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Planar percolation with a glimpse of Schramm–Loewner evolution 平面渗流与Schramm-Loewner演化的一瞥
IF 1.6
Probability Surveys Pub Date : 2011-07-01 DOI: 10.1214/11-PS186
V. Beffara, H. Duminil-Copin
{"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":null,"pages":null},"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}
引用次数: 26
Interpolation in de Branges-Rovnyak spaces de Branges-Rovnyak空间中的插值
IF 1.6
Probability Surveys Pub Date : 2011-02-01 DOI: 10.1090/S0002-9939-2010-10505-1
J. Ball, V. Bolotnikov, S. Horst
{"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":null,"pages":null},"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}
引用次数: 14
Reviewing alternative characterizations of Meixner process 回顾迈克斯纳工艺的不同特征
IF 1.6
Probability Surveys Pub Date : 2011-01-01 DOI: 10.1214/11-PS177
E. Mazzola, P. Muliere
{"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":null,"pages":null},"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}
引用次数: 9
A basic theory of Benford's Law ∗ 本福德定律的基本理论
IF 1.6
Probability Surveys Pub Date : 2011-01-01 DOI: 10.1214/11-PS175
A. Berger, T. Hill
{"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":null,"pages":null},"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}
引用次数: 93
Addendum to Moments of Gamma type and the Brownian supremum process area 伽玛型矩和布朗最高过程域的附录
IF 1.6
Probability Surveys Pub Date : 2010-12-13 DOI: 10.1214/10-PS169
S. Janson
{"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":null,"pages":null},"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}
引用次数: 1
Combinatorics and cluster expansions 组合学和簇展开
IF 1.6
Probability Surveys Pub Date : 2010-06-07 DOI: 10.1214/10-PS159
William G. Faris
{"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":null,"pages":null},"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}
引用次数: 30
Symbolic extensions of smooth interval maps 平滑区间映射的符号扩展
IF 1.6
Probability Surveys Pub Date : 2010-05-18 DOI: 10.1214/10-PS164
T. Downarowicz, Poland
{"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":null,"pages":null},"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}
引用次数: 4
Moments of Gamma type and the Brownian supremum process area 伽玛型矩和布朗最高过程域
IF 1.6
Probability Surveys Pub Date : 2010-02-22 DOI: 10.1214/10-PS160
S. Janson
{"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":null,"pages":null},"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}
引用次数: 48
Limit theorems for discrete-time metapopulation models 离散时间元人口模型的极限定理
IF 1.6
Probability Surveys Pub Date : 2010-01-01 DOI: 10.1214/10-PS158
F. Buckley, P. Pollett
{"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":null,"pages":null},"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}
引用次数: 46
Conformally invariant scaling limits in planar critical percolation 平面临界渗流的保形不变尺度极限
IF 1.6
Probability Surveys Pub Date : 2009-11-02 DOI: 10.1214//11-PS180
Nike Sun
{"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":null,"pages":null},"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}
引用次数: 28
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