In this paper, it is shown that $alpha$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $alpha$-permanents of the block matrices obtained from matrices associated to $*$-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.
{"title":"A note on α-permanent and loop soup","authors":"Xiaodan Li, Yushu Zheng","doi":"10.1214/23-ECP530","DOIUrl":"https://doi.org/10.1214/23-ECP530","url":null,"abstract":"In this paper, it is shown that $alpha$-permanent in algebra is closely related to loop soup in probability. We give explicit expansions of $alpha$-permanents of the block matrices obtained from matrices associated to $*$-forests, which are a special class of matrices containing tridiagonal matrices. It is proved in two ways, one is the direct combinatorial proof, and the other is the probabilistic proof via loop soup.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47950495","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $varphi$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.
{"title":"Investigation of Airy equations with random initial conditions","authors":"L. Sakhno","doi":"10.1214/23-ecp522","DOIUrl":"https://doi.org/10.1214/23-ecp522","url":null,"abstract":"The paper investigates properties of mean-square solutions to the Airy equation with random initial data given by stationary processes. The result on the modulus of continiuty of the solution is stated and properties of the covariance function are described. Bounds for the distributions of the suprema of solutions under $varphi$-sub-Gaussian initial conditions are presented. Several examples are provided to illustrate the results. Extension of the results to the case of fractional Airy equation is given.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48209881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Hierarchical Dirichlet process is a discrete random measure serving as an im-portant prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity. The large deviation rate functions are identified explicitly. The rate function for the hierarchical Dirichlet process consists of two terms corresponding to the relative entropies at each level. It is less than the rate function for the Dirichlet process, which reflects the fact that the number of clusters under the hierarchical Dirichlet process has a slower growth rate than under the Dirichlet process.
{"title":"Hierarchical Dirichlet process and relative entropy","authors":"S. Feng","doi":"10.1214/23-ecp511","DOIUrl":"https://doi.org/10.1214/23-ecp511","url":null,"abstract":"The Hierarchical Dirichlet process is a discrete random measure serving as an im-portant prior in Bayesian non-parametrics. It is motivated with the study of groups of clustered data. Each group is modelled through a level two Dirichlet process and all groups share the same base distribution which itself is a drawn from a level one Dirichlet process. It has two concentration parameters with one at each level. The main results of the paper are the law of large numbers and large deviations for the hierarchical Dirichlet process and its mass when both concentration parameters converge to infinity. The large deviation rate functions are identified explicitly. The rate function for the hierarchical Dirichlet process consists of two terms corresponding to the relative entropies at each level. It is less than the rate function for the Dirichlet process, which reflects the fact that the number of clusters under the hierarchical Dirichlet process has a slower growth rate than under the Dirichlet process.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43022214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We construct Dyson Brownian motion for $beta in (0,infty]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When $beta$ is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.
将黎曼流形上布朗运动的外在构造适应于厄米矩阵空间内群轨道的几何构造，构造了$beta in (0,infty]$的戴森布朗运动。当$beta$为无穷大时，特征值通过库仑斥力演化，群轨道通过(- 1 / 2)平均曲率的运动演化。
{"title":"Motion by mean curvature and Dyson Brownian Motion","authors":"Ching-Peng Huang, D. Inauen, Govind Menon","doi":"10.1214/23-ecp540","DOIUrl":"https://doi.org/10.1214/23-ecp540","url":null,"abstract":"We construct Dyson Brownian motion for $beta in (0,infty]$ by adapting the extrinsic construction of Brownian motion on Riemannian manifolds to the geometry of group orbits within the space of Hermitian matrices. When $beta$ is infinite, the eigenvalues evolve by Coulombic repulsion and the group orbits evolve by motion by (minus one half times) mean curvature.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46337664","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this short note, we prove that $v(-epsilon)=-v(epsilon)$. Here, $v(epsilon)$ is the speed of a one-dimensional random walk in a dynamic emph{reversible} random environment, that jumps to the right (resp. to the left) with probability $1/2+epsilon$ (resp. $1/2-epsilon$) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where $v(epsilon), v(-epsilon)$ are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.
{"title":"A note on the antisymmetry in the speed of a random walk in reversible dynamic random environment","authors":"O. Blondel","doi":"10.1214/23-ecp514","DOIUrl":"https://doi.org/10.1214/23-ecp514","url":null,"abstract":"In this short note, we prove that $v(-epsilon)=-v(epsilon)$. Here, $v(epsilon)$ is the speed of a one-dimensional random walk in a dynamic emph{reversible} random environment, that jumps to the right (resp. to the left) with probability $1/2+epsilon$ (resp. $1/2-epsilon$) if it stands on an occupied site, and vice-versa on an empty site. We work in any setting where $v(epsilon), v(-epsilon)$ are well-defined, i.e. a weak LLN holds. The proof relies on a simple coupling argument that holds only in the discrete setting.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47724059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Junge, Arturo Ortiz San Miguel, Lily Reeves, Cynthia Rivera S'anchez
In ballistic annihilation, infinitely many particles with randomly assigned velocities move across the real line and mutually annihilate upon contact. We introduce a variant with superimposed clusters of multiple stationary particles. Our main result is that the critical initial cluster density to ensure species survival depends on both the mean and variance of the cluster size. Our result contrasts with recent ballistic annihilation universality findings with respect to particle spacings. A corollary of our theorem resolves an open question for coalescing ballistic annihilation.
{"title":"Non-universality in clustered ballistic annihilation","authors":"M. Junge, Arturo Ortiz San Miguel, Lily Reeves, Cynthia Rivera S'anchez","doi":"10.1214/23-ecp529","DOIUrl":"https://doi.org/10.1214/23-ecp529","url":null,"abstract":"In ballistic annihilation, infinitely many particles with randomly assigned velocities move across the real line and mutually annihilate upon contact. We introduce a variant with superimposed clusters of multiple stationary particles. Our main result is that the critical initial cluster density to ensure species survival depends on both the mean and variance of the cluster size. Our result contrasts with recent ballistic annihilation universality findings with respect to particle spacings. A corollary of our theorem resolves an open question for coalescing ballistic annihilation.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48172067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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 consider stationary Markov chains with trivial two-sided tail sigma field, and prove that additive functionals satisfy the central limit theorem provided the variance of partial sums divided by n is bounded.
{"title":"On the CLT for stationary Markov chains with trivial tail sigma field","authors":"M. Peligrad","doi":"10.1214/23-ecp509","DOIUrl":"https://doi.org/10.1214/23-ecp509","url":null,"abstract":"In this paper we consider stationary Markov chains with trivial two-sided tail sigma field, and prove that additive functionals satisfy the central limit theorem provided the variance of partial sums divided by n is bounded.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43211843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let B = { B t } t ≥ 0 be a one-dimensional standard Brownian motion and denote by A t , t ≥ 0, the quadratic variation of e B t , t ≥ 0. The celebrated Bougerol’s identity in law (1983) asserts that, if β = { β t } t ≥ 0 is another Brownian motion independent of B , then β A t has the same law as sinh B t for every fixed t > 0. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving as the second coordinates the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.
设B＝｛B t｝t≥0是一维标准布朗运动，并用a t，t≥0表示e B t，t的二次变分。著名的Bougerol定律恒等式（1983）断言，如果β={βt}t≥0是另一个独立于B的布朗运动，那么对于每个固定t>0，β-t与sinh bt具有相同的定律。Bertoin、Dufresne和Yor（2013）获得了恒等式的二维扩展，涉及B和β在零级的局部时间作为第二坐标。在本文中，我们在这些局部时间的能级不限于零的情况下，给出了它们的推广。我们的论点为最初的扩展提供了一个简短的基本证明，并为这个微妙的身份提供了新的线索。
{"title":"On two-dimensional extensions of Bougerol’s identity in law","authors":"Yuu Hariya, Yohei Matsumura","doi":"10.1214/23-ECP510","DOIUrl":"https://doi.org/10.1214/23-ECP510","url":null,"abstract":"Let B = { B t } t ≥ 0 be a one-dimensional standard Brownian motion and denote by A t , t ≥ 0, the quadratic variation of e B t , t ≥ 0. The celebrated Bougerol’s identity in law (1983) asserts that, if β = { β t } t ≥ 0 is another Brownian motion independent of B , then β A t has the same law as sinh B t for every fixed t > 0. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving as the second coordinates the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41865345","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. Our main result is that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, we provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.
{"title":"Lipschitz continuity of the Wasserstein projections in the convex order on the line","authors":"B. Jourdain, W. Margheriti, G. Pammer","doi":"10.1214/23-ecp525","DOIUrl":"https://doi.org/10.1214/23-ecp525","url":null,"abstract":"Wasserstein projections in the convex order were first considered in the framework of weak optimal transport, and found application in various problems such as concentration inequalities and martingale optimal transport. In dimension one, it is well-known that the set of probability measures with a given mean is a lattice w.r.t. the convex order. Our main result is that, contrary to the minimum and maximum in the convex order, the Wasserstein projections are Lipschitz continuity w.r.t. the Wasserstein distance in dimension one. Moreover, we provide examples that show sharpness of the obtained bounds for the 1-Wasserstein distance.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46024252","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in $mathbb{Z}^d$, $d>2$, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from Betz and Taggi (2019) and Taggi (2021) that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when $d>2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.
{"title":"Macroscopic loops in the 3d double-dimer model","authors":"A. Quitmann, L. Taggi","doi":"10.1214/23-ecp536","DOIUrl":"https://doi.org/10.1214/23-ecp536","url":null,"abstract":"The double dimer model is defined as the superposition of two independent uniformly distributed dimer covers of a graph. Its configurations can be viewed as disjoint collections of self-avoiding loops. Our first result is that in $mathbb{Z}^d$, $d>2$, the loops in the double dimer model are macroscopic. These are shown to behave qualitatively differently than in two dimensions. In particular, we show that, given two distant points of a large box, with uniformly positive probability there exists a loop visiting both points. Our second result involves the monomer double-dimer model, namely the double-dimer model in the presence of a density of monomers. These are vertices which are not allowed to be touched by any loop. This model depends on a parameter, the monomer activity, which controls the density of monomers. It is known from Betz and Taggi (2019) and Taggi (2021) that a finite critical threshold of the monomer activity exists, below which a self-avoiding walk forced through the system is macroscopic. Our paper shows that, when $d>2$, such a critical threshold is strictly positive. In other words, the self-avoiding walk is macroscopic even in the presence of a positive density of monomers.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":" ","pages":""},"PeriodicalIF":0.5,"publicationDate":"2022-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49534287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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