These notes are from my mini-courses given at the PIMS summer school in 2010 at the University of Washington and at the Cornell probability summer school in 2011. The goal was to give an introduction to the Schramm-Loewner evolution to graduate students with background in probability. This is not intended to be a comprehensive survey of SLE.
{"title":"Scaling limits and the Schramm-Loewner evolution","authors":"G. Lawler","doi":"10.1214/11-PS189","DOIUrl":"https://doi.org/10.1214/11-PS189","url":null,"abstract":"These notes are from my mini-courses given at the PIMS summer school in 2010 at the University of Washington and at the Cornell probability summer school in 2011. The goal was to give an introduction to the Schramm-Loewner evolution to graduate students with background in probability. This is not intended to be a comprehensive survey of SLE.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"442-495"},"PeriodicalIF":1.6,"publicationDate":"2011-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966270","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 survey concerns the study of quasi-stationary distributions with a specific focus on models derived from ecology and population dynamics. We are concerned with the long time behavior of different stochastic population size processes when 0 is an absorbing point almost surely attained by the process. The hitting time of this point, namely the extinction time, can be large compared to the physical time and the population size can fluctuate for large amount of time before extinction actually occurs. This phenomenon can be understood by the study of quasi-limiting distributions. In this paper, general results on quasi-stationarity are given and examples developed in detail. One shows in particular how this notion is related to the spectral properties of the semi-group of the process killed at 0. Then we study different stochastic population models including nonlinear terms modeling the regulation of the population. These models will take values in countable sets (as birth and death processes) or in continuous spaces (as logistic Feller diffusion processes or stochastic Lotka-Volterra processes). In all these situations we study in detail the quasi-stationarity properties. We also develop an algorithm based on Fleming-Viot particle systems and show a lot of numerical pictures.
{"title":"Quasi-stationary distributions and population processes","authors":"S. M'el'eard, D. Villemonais","doi":"10.1214/11-PS191","DOIUrl":"https://doi.org/10.1214/11-PS191","url":null,"abstract":"This survey concerns the study of quasi-stationary distributions with \u0000a specific focus on models derived from ecology and population \u0000dynamics. We are concerned with the long time behavior of different \u0000stochastic population size processes when 0 is an absorbing point \u0000almost surely attained by the process. The hitting time of this point, \u0000namely the extinction time, can be large compared to the physical time \u0000and the population size can fluctuate for large amount of time before \u0000extinction actually occurs. This phenomenon can be understood by the \u0000study of quasi-limiting distributions. In this paper, general results \u0000on quasi-stationarity are given and examples developed in detail. One \u0000shows in particular how this notion is related to the spectral \u0000properties of the semi-group of the process killed at 0. Then we \u0000study different stochastic population models including nonlinear terms \u0000modeling the regulation of the population. These models will take \u0000values in countable sets (as birth and death processes) or in \u0000continuous spaces (as logistic Feller diffusion processes or \u0000stochastic Lotka-Volterra processes). In all these situations we study \u0000in detail the quasi-stationarity properties. We also develop an \u0000algorithm based on Fleming-Viot particle systems and show a lot of \u0000numerical pictures.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"18 1","pages":"340-410"},"PeriodicalIF":1.6,"publicationDate":"2011-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS191","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966288","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 these notes, we discuss a selection of topics on several models of planar statistical mechanics. We consider the Ising, Potts, and more generally abelian spin models; the discrete Gaussian free field; the random cluster model; and the six-vertex model. Emphasis is put on duality, order, disorder and spinor variables, and on mappings between these models.
{"title":"Topics on abelian spin models and related problems","authors":"Julien Dub'edat","doi":"10.1214/11-PS187","DOIUrl":"https://doi.org/10.1214/11-PS187","url":null,"abstract":"In these notes, we discuss a selection of topics on several models of \u0000planar statistical mechanics. We consider the Ising, Potts, and more \u0000generally abelian spin models; the discrete Gaussian free field; the \u0000random cluster model; and the six-vertex model. Emphasis is put on \u0000duality, order, disorder and spinor variables, and on mappings between \u0000these models.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"374-402"},"PeriodicalIF":1.6,"publicationDate":"2011-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966108","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 give a unified treatment of the limit, as the size tends to infinity, of simply generated random trees, including both the well-known result in the standard case of critical Galton–Watson trees and similar but less well-known results in the other cases (i.e., when no equivalent critical Galton–Watson tree exists). There is a well-defined limit in the form of an infinite random tree in all cases; for critical Galton–Watson trees this tree is locally finite but for the other cases the random limit has exactly one node of infinite degree. The proofs use a well-known connection to a random allocation model that we call balls-in-boxes, and we prove corresponding theorems for this model. This survey paper contains many known results from many different sources, together with some new results.
{"title":"Simply generated trees, conditioned Galton–Watson trees, random allocations and condensation","authors":"S. Janson","doi":"10.1214/11-PS188","DOIUrl":"https://doi.org/10.1214/11-PS188","url":null,"abstract":"We give a unified treatment of the limit, as the size tends to infinity, \u0000of simply generated random trees, \u0000including both \u0000the well-known result in the standard case \u0000of critical Galton–Watson trees \u0000and similar but less well-known results in the \u0000other cases (i.e., when no equivalent critical Galton–Watson tree exists). \u0000There is a well-defined limit in the form of an infinite \u0000random tree in all cases; \u0000for critical Galton–Watson trees this tree is locally finite but for the \u0000other cases the random limit has exactly one node of infinite degree. \u0000 \u0000The proofs \u0000use a well-known connection to \u0000a random allocation model that we call balls-in-boxes, and \u0000we prove corresponding theorems for this model. \u0000 \u0000This survey paper contains \u0000many known results from many different sources, together \u0000with some new results.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"103-252"},"PeriodicalIF":1.6,"publicationDate":"2011-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS188","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966130","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}
Recent progress on the understanding of the Random Conductance Model is reviewed and commented. A particular emphasis is on the results on the scaling limit of the random walk among random conductances for almost every realization of the environment, observations on the behavior of the effective resistance as well as the scaling limit of certain models of gradient fields with non-convex interactions. The text is an expanded version of the lecture notes for a course delivered at the 2011 Cornell Summer School on Probability.
{"title":"Recent progress on the Random Conductance Model","authors":"M. Biskup","doi":"10.1214/11-PS190","DOIUrl":"https://doi.org/10.1214/11-PS190","url":null,"abstract":"Recent progress on the understanding of the Random Conductance Model is \u0000reviewed and commented. A particular emphasis is on the results on the \u0000scaling limit of the random walk among random conductances for almost \u0000every realization of the environment, observations on the behavior of \u0000the effective resistance as well as the scaling limit of certain models \u0000of gradient fields with non-convex interactions. The text is an \u0000expanded version of the lecture notes for a course delivered at the \u00002011 Cornell Summer School on Probability.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"294-373"},"PeriodicalIF":1.6,"publicationDate":"2011-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS190","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966279","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}
These notes are focused on three recent results in discrete ran- dom geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is p 2 + √ 2; the proof by the author and Manolescu of the universality of inhomogeneous bond percola- tion on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on Z 2 is √ q/(1 + √ q). Background information on the relevant random pro- cesses is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs. AMS 2000 subject classifications: Primary 60K35; secondary 82B43. Keywords and phrases: Self-avoiding walk, connective constant, per- colation, random-cluster model, Ising model, star-triangle transformation, Yang-Baxter equation, critical exponent, universality, isoradiality.
{"title":"Three theorems in discrete random geometry","authors":"G. Grimmett","doi":"10.1214/11-PS185","DOIUrl":"https://doi.org/10.1214/11-PS185","url":null,"abstract":"These notes are focused on three recent results in discrete ran- dom geometry, namely: the proof by Duminil-Copin and Smirnov that the connective constant of the hexagonal lattice is p 2 + √ 2; the proof by the author and Manolescu of the universality of inhomogeneous bond percola- tion on the square, triangular, and hexagonal lattices; the proof by Beffara and Duminil-Copin that the critical point of the random-cluster model on Z 2 is √ q/(1 + √ q). Background information on the relevant random pro- cesses is presented on route to these theorems. The emphasis is upon the communication of ideas and connections as well as upon the detailed proofs. AMS 2000 subject classifications: Primary 60K35; secondary 82B43. Keywords and phrases: Self-avoiding walk, connective constant, per- colation, random-cluster model, Ising model, star-triangle transformation, Yang-Baxter equation, critical exponent, universality, isoradiality.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"403-441"},"PeriodicalIF":1.6,"publicationDate":"2011-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65966314","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}
These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.
{"title":"Around the circular law","authors":"C. Bordenave, Djalil CHAFAÏ","doi":"10.1214/11-PS183","DOIUrl":"https://doi.org/10.1214/11-PS183","url":null,"abstract":"These expository notes are centered around the circular law theorem, which states that the empirical spectral distribution of a nxn random matrix with i.i.d. entries of variance 1/n tends to the uniform law on the unit disc of the complex plane as the dimension $n$ tends to infinity. This phenomenon is the non-Hermitian counterpart of the semi circular limit for Wigner random Hermitian matrices, and the quarter circular limit for Marchenko-Pastur random covariance matrices. We present a proof in a Gaussian case, due to Silverstein, based on a formula by Ginibre, and a proof of the universal case by revisiting the approach of Tao and Vu, based on the Hermitization of Girko, the logarithmic potential, and the control of the small singular values. Beyond the finite variance model, we also consider the case where the entries have heavy tails, by using the objective method of Aldous and Steele borrowed from randomized combinatorial optimization. The limiting law is then no longer the circular law and is related to the Poisson weighted infinite tree. We provide a weak control of the smallest singular value under weak assumptions, using asymptotic geometric analysis tools. We also develop a quaternionic Cauchy-Stieltjes transform borrowed from the Physics literature.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"1-89"},"PeriodicalIF":1.6,"publicationDate":"2011-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS183","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965884","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 survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning raduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein's method literature.
{"title":"Fundamentals of Stein's method","authors":"Nathan Ross","doi":"10.1214/11-PS182","DOIUrl":"https://doi.org/10.1214/11-PS182","url":null,"abstract":"This survey article discusses the main concepts and techniques of Stein's method for distributional approximation by the normal, Poisson, exponential, and geometric distributions, and also its relation to concentration of measure inequalities. The material is presented at a level accessible to beginning raduate students studying probability with the main emphasis on the themes that are common to these topics and also to much of the Stein's method literature.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"8 1","pages":"210-293"},"PeriodicalIF":1.6,"publicationDate":"2011-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS182","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965720","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}
The theory of orthogonal polynomials on the unit circle (OPUC) dates back to Szego's work of 1915-21, and has been given a great impetus by the recent work of Simon, in particular his survey paper and three recent books; we allude to the title of the third of these, Szego's theorem and its descendants , in ours. Simon's motivation comes from spectral theory and analysis. Another major area of application of OPUC comes from probability, statistics, time series and prediction theory; see for instance the classic book by Grenander and Szego, Toeplitz forms and their applications . Coming to the subject from this background, our aim here is to complement this recent work by giving some probabilistically motivated results. We also advocate a new definition of long-range dependence.
{"title":"Szegö's theorem and its probabilistic descendants","authors":"N. Bingham","doi":"10.1214/11-PS178","DOIUrl":"https://doi.org/10.1214/11-PS178","url":null,"abstract":"The theory of orthogonal polynomials on the unit circle (OPUC) \u0000dates back to Szego's work of 1915-21, and has been given a \u0000great impetus by the recent work of Simon, in particular his \u0000survey paper and three recent books; we allude to the title of the \u0000third of these, \u0000 Szego's theorem and its descendants , in ours. \u0000Simon's motivation comes from spectral theory and analysis. Another major \u0000area of application of OPUC comes from probability, statistics, \u0000time series and prediction theory; see for instance the classic book by \u0000Grenander and Szego, Toeplitz forms and their applications . \u0000Coming to the subject from this \u0000background, our aim here is to complement this recent work by giving some \u0000probabilistically motivated results. We also advocate a new definition \u0000of long-range dependence.","PeriodicalId":46216,"journal":{"name":"Probability Surveys","volume":"9 1","pages":"287-324"},"PeriodicalIF":1.6,"publicationDate":"2011-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/11-PS178","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"65965526","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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