In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy--Smirnov formula. This theorem, together with the introduction of Schramm--Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $theta(p)$ for site percolation on the triangular lattice behaves like $(p-p_c)^{5/36+o(1)}$ as $psearrow p_c=1/2$.
{"title":"Planar percolation with a glimpse of Schramm–Loewner evolution","authors":"V. Beffara, H. Duminil-Copin","doi":"10.1214/11-PS186","DOIUrl":"https://doi.org/10.1214/11-PS186","url":null,"abstract":"In recent years, important progress has been made in the field of two-dimensional statistical physics. One of the most striking achievements is the proof of the Cardy--Smirnov formula. This theorem, together with the introduction of Schramm--Loewner Evolution and techniques developed over the years in percolation, allow precise descriptions of the critical and near-critical regimes of the model. This survey aims to describe the different steps leading to the proof that the infinite-cluster density $theta(p)$ for site percolation on the triangular lattice behaves like $(p-p_c)^{5/36+o(1)}$ as $psearrow p_c=1/2$.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"10 1","pages":"1-50"},"PeriodicalIF":1.6,"publicationDate":"2011-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS186","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966460","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 : 2011-02-01DOI: 10.1090/S0002-9939-2010-10505-1
J. Ball, V. Bolotnikov, S. Horst
A general interpolation problem with operator argument is studied for functions from the de Branges-Rovnyak space associated with an analytic function mapping the open unit disk into the closed unit disk. The interpolation condition is taken in the Rosenblum-Rovnyak form (with a suitable interpretation of ) for given Hilbert space operator and two vectors from the same space.
{"title":"Interpolation in de Branges-Rovnyak spaces","authors":"J. Ball, V. Bolotnikov, S. Horst","doi":"10.1090/S0002-9939-2010-10505-1","DOIUrl":"https://doi.org/10.1090/S0002-9939-2010-10505-1","url":null,"abstract":"A general interpolation problem with operator argument is studied for functions from the de Branges-Rovnyak space associated with an analytic function mapping the open unit disk into the closed unit disk. The interpolation condition is taken in the Rosenblum-Rovnyak form (with a suitable interpretation of ) for given Hilbert space operator and two vectors from the same space.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"1 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2011-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1090/S0002-9939-2010-10505-1","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"60558324","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}
Abstract: Based on the first author’s recent PhD thesis entitled “Profiling processes of Meixner type”, [50] a review of the main characteristics and characterizations of such particular Lévy processes is extracted, emphasizing the motivations for their introduction in literature as reliable financial models. An insight on orthogonal polynomials is also provided, together with an alternative path for defining the same processes. Also, an attempt of simulation of their trajectories is introduced by means of an original R simulation routine.
摘要:基于第一作者最近发表的博士论文“Profiling processes of Meixner type”，本文综述了这类特殊的l)过程的主要特征和特征，强调了它们作为可靠的金融模型被引入文献的动机。还提供了对正交多项式的见解，以及定义相同过程的替代路径。此外，还介绍了利用原始的R仿真程序对其轨迹进行仿真的尝试。
{"title":"Reviewing alternative characterizations of Meixner process","authors":"E. Mazzola, P. Muliere","doi":"10.1214/11-PS177","DOIUrl":"https://doi.org/10.1214/11-PS177","url":null,"abstract":"Abstract: Based on the first author’s recent PhD thesis entitled “Profiling processes of Meixner type”, [50] a review of the main characteristics and characterizations of such particular Lévy processes is extracted, emphasizing the motivations for their introduction in literature as reliable financial models. An insight on orthogonal polynomials is also provided, together with an alternative path for defining the same processes. Also, an attempt of simulation of their trajectories is introduced by means of an original R simulation routine.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"127-154"},"PeriodicalIF":1.6,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS177","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965966","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}
Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford's Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.
{"title":"A basic theory of Benford's Law ∗","authors":"A. Berger, T. Hill","doi":"10.1214/11-PS175","DOIUrl":"https://doi.org/10.1214/11-PS175","url":null,"abstract":"Drawing from a large, diverse body of work, this survey presents a comprehensive and unified introduction to the mathematics underlying the prevalent logarithmic distribution of significant digits and significands, often referred to as Benford's Law (BL) or, in a special case, as the First Digit Law. The invariance properties that characterize BL are developed in detail. Special attention is given to the emergence of BL in a wide variety of deterministic and random processes. Though mainly expository in nature, the article also provides strengthened versions of, and simplified proofs for, many key results in the literature. Numerous intriguing problems for future research arise naturally.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"1-126"},"PeriodicalIF":1.6,"publicationDate":"2011-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS175","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965852","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}
Supplementary references and material are provided to the paper entitled `Moments of Gamma type and the Brownian supremum process area', published in Probability Surveys 7 (2010) 1–52 .
{"title":"Addendum to Moments of Gamma type and the Brownian supremum process area","authors":"S. Janson","doi":"10.1214/10-PS169","DOIUrl":"https://doi.org/10.1214/10-PS169","url":null,"abstract":"Supplementary references and material are provided to the paper entitled \u0000 `Moments of Gamma type and the Brownian supremum process area', published in Probability Surveys 7 (2010) 1–52 .","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"7 1","pages":"207-208"},"PeriodicalIF":1.6,"publicationDate":"2010-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65949099","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}
This article is about the connection between enumerative combinatorics and equilibrium statistical mechanics. The combinatorics side concerns species of combinatorial structures and the associated exponential generating functions. The passage from species to generating functions is a combinatorial analog of the Fourier transform. Indeed, there is a convolution multiplication on species that is mapped to a pointwise multiplication of the exponential generating functions. The statistical mechanics side deals with a probability model of an equilibrium gas. The cluster expansion that gives the density of the gas is the exponential generating function for the species of rooted connected graphs. The main results of the theory are simple criteria that guarantee the convergence of this expansion. It turns out that other problems in combinatorics and statistical mechanics can be translated to this gas setting, so it is a universal prescription for dealing with systems of high dimension.
{"title":"Combinatorics and cluster expansions","authors":"William G. Faris","doi":"10.1214/10-PS159","DOIUrl":"https://doi.org/10.1214/10-PS159","url":null,"abstract":"This article is about the connection between enumerative combinatorics and equilibrium statistical mechanics. The combinatorics side \u0000concerns species of combinatorial structures and the associated exponential generating functions. The passage from species to generating functions \u0000is a combinatorial analog of the Fourier transform. Indeed, there is a convolution multiplication on species that is mapped to a pointwise multiplication of the exponential generating functions. The statistical mechanics side \u0000deals with a probability model of an equilibrium gas. The cluster expansion \u0000that gives the density of the gas is the exponential generating function for \u0000the species of rooted connected graphs. The main results of the theory are \u0000simple criteria that guarantee the convergence of this expansion. It turns \u0000out that other problems in combinatorics and statistical mechanics can be \u0000translated to this gas setting, so it is a universal prescription for dealing \u0000with systems of high dimension.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"7 1","pages":"157-206"},"PeriodicalIF":1.6,"publicationDate":"2010-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/10-PS159","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65949173","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 this course we will present the full proof of the fact that every smooth dynamical system on the interval or circle X , constituted by the forward iterates of a function f : X → X which is of class C r with r > 1, admits a symbolic extension, i.e., there exists a bilateral subshift ( Y , S ) with Y a closed shift-invariant subset of Λ ℤ , where Λ is a finite alphabet, and a continuous surjection π : Y → X which intertwines the action of f (on X ) with that of the shift map S (on Y ). Moreover, we give a precise estimate (from above) on the entropy of each invariant measure ν supported by Y in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure μ on X , the "Lyapunov exponent" of μ (the genuine Lyapunov exponent for ergodic μ, otherwise its analog), and the smoothness parameter r . This estimate agrees with a conjecture formulated in [15] around 2003 for smooth dynamical systems on manifolds.
在本课程中我们将完整的证明,每一个区间或圆X光滑动力系统,由正向迭代函数f:→X的类C r r > 1,承认一个象征性的扩展,也就是说,存在一个双边构造(Y, S)与Y的一个封闭的移不变的子集Λℤ,Λ有限字母表,和一个连续满射π:Y→X的行动与f (X)与转变地图(Y)。此外，我们给出了在优化的符号扩展中Y支持的每个不变测度ν的熵的精确估计(从上面)。这个估计取决于底层度量μ在X上的熵，μ的“Lyapunov指数”(遍历μ的真正Lyapunov指数，否则它的模拟)和平滑参数r。这个估计与[15]在2003年左右对流形上的光滑动力系统提出的一个猜想一致。
{"title":"Symbolic extensions of smooth interval maps","authors":"T. Downarowicz, Poland","doi":"10.1214/10-PS164","DOIUrl":"https://doi.org/10.1214/10-PS164","url":null,"abstract":"In this course we will present the full proof of the fact that every \u0000smooth dynamical system on the interval or circle X , constituted by the \u0000forward iterates of a function f : X → X which is of class C r with r > 1, \u0000admits a symbolic extension, i.e., there exists a bilateral subshift ( Y , S ) with \u0000 Y a closed shift-invariant subset of Λ ℤ , where Λ is a finite alphabet, and a \u0000continuous surjection π : Y → X which intertwines the action of f (on X ) \u0000with that of the shift map S (on Y ). Moreover, we give a precise estimate \u0000(from above) on the entropy of each invariant measure ν supported by Y \u0000in an optimized symbolic extension. This estimate depends on the entropy \u0000of the underlying measure μ on X , the \"Lyapunov exponent\" of μ (the \u0000genuine Lyapunov exponent for ergodic μ, otherwise its analog), and the \u0000smoothness parameter r . This estimate agrees with a conjecture formulated \u0000in [15] around 2003 for smooth dynamical systems on manifolds.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"16 1","pages":"84-104"},"PeriodicalIF":1.6,"publicationDate":"2010-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/10-PS164","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65949014","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 positive random variables whose moments can be expressed by products and quotients of Gamma functions; this includes many standard distributions. General results are given on existence, series expansion and asymptotics of density functions. It is shown that the integral of the supremum process of Brownian motion has moments of this type, as well as a related random variable occurring in the study of hashing with linear displacement, and the general results are applied to these variables. Addendum: An addendum is published in Probability Surveys 7 (2010) 207–208 .
{"title":"Moments of Gamma type and the Brownian supremum process area","authors":"S. Janson","doi":"10.1214/10-PS160","DOIUrl":"https://doi.org/10.1214/10-PS160","url":null,"abstract":"We study positive random variables whose moments can be \u0000expressed by products and quotients of Gamma functions; this includes \u0000many standard distributions. General results are given on existence, series \u0000expansion and asymptotics of density functions. It is shown that the integral \u0000of the supremum process of Brownian motion has moments of this type, as \u0000well as a related random variable occurring in the study of hashing with \u0000linear displacement, and the general results are applied to these variables. Addendum: An addendum is published in Probability Surveys 7 (2010) 207–208 .","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"7 1","pages":"1-52"},"PeriodicalIF":1.6,"publicationDate":"2010-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/10-PS160","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65949236","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 a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.
{"title":"Limit theorems for discrete-time metapopulation models","authors":"F. Buckley, P. Pollett","doi":"10.1214/10-PS158","DOIUrl":"https://doi.org/10.1214/10-PS158","url":null,"abstract":"We describe a class of one-dimensional chain binomial models of use in studying metapopulations (population networks). Limit theorems are established for time-inhomogeneous Markov chains that share the salient features of these models. We prove a law of large numbers, which can be used to identify an approximating deterministic trajectory, and a central limit theorem, which establishes that the scaled fluctuations about this trajectory have an approximating autoregressive structure.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"7 1","pages":"53-83"},"PeriodicalIF":1.6,"publicationDate":"2010-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65949159","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}
This is an introductory account of the emergence of conformal invariance in the scaling limit of planar critical percolation. We give an exposition of Smirnov's theorem (2001) on the conformal invariance of crossing probabilities in site percolation on the triangular lattice. We also give an introductory account of Schramm-Loewner evolutions (SLE ĸ ), a one-parameter family of conformally invariant random curves discovered by Schramm (2000). The article is organized around the aim of proving the result, due to Smirnov (2001) and to Camia and Newman (2007), that the percolation exploration path converges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.
{"title":"Conformally invariant scaling limits in planar critical percolation","authors":"Nike Sun","doi":"10.1214//11-PS180","DOIUrl":"https://doi.org/10.1214//11-PS180","url":null,"abstract":"This is an introductory account of the emergence of conformal \u0000invariance in the scaling limit of planar critical percolation. We give \u0000an exposition of Smirnov's theorem (2001) on the conformal invariance \u0000of crossing probabilities in site percolation on the triangular \u0000lattice. We also give an introductory account of Schramm-Loewner \u0000evolutions (SLE ĸ ), a one-parameter family of conformally \u0000invariant random curves discovered by Schramm (2000). The article is \u0000organized around the aim of proving the result, due to Smirnov (2001) \u0000and to Camia and Newman (2007), that the percolation exploration path \u0000converges in the scaling limit to chordal SLE 6 . No prior knowledge is assumed beyond some general complex analysis and probability theory.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"155-209"},"PeriodicalIF":1.6,"publicationDate":"2009-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66432054","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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