Suppose $N$ is a compact Riemannian manifold, in this paper we will introduce the definition of $N$-valued BSDE and $L^2(mathbb{T}^m;N)$-valued BSDE for which the solution are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to $L^2(mathbb{T}^m;N)$-valued BSDE will be proved without any convexity condition on $N$.
{"title":"A study of backward stochastic differential equation on a Riemannian manifold","authors":"Xin Chen, Wenjie Ye","doi":"10.1214/21-ejp649","DOIUrl":"https://doi.org/10.1214/21-ejp649","url":null,"abstract":"Suppose $N$ is a compact Riemannian manifold, in this paper we will introduce the definition of $N$-valued BSDE and $L^2(mathbb{T}^m;N)$-valued BSDE for which the solution are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to $L^2(mathbb{T}^m;N)$-valued BSDE will be proved without any convexity condition on $N$.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"132 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89124467","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 leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in $mathbb{Z}^2$ and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. We compute the limit shape as a function of $d$ in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as $dto 1$ and a diamond as $dtoinfty$. We compute the limit shape by comparing the odometer function at a site to the probability that a killed random walk dies at that site. When $dto 1$ the Leaky-ASM converges to the abelian sandpile model (ASM) with a modified initial configuration. We also prove the limit shape is a circle when simultaneously with $ntoinfty$ we have that $d=d_n$ converges to $1$ slower than any power of $n$. To gain information about the ASM faster convergence is necessary.
{"title":"The Limit Shape of the Leaky Abelian Sandpile Model","authors":"Ian Alevy, S. Mkrtchyan","doi":"10.1093/IMRN/RNAB124","DOIUrl":"https://doi.org/10.1093/IMRN/RNAB124","url":null,"abstract":"The leaky abelian sandpile model (Leaky-ASM) is a growth model in which $n$ grains of sand start at the origin in $mathbb{Z}^2$ and diffuse along the vertices according to a toppling rule. A site can topple if its amount of sand is above a threshold. In each topple a site sends some sand to each neighbor and leaks a portion $1-1/d$ of its sand. We compute the limit shape as a function of $d$ in the symmetric case where each topple sends an equal amount of sand to each neighbor. The limit shape converges to a circle as $dto 1$ and a diamond as $dtoinfty$. We compute the limit shape by comparing the odometer function at a site to the probability that a killed random walk dies at that site. When $dto 1$ the Leaky-ASM converges to the abelian sandpile model (ASM) with a modified initial configuration. We also prove the limit shape is a circle when simultaneously with $ntoinfty$ we have that $d=d_n$ converges to $1$ slower than any power of $n$. To gain information about the ASM faster convergence is necessary.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"40 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78983770","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 show that the last zero before time $t$ of a recurrent Bessel process with drift starting at $0$ has the same distribution as the product of an independent right censored exponential random variable and a beta random variable. This extends a recent result of Schulte-Geers and Stadje (2017) from Brownian motion with drift to recurrent Bessel processes with drift. Our proof is intuitive and direct while avoiding heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.
{"title":"Independent factorization of the last zero arcsine law for Bessel processes with drift","authors":"Hugo Panzo","doi":"10.1214/21-ecp405","DOIUrl":"https://doi.org/10.1214/21-ecp405","url":null,"abstract":"We show that the last zero before time $t$ of a recurrent Bessel process with drift starting at $0$ has the same distribution as the product of an independent right censored exponential random variable and a beta random variable. This extends a recent result of Schulte-Geers and Stadje (2017) from Brownian motion with drift to recurrent Bessel processes with drift. Our proof is intuitive and direct while avoiding heavy computations. For this we develop a novel additive decomposition for the square of a Bessel process with drift that may be of independent interest.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"2015 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89885323","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 : 2020-09-28DOI: 10.1142/s2010326321500398
C. Charlier
The Pearcey process is a universal point process in random matrix theory and depends on a parameter $rho in mathbb{R}$. Let $N(x)$ be the random variable that counts the number of points in this process that fall in the interval $[-x,x]$. In this note, we establish the following global rigidity upper bound: begin{align*} lim_{s to infty}mathbb Pleft(sup_{x> s}left|frac{N(x)-big( frac{3sqrt{3}}{4pi}x^{frac{4}{3}}-frac{sqrt{3}rho}{2pi}x^{frac{2}{3}} big)}{log x}right| leq frac{4sqrt{2}}{3pi} + epsilon right) = 1, end{align*} where $epsilon > 0$ is arbitrary. We also obtain a similar upper bound for the maximum deviation of the points, and a central limit theorem for the individual fluctuations. The proof is short and combines a recent result of Dai, Xu and Zhang with another result of Charlier and Claeys.
{"title":"Upper bounds for the maximum deviation of the Pearcey process","authors":"C. Charlier","doi":"10.1142/s2010326321500398","DOIUrl":"https://doi.org/10.1142/s2010326321500398","url":null,"abstract":"The Pearcey process is a universal point process in random matrix theory and depends on a parameter $rho in mathbb{R}$. Let $N(x)$ be the random variable that counts the number of points in this process that fall in the interval $[-x,x]$. In this note, we establish the following global rigidity upper bound: begin{align*} lim_{s to infty}mathbb Pleft(sup_{x> s}left|frac{N(x)-big( frac{3sqrt{3}}{4pi}x^{frac{4}{3}}-frac{sqrt{3}rho}{2pi}x^{frac{2}{3}} big)}{log x}right| leq frac{4sqrt{2}}{3pi} + epsilon right) = 1, end{align*} where $epsilon > 0$ is arbitrary. We also obtain a similar upper bound for the maximum deviation of the points, and a central limit theorem for the individual fluctuations. The proof is short and combines a recent result of Dai, Xu and Zhang with another result of Charlier and Claeys.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"14 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91478968","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 generalize the concept of extremal index of a stationary random sequence to the series scheme of identically distributed random variables with random series sizes tending to infinity in probability. We introduce new extremal indices through two definitions generalizing the basic properties of the classical extremal index. We prove some useful properties of the new extremal indices. We show how the behavior of aggregate activity maxima on random graphs (in information network models) and the behavior of maxima of random particle scores in branching processes (in biological population models) can be described in terms of the new extremal indices. We also obtain new results on models with copulas and threshold models. We show that the new indices can take different values for the same system, as well as values greater than one.
{"title":"Extremal Indices in the Series Scheme and their Applications","authors":"A. Lebedev","doi":"10.14357/19922264150305","DOIUrl":"https://doi.org/10.14357/19922264150305","url":null,"abstract":"We generalize the concept of extremal index of a stationary random sequence to the series scheme of identically distributed random variables with random series sizes tending to infinity in probability. We introduce new extremal indices through two definitions generalizing the basic properties of the classical extremal index. We prove some useful properties of the new extremal indices. We show how the behavior of aggregate activity maxima on random graphs (in information network models) and the behavior of maxima of random particle scores in branching processes (in biological population models) can be described in terms of the new extremal indices. We also obtain new results on models with copulas and threshold models. We show that the new indices can take different values for the same system, as well as values greater than one.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"43 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86713835","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 power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.
利用概率分布的Laplace-Stieltjes变换研究了功率混合型泛函方程。这些方程是在研究Z = X + TZ类型的分布方程时产生的，其中T是已知的随机变量，而变量Z是通过X定义的，我们想要“找到”X。我们提供了这种泛函方程具有唯一解的充分必要条件。唯一性等价于一个概率分布的表征性质。本文给出了一些关于复合指数型和复合泊松型泛函方程的新结果或推广和改进了以往的结果。特别是，我们对J. Pitman和M. Yor在2003年提出的一个问题给出了另一个肯定的答案。我们提供了明确的说明性的例子，并处理相关的主题。
{"title":"Characterization of Probability Distributions via Functional Equations of Power-Mixture Type","authors":"Chin-yuan Hu, G. D. Lin, J. Stoyanov","doi":"10.3390/MATH9030271","DOIUrl":"https://doi.org/10.3390/MATH9030271","url":null,"abstract":"We study power-mixture type functional equations in terms of Laplace-Stieltjes transforms of probability distributions. These equations arise when studying distributional equations of the type Z = X + TZ, where T is a known random variable, while the variable Z is defined via X, and we want to `find' X. We provide necessary and sufficient conditions for such functional equations to have unique solutions. The uniqueness is equivalent to a characterization property of a probability distribution. We present results which are either new or extend and improve previous results about functional equations of compound-exponential and compound-Poisson types. In particular, we give another affirmative answer to a question posed by J. Pitman and M. Yor in 2003. We provide explicit illustrative examples and deal with related topics.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"27 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78182347","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}
Neretin constructed an analogue of the Hua measures on the infinite $p$-adic matrices $Matleft(mathbb{N},mathbb{Q}_pright)$. Bufetov and Qiu classified the ergodic measures on $Matleft(mathbb{N},mathbb{Q}_pright)$ that are invariant under the natural action of $GL(infty,mathbb{Z}_p)times GL(infty,mathbb{Z}_p)$. In this paper we solve the problem of ergodic decomposition for the $p$-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.
{"title":"Infinite 𝑝-adic random matrices and ergodic decomposition of 𝑝-adic Hua measures","authors":"T. Assiotis","doi":"10.1090/tran/8526","DOIUrl":"https://doi.org/10.1090/tran/8526","url":null,"abstract":"Neretin constructed an analogue of the Hua measures on the infinite $p$-adic matrices $Matleft(mathbb{N},mathbb{Q}_pright)$. Bufetov and Qiu classified the ergodic measures on $Matleft(mathbb{N},mathbb{Q}_pright)$ that are invariant under the natural action of $GL(infty,mathbb{Z}_p)times GL(infty,mathbb{Z}_p)$. In this paper we solve the problem of ergodic decomposition for the $p$-adic Hua measures introduced by Neretin. We prove that the probability measure governing the ergodic decomposition has an explicit expression which identifies it with a Hall-Littlewood measure on partitions. Our arguments involve certain Markov chains.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"52 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78913633","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}
$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $beta$-Hermite, $beta$-Laguerre, and $beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $betatoinfty$ in the $beta$-Hermite and $beta$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $Sigma_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $Sigma_N^{-1}$ with formulas for the eigenvalues and eigenvectors of $Sigma_N^{-1}$ and thus of $Sigma_N$. In the present paper we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for $Sigma_N$ from $Sigma_N^{-1}$ where, for $beta$-Hermite and $beta$-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for $Ntoinfty$ in terms of the Airy function. For $beta$-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.
{"title":"Limit theorems and soft edge of freezing random matrix models via dual orthogonal polynomials","authors":"Sergio Andraus, K. Hermann, M. Voit","doi":"10.1063/5.0028706","DOIUrl":"https://doi.org/10.1063/5.0028706","url":null,"abstract":"$N$-dimensional Bessel and Jacobi processes describe interacting particle systems with $N$ particles and are related to $beta$-Hermite, $beta$-Laguerre, and $beta$-Jacobi ensembles. For fixed $N$ there exist associated weak limit theorems (WLTs) in the freezing regime $betatoinfty$ in the $beta$-Hermite and $beta$-Laguerre case by Dumitriu and Edelman (2005) with explicit formulas for the covariance matrices $Sigma_N$ in terms of the zeros of associated orthogonal polynomials. Recently, the authors derived these WLTs in a different way and computed $Sigma_N^{-1}$ with formulas for the eigenvalues and eigenvectors of $Sigma_N^{-1}$ and thus of $Sigma_N$. In the present paper we use these data and the theory of finite dual orthogonal polynomials of de Boor and Saff to derive formulas for $Sigma_N$ from $Sigma_N^{-1}$ where, for $beta$-Hermite and $beta$-Laguerre ensembles, our formulas are simpler than those of Dumitriu and Edelman. We use these polynomials to derive asymptotic results for the soft edge in the freezing regime for $Ntoinfty$ in terms of the Airy function. For $beta$-Hermite ensembles, our limit expressions are different from those of Dumitriu and Edelman.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75095474","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 consider the asymmetric simple exclusion process (ASEP) on $mathbb{Z}$. For continuous densities, ASEP is in local equilibrium for large times, at discontinuities however, one expects to see a dynamical phase transition, i.e. a mixture of different equilibriums. We consider ASEP with deterministic initial data such that at large times, two rarefactions come together at the origin, and the density jumps from $0$ to $1$. Shifting the measure on the KPZ $1/3$ scale, we show that ASEP converges to a mixture of the Dirac measures with only holes resp. only particles. The parameter of that mixture is the probability that the second class particle, which is distributed as the difference of two independent GUEs, stays to the left of the shift. This should be compared with the results of Ferrari and Fontes from 1994 cite{FF94b}, who obtained a mixture of Bernoulli product measures at discontinuities created by random initial data, where instead of independent GUEs, independent Gaussians determine the parameter of the mixture.
{"title":"Dynamical phase transition of ASEP in the KPZ regime","authors":"Peter Nejjar","doi":"10.1214/21-EJP642","DOIUrl":"https://doi.org/10.1214/21-EJP642","url":null,"abstract":"We consider the asymmetric simple exclusion process (ASEP) on $mathbb{Z}$. For continuous densities, ASEP is in local equilibrium for large times, at discontinuities however, one expects to see a dynamical phase transition, i.e. a mixture of different equilibriums. We consider ASEP with deterministic initial data such that at large times, two rarefactions come together at the origin, and the density jumps from $0$ to $1$. Shifting the measure on the KPZ $1/3$ scale, we show that ASEP converges to a mixture of the Dirac measures with only holes resp. only particles. The parameter of that mixture is the probability that the second class particle, which is distributed as the difference of two independent GUEs, stays to the left of the shift. This should be compared with the results of Ferrari and Fontes from 1994 cite{FF94b}, who obtained a mixture of Bernoulli product measures at discontinuities created by random initial data, where instead of independent GUEs, independent Gaussians determine the parameter of the mixture.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"222 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79640885","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 article we investigate the Uniform Spanning Forest ($mathsf{USF}$) in the nearest-neighbour integer lattice $mathbf{Z}^{d+1} = mathbf{Z}times mathbf{Z}^d$ with an assignment of conductances that makes the underlying (Network) Random Walk ($mathsf{NRW}$) drifted towards the right of the first coordinate. This assignment of conductances has exponential growth and decay; in particular, the measure of balls can be made arbitrarily close to zero or arbitrarily large. We establish upper and lower bounds for its Green's function. We show that in dimension $d = 1, 2$ the $mathsf{USF}$ consists of a single tree while in $d geq 3,$ there are infinitely many trees. We then show, by an intricate study of multiple $mathsf{NRW}$s, that in every dimension the trees are one-ended; the technique for $d = 2$ is completely new, while the technique for $d geq 3$ is a major makeover of the technique for the proof of the same result for the graph $mathbf{Z}^d.$ We finally establish the probability that two or more vertices are $mathsf{USF}$-connected and study the distance between different trees.
{"title":"Uniform spanning forest on the integer lattice with drift in one coordinate.","authors":"Guillermo Martinez Dibene","doi":"10.14288/1.0392676","DOIUrl":"https://doi.org/10.14288/1.0392676","url":null,"abstract":"In this article we investigate the Uniform Spanning Forest ($mathsf{USF}$) in the nearest-neighbour integer lattice $mathbf{Z}^{d+1} = mathbf{Z}times mathbf{Z}^d$ with an assignment of conductances that makes the underlying (Network) Random Walk ($mathsf{NRW}$) drifted towards the right of the first coordinate. This assignment of conductances has exponential growth and decay; in particular, the measure of balls can be made arbitrarily close to zero or arbitrarily large. We establish upper and lower bounds for its Green's function. We show that in dimension $d = 1, 2$ the $mathsf{USF}$ consists of a single tree while in $d geq 3,$ there are infinitely many trees. We then show, by an intricate study of multiple $mathsf{NRW}$s, that in every dimension the trees are one-ended; the technique for $d = 2$ is completely new, while the technique for $d geq 3$ is a major makeover of the technique for the proof of the same result for the graph $mathbf{Z}^d.$ We finally establish the probability that two or more vertices are $mathsf{USF}$-connected and study the distance between different trees.","PeriodicalId":8470,"journal":{"name":"arXiv: Probability","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83067591","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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