measure. This enables us to explain and extend some recent results of Kuba and Panholzer [15] on multiple isolation of nodes in random recursive trees.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"1 1","pages":"478-488"},"PeriodicalIF":1.5,"publicationDate":"2015-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91247750","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 construct, on a single probability space, a class of regenerative sets R, indexed by all measurable functions α : [0, 1] → [0, 1]. For each function α, R, has the law of the range of a special subordinator. Constant functions correspond to stable subordinators. If α ≤ β, then R ⊂ R. Other examples of special subordinators are given in the lattice case.
{"title":"A class of special subordinators with nested ranges","authors":"P. Marchal","doi":"10.1214/13-AIHP595","DOIUrl":"https://doi.org/10.1214/13-AIHP595","url":null,"abstract":"We construct, on a single probability space, a class of regenerative sets R, indexed by all measurable functions α : [0, 1] → [0, 1]. For each function α, R, has the law of the range of a special subordinator. Constant functions correspond to stable subordinators. If α ≤ β, then R ⊂ R. Other examples of special subordinators are given in the lattice case.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"84 1","pages":"533-544"},"PeriodicalIF":1.5,"publicationDate":"2015-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83862871","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}
{"title":"Three examples of Brownian flows on $mathbb{R}$","authors":"Y. Jan, Olivier Raimond","doi":"10.1214/13-AIHP541","DOIUrl":"https://doi.org/10.1214/13-AIHP541","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"11 1","pages":"1323-1346"},"PeriodicalIF":1.5,"publicationDate":"2014-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75294276","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 establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.
{"title":"Path-dependent infinite-dimensional SDE with non-regular drift : an existence result","authors":"D. Dereudre, S. Roelly","doi":"10.1214/15-AIHP728","DOIUrl":"https://doi.org/10.1214/15-AIHP728","url":null,"abstract":"We establish in this paper the existence of weak solutions of infinite-dimensional shift invariant stochastic differential equations driven by a Brownian term. The drift function is very general, in the sense that it is supposed to be neither small or continuous, nor Markov. On the initial law we only assume that it admits a finite specific entropy. Our result strongly improves the previous ones obtained for free dynamics with a small perturbative drift. The originality of our method leads in the use of the specific entropy as a tightness tool and on a description of such stochastic differential equation as solution of a variational problem on the path space.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"4 1","pages":"641-657"},"PeriodicalIF":1.5,"publicationDate":"2014-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87980466","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 Conformal Loop Ensemble (CLEκ ) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLEκ for which the loops are simple, i.e. κ ∈ (8/3,4]. In [SW12], simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLEs, along with the corresponding exploration processes.
{"title":"Simple CLE in doubly connected domains","authors":"S. Sheffield, Samuel S. Watson, Hao Wu","doi":"10.1214/15-AIHP726","DOIUrl":"https://doi.org/10.1214/15-AIHP726","url":null,"abstract":"We study Conformal Loop Ensemble (CLEκ ) in doubly connected domains: annuli, the punctured disc, and the punctured plane. We restrict attention to CLEκ for which the loops are simple, i.e. κ ∈ (8/3,4]. In [SW12], simple CLE in the unit disc is introduced and constructed as the collection of outer boundaries of outermost clusters of the Brownian loop soup. For simple CLE in the unit disc, any fixed interior point is almost surely surrounded by some loop of CLE. The gasket of the collection of loops in CLE, i.e. the set of points that are not surrounded by any loop, almost surely has Lebesgue measure zero. In the current paper, simple CLE in an annulus is constructed similarly: it is the collection of outer boundaries of outermost clusters of the Brownian loop soup conditioned on the event that there is no cluster disconnecting the two components of the boundary of the annulus. Simple CLE in the punctured disc can be viewed as simple CLE in the unit disc conditioned on the event that the origin is in the gasket. Simple CLE in the punctured plane can be viewed as simple CLE in the whole plane conditioned on the event that both the origin and infinity are in the gasket. We construct and study these three kinds of CLEs, along with the corresponding exploration processes.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"29 1","pages":"594-615"},"PeriodicalIF":1.5,"publicationDate":"2014-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89356558","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 examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial $R$-transforms in two settings relating the moments and cumulants of a left-right pair of operators.
{"title":"Independences and partial $R$-transforms in bi-free probability","authors":"P. Skoufranis","doi":"10.1214/15-AIHP691","DOIUrl":"https://doi.org/10.1214/15-AIHP691","url":null,"abstract":"In this paper, we examine how various notions of independence in non-commutative probability theory arise in bi-free probability. We exhibit how Boolean and monotone independence occur from bi-free pairs of faces and establish a Kac/Loeve Theorem for bi-free independence. In addition, we prove that bi-freeness is preserved under tensoring with matrices. Finally, via combinatorial arguments, we construct partial $R$-transforms in two settings relating the moments and cumulants of a left-right pair of operators.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"82 1","pages":"1437-1473"},"PeriodicalIF":1.5,"publicationDate":"2014-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78990502","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 prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin (J. Stat. Phys. 98 (2000) 131-244). We do so by establishing the convergence of certain exploration processes towards SLE(3, -3/2, -3/2). We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield (Duke Math. J. 147 (2009) 79-129).
{"title":"Conformal invariance of crossing probabilities for the Ising model with free boundary conditions","authors":"S. Benoist, H. Duminil-Copin, C. Hongler","doi":"10.1214/15-AIHP698","DOIUrl":"https://doi.org/10.1214/15-AIHP698","url":null,"abstract":"We prove that crossing probabilities for the critical planar Ising model with free boundary conditions are conformally invariant in the scaling limit, a phenomenon first investigated numerically by Langlands, Lewis and Saint-Aubin (J. Stat. Phys. 98 (2000) 131-244). We do so by establishing the convergence of certain exploration processes towards SLE(3, -3/2, -3/2). We also construct an exploration tree for free boundary conditions, analogous to the one introduced by Sheffield (Duke Math. J. 147 (2009) 79-129).","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"1 1","pages":"1784-1798"},"PeriodicalIF":1.5,"publicationDate":"2014-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76705481","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}
Let $(varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T= sum_{t=s+1}^{s+T}varepsilon_tvarepsilon^T_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $varepsilon_t$. Since $X_T$ is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of $X_TX^T_T$. Therefore, the purpose of this paper is to investigate the limiting behaviors of the eigenvalues of $X_TX^T_T$ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit $F$. Second, we establish the convergence of its largest eigenvalue to the right edge of $F$. Both results are derived using moment methods.
{"title":"Moment approach for singular values distribution of a large auto-covariance matrix","authors":"Qinwen Wang, Jianfeng Yao","doi":"10.1214/15-AIHP693","DOIUrl":"https://doi.org/10.1214/15-AIHP693","url":null,"abstract":"Let $(varepsilon_{t})_{t>0}$ be a sequence of independent real random vectors of $p$-dimension and let $X_T= sum_{t=s+1}^{s+T}varepsilon_tvarepsilon^T_{t-s}/T$ be the lag-$s$ ($s$ is a fixed positive integer) auto-covariance matrix of $varepsilon_t$. Since $X_T$ is not symmetric, we consider its singular values, which are the square roots of the eigenvalues of $X_TX^T_T$. Therefore, the purpose of this paper is to investigate the limiting behaviors of the eigenvalues of $X_TX^T_T$ in two aspects. First, we show that the empirical spectral distribution of its eigenvalues converges to a nonrandom limit $F$. Second, we establish the convergence of its largest eigenvalue to the right edge of $F$. Both results are derived using moment methods.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"135 1","pages":"1641-1666"},"PeriodicalIF":1.5,"publicationDate":"2014-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86377332","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}
Let $X_1, X_2, ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: mathbb R to mathbb R_+$. We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{lfloor N t rfloor})_{t in mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t in mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N to infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.
设$X_1, X_2, ldots$为均值为零的i.i.d实值随机变量序列，并考虑形式为$Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$的缩放随机游走，其中$a_N: mathbb R to mathbb R_+$。我们表明，在对$X_i$定律的温和假设下，我们可以这样选择尺度因子$a_N$，使过程$(Y^N_{lfloor N t rfloor})_{t in mathbb R_+}$在分布上收敛于给定的扩散$(M_t)_{t in mathbb R_+}$，求解一个可能具有不规则系数的随机微分方程，如$N to infty$。为此，我们将缩放的随机漫步嵌入到扩散中$M$，并具有期望时间步长$1/N$的停止时间序列。
{"title":"A functional limit theorem for irregular SDEs","authors":"S. Ankirchner, T. Kruse, M. Urusov","doi":"10.1214/16-AIHP760","DOIUrl":"https://doi.org/10.1214/16-AIHP760","url":null,"abstract":"Let $X_1, X_2, ldots$ be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the form $Y^N_{k+1} = Y^N_{k} + a_N(Y^N_k) X_{k+1}$, where $a_N: mathbb R to mathbb R_+$. We show, under mild assumptions on the law of $X_i$, that one can choose the scale factor $a_N$ in such a way that the process $(Y^N_{lfloor N t rfloor})_{t in mathbb R_+}$ converges in distribution to a given diffusion $(M_t)_{t in mathbb R_+}$ solving a stochastic differential equation with possibly irregular coefficients, as $N to infty$. To this end we embed the scaled random walks into the diffusion $M$ with a sequence of stopping times with expected time step $1/N$.","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"9 1","pages":"1438-1457"},"PeriodicalIF":1.5,"publicationDate":"2014-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74321672","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}
{"title":"Odd cutsets and the hard-core model on $mathbb{Z}^{d}$","authors":"R. Peled, W. Samotij","doi":"10.1214/12-AIHP535","DOIUrl":"https://doi.org/10.1214/12-AIHP535","url":null,"abstract":"","PeriodicalId":7902,"journal":{"name":"Annales De L Institut Henri Poincare-probabilites Et Statistiques","volume":"316 1","pages":"975-998"},"PeriodicalIF":1.5,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76454341","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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