We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a $sigma$-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a L'evy-It^o representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.
{"title":"The cut-and-paste process","authors":"Harry Crane","doi":"10.1214/14-AOP922","DOIUrl":"https://doi.org/10.1214/14-AOP922","url":null,"abstract":"We characterize the class of exchangeable Feller processes evolving on partitions with boundedly many blocks. In continuous-time, the jump measure decomposes into two parts: a $sigma$-finite measure on stochastic matrices and a collection of nonnegative real constants. This decomposition prompts a L'evy-It^o representation. In discrete-time, the evolution is described more simply by a product of independent, identically distributed random matrices.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"42 1","pages":"1952-1979"},"PeriodicalIF":2.3,"publicationDate":"2014-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/14-AOP922","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66006877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron–Stein inequalities developed by Boucheron et al. The proofs rely on the method of exchangeable pairs.
{"title":"Efron–Stein inequalities for random matrices","authors":"D. Paulin, Lester W. Mackey, J. Tropp","doi":"10.1214/15-AOP1054","DOIUrl":"https://doi.org/10.1214/15-AOP1054","url":null,"abstract":"This paper establishes new concentration inequalities for random matrices constructed from independent random variables. These results are analogous with the generalized Efron–Stein inequalities developed by Boucheron et al. The proofs rely on the method of exchangeable pairs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3431-3473"},"PeriodicalIF":2.3,"publicationDate":"2014-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Elisabetta Candellero, S. Ganguly, C. Hoffman, Lionel Levine
We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting withn oil" andn water" particles at the origin, the particles diuse in Z according to the following rule: whenever some site x2 Z has at least 1 oil and at least 1 water particle present, it res by sending 1 oil particle and 1 water particle each to an independent random neighbor x 1. Firing continues until every site has at most one type of particles. We establish the correct order for several statistics of this model and identify the scaling limit under assumption of existence.
{"title":"Oil and water: A two-type internal aggregation model","authors":"Elisabetta Candellero, S. Ganguly, C. Hoffman, Lionel Levine","doi":"10.1214/16-AOP1157","DOIUrl":"https://doi.org/10.1214/16-AOP1157","url":null,"abstract":"We introduce a two-type internal DLA model which is an example of a non-unary abelian network. Starting withn oil\" andn water\" particles at the origin, the particles diuse in Z according to the following rule: whenever some site x2 Z has at least 1 oil and at least 1 water particle present, it res by sending 1 oil particle and 1 water particle each to an independent random neighbor x 1. Firing continues until every site has at most one type of particles. We establish the correct order for several statistics of this model and identify the scaling limit under assumption of existence.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"4019-4070"},"PeriodicalIF":2.3,"publicationDate":"2014-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1157","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047864","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We extend the LpLp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of WW-random graphs.
{"title":"An $L^{p}$ theory of sparse graph convergence II: LD convergence, quotients and right convergence","authors":"C. Borgs, J. Chayes, Henry Cohn, Yufei Zhao","doi":"10.1214/17-AOP1187","DOIUrl":"https://doi.org/10.1214/17-AOP1187","url":null,"abstract":"We extend the LpLp theory of sparse graph limits, which was introduced in a companion paper, by analyzing different notions of convergence. Under suitable restrictions on node weights, we prove the equivalence of metric convergence, quotient convergence, microcanonical ground state energy convergence, microcanonical free energy convergence and large deviation convergence. Our theorems extend the broad applicability of dense graph convergence to all sparse graphs with unbounded average degree, while the proofs require new techniques based on uniform upper regularity. Examples to which our theory applies include stochastic block models, power law graphs and sparse versions of WW-random graphs.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"14 1","pages":"337-396"},"PeriodicalIF":2.3,"publicationDate":"2014-08-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/17-AOP1187","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66060699","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.
{"title":"Mean field games with common noise","authors":"R. Carmona, F. Delarue, D. Lacker","doi":"10.1214/15-AOP1060","DOIUrl":"https://doi.org/10.1214/15-AOP1060","url":null,"abstract":"A theory of existence and uniqueness is developed for general stochastic differential mean field games with common noise. The concepts of strong and weak solutions are introduced in analogy with the theory of stochastic differential equations, and existence of weak solutions for mean field games is shown to hold under very general assumptions. Examples and counter-examples are provided to enlighten the underpinnings of the existence theory. Finally, an analog of the famous result of Yamada and Watanabe is derived, and it is used to prove existence and uniqueness of a strong solution under additional assumptions.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3740-3803"},"PeriodicalIF":2.3,"publicationDate":"2014-07-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1060","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.
{"title":"Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients","authors":"Longjie Xie, Xicheng Zhang","doi":"10.1214/15-AOP1057","DOIUrl":"https://doi.org/10.1214/15-AOP1057","url":null,"abstract":"By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"3661-3687"},"PeriodicalIF":2.3,"publicationDate":"2014-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1057","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032574","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β>0β>0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph.We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree dd, the Ising model has cutoff provided that β<κ/dβ<κ/d for some absolute constant κκ (a result which, up to the value of κκ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1)O(1), just as when β=0β=0.Finally, the mixing time from almost every initial state is not more than a factor of 1+eβ1+eβ faster then the worst one (with eβ→0eβ→0 as β→0β→0), whereas the uniform starting state is at least 2−eβ2−eβ times faster.
{"title":"Universality of cutoff for the ising model","authors":"E. Lubetzky, A. Sly","doi":"10.1214/16-AOP1146","DOIUrl":"https://doi.org/10.1214/16-AOP1146","url":null,"abstract":"On any locally-finite geometry, the stochastic Ising model is known to be contractive when the inverse-temperature ββ is small enough, via classical results of Dobrushin and of Holley in the 1970s. By a general principle proposed by Peres, the dynamics is then expected to exhibit cutoff. However, so far cutoff for the Ising model has been confirmed mainly for lattices, heavily relying on amenability and log Sobolev inequalities. Without these, cutoff was unknown at any fixed β>0β>0, no matter how small, even in basic examples such as the Ising model on a binary tree or a random regular graph.We use the new framework of information percolation to show that, in any geometry, there is cutoff for the Ising model at high enough temperatures. Precisely, on any sequence of graphs with maximum degree dd, the Ising model has cutoff provided that β<κ/dβ<κ/d for some absolute constant κκ (a result which, up to the value of κκ, is best possible). Moreover, the cutoff location is established as the time at which the sum of squared magnetizations drops to 1, and the cutoff window is O(1)O(1), just as when β=0β=0.Finally, the mixing time from almost every initial state is not more than a factor of 1+eβ1+eβ faster then the worst one (with eβ→0eβ→0 as β→0β→0), whereas the uniform starting state is at least 2−eβ2−eβ times faster.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"45 1","pages":"3664-3696"},"PeriodicalIF":2.3,"publicationDate":"2014-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/16-AOP1146","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66047787","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.
{"title":"Bulk universality for deformed Wigner matrices","authors":"J. Lee, Kevin Schnelli, Ben Stetler, H. Yau","doi":"10.1214/15-AOP1023","DOIUrl":"https://doi.org/10.1214/15-AOP1023","url":null,"abstract":"We consider N×N random matrices of the form H=W+V where W is a real symmetric or complex Hermitian Wigner matrix and V is a random or deterministic, real, diagonal matrix whose entries are independent of W. We assume subexponential decay for the matrix entries of W, and we choose V so that the eigenvalues of W and V are typically of the same order. For a large class of diagonal matrices V, we show that the local statistics in the bulk of the spectrum are universal in the limit of large N.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"2349-2425"},"PeriodicalIF":2.3,"publicationDate":"2014-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031634","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Consider a discrete-time martingale {Xt}{Xt} taking values in a Hilbert space HH. We show that if for some L≥1L≥1, the bounds E[∥Xt+1−Xt∥2H|Xt]=1E[‖Xt+1−Xt‖H2|Xt]=1 and ∥Xt+1−Xt∥H≤L‖Xt+1−Xt‖H≤L are satisfied for all times t≥0t≥0, then there is a constant c=c(L)c=c(L) such that for 1≤R≤t√1≤R≤t, P(∥Xt−X0∥H≤R)≤cRt√. P(‖Xt−X0‖H≤R)≤cRt. Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph GG with bounded degree, there is a constant CG>0CG>0 such that if {Zt}{Zt} is the simple random walk on GG, then for every e>0e>0 and t≥1/e2t≥1/e2, P(distG(Zt,Z0)≤et√)≤CGe, P(distG(Zt,Z0)≤et)≤CGe, where distGdistG denotes the graph distance in GG.
{"title":"A Gaussian upper bound for martingale small-ball probabilities","authors":"James R. Lee, Y. Peres, Charles K. Smart","doi":"10.1214/15-AOP1073","DOIUrl":"https://doi.org/10.1214/15-AOP1073","url":null,"abstract":"Consider a discrete-time martingale {Xt}{Xt} taking values in a Hilbert space HH. We show that if for some L≥1L≥1, the bounds E[∥Xt+1−Xt∥2H|Xt]=1E[‖Xt+1−Xt‖H2|Xt]=1 and ∥Xt+1−Xt∥H≤L‖Xt+1−Xt‖H≤L are satisfied for all times t≥0t≥0, then there is a constant c=c(L)c=c(L) such that for 1≤R≤t√1≤R≤t, \u0000 \u0000P(∥Xt−X0∥H≤R)≤cRt√. \u0000P(‖Xt−X0‖H≤R)≤cRt. \u0000Following Lee and Peres [Ann. Probab. 41 (2013) 3392–3419], this estimate has applications to small-ball estimates for random walks on vertex-transitive graphs: We show that for every infinite, connected, vertex-transitive graph GG with bounded degree, there is a constant CG>0CG>0 such that if {Zt}{Zt} is the simple random walk on GG, then for every e>0e>0 and t≥1/e2t≥1/e2, \u0000 \u0000P(distG(Zt,Z0)≤et√)≤CGe, \u0000P(distG(Zt,Z0)≤et)≤CGe, \u0000where distGdistG denotes the graph distance in GG.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"4184-4197"},"PeriodicalIF":2.3,"publicationDate":"2014-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1073","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66032828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.
{"title":"Intermittency for branching random walk in Pareto environment","authors":"Marcel Ortgiese, Matthew I. Roberts","doi":"10.1214/15-AOP1021","DOIUrl":"https://doi.org/10.1214/15-AOP1021","url":null,"abstract":"We consider a branching random walk on the lattice, where the branching rates are given by an i.i.d. Pareto random potential. We describe the process, including a detailed shape theorem, in terms of a system of growing lilypads. As an application we show that the branching random walk is intermittent, in the sense that most particles are concentrated on one very small island with large potential. Moreover, we compare the branching random walk to the parabolic Anderson model and observe that although the two systems show similarities, the mechanisms that control the growth are fundamentally different.","PeriodicalId":50763,"journal":{"name":"Annals of Probability","volume":"44 1","pages":"2198-2263"},"PeriodicalIF":2.3,"publicationDate":"2014-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1214/15-AOP1021","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66031549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"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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