We study the anisotropic version of the Hastings-Levitov model AHL$(nu)$. Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window in which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable fixed point, and we show that the full trajectory can be characterised in terms of a single Gaussian random variable.
{"title":"Scaling limits of anisotropic growth on logarithmic time-scales","authors":"George Liddle, Amanda G. Turner","doi":"10.1214/23-ejp964","DOIUrl":"https://doi.org/10.1214/23-ejp964","url":null,"abstract":"We study the anisotropic version of the Hastings-Levitov model AHL$(nu)$. Previous results have shown that on bounded time-scales the harmonic measure on the boundary of the cluster converges, in the small-particle limit, to the solution of a deterministic ordinary differential equation. We consider the evolution of the harmonic measure on time-scales which grow logarithmically as the particle size converges to zero and show that, over this time-scale, the leading order behaviour of the harmonic measure becomes random. Specifically, we show that there exists a critical logarithmic time window in which the harmonic measure flow, started from the unstable fixed point, moves stochastically from the unstable point towards a stable fixed point, and we show that the full trajectory can be characterised in terms of a single Gaussian random variable.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47251865","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.
{"title":"Large deviations for the volume of k-nearest neighbor balls","authors":"C. Hirsch, Taegyu Kang, Takashi Owada","doi":"10.1214/23-ejp965","DOIUrl":"https://doi.org/10.1214/23-ejp965","url":null,"abstract":"This paper develops the large deviations theory for the point process associated with the Euclidean volume of $k$-nearest neighbor balls centered around the points of a homogeneous Poisson or a binomial point processes in the unit cube. Two different types of large deviation behaviors of such point processes are investigated. Our first result is the Donsker-Varadhan large deviation principle, under the assumption that the centering terms for the volume of $k$-nearest neighbor balls grow to infinity more slowly than those needed for Poisson convergence. Additionally, we also study large deviations based on the notion of $mathcal M_0$-topology, which takes place when the centering terms tend to infinity sufficiently fast, compared to those for Poisson convergence. As applications of our main theorems, we discuss large deviations for the number of Poisson or binomial points of degree at most $k$ in a geometric graph in the dense regime.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48139140","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type one of two types, live in colonies, and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group endowed with the discrete topology. In earlier work we showed that the system has a one-parameter family of equilibria controlled by the relative density of the two types. Moreover, these equilibria exhibit a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identified the parameter regimes for which these two phases occur, and found that these regimes are different when the mean wake-up time of a dormant individual is finite or infinite. The goal of the present paper is to establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together. If the wake-up time has finite mean, then there is a single universality class for the scaling limit. On the other hand, if the wake-up time has infinite mean, then there are two universality classes depending on how fast the truncation level of the seed-bank grows compared to the truncation level of the geographic space.
{"title":"Spatial populations with seed-bank: finite-systems scheme","authors":"A. Greven, F. Hollander","doi":"10.1214/23-ejp974","DOIUrl":"https://doi.org/10.1214/23-ejp974","url":null,"abstract":"We consider a system of interacting Fisher-Wright diffusions with seed-bank. Individuals carry type one of two types, live in colonies, and are subject to resampling and migration as long as they are active. Each colony has a structured seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. As geographic space labelling the colonies we consider a countable Abelian group endowed with the discrete topology. In earlier work we showed that the system has a one-parameter family of equilibria controlled by the relative density of the two types. Moreover, these equilibria exhibit a dichotomy of coexistence (= locally multi-type equilibrium) versus clustering (= locally mono-type equilibrium). We identified the parameter regimes for which these two phases occur, and found that these regimes are different when the mean wake-up time of a dormant individual is finite or infinite. The goal of the present paper is to establish the finite-systems scheme, i.e., identify how a finite truncation of the system (both in the geographic space and in the seed-bank) behaves as both the time and the truncation level tend to infinity, properly tuned together. If the wake-up time has finite mean, then there is a single universality class for the scaling limit. On the other hand, if the wake-up time has infinite mean, then there are two universality classes depending on how fast the truncation level of the seed-bank grows compared to the truncation level of the geographic space.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44434244","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $X_1, X_2, ldots$ be i.i.d. random variables with values in $mathbb{Z}^d$ satisfying $mathbb{P} left(X_1=xright) = mathbb{P} left(X_1=-xright) = Theta left(|x|^{-s}right)$ for some $s>d$. We show that the random walk defined by $S_n = sum_{k=1}^{n} X_k$ is recurrent for $din {1,2}$ and $s geq 2d$, and transient otherwise. This also shows that for an electric network in dimension $din {1,2}$ the condition $c_{{x,y}} leq C |x-y|^{-2d}$ implies recurrence, whereas $c_{{x,y}} geq c |x-y|^{-s}$ for some $c>0$ and $s<2d$ implies transience. This fact was already previously known, but we give a new proof of it that uses only electric networks. We also use these results to show the recurrence of random walks on certain long-range percolation clusters. In particular, we show recurrence for several cases of the two-dimensional weight-dependent random connection model, which was previously studied by Gracar et al. [Electron. J. Probab. 27. 1-31 (2022)].
{"title":"Recurrence and transience of symmetric random walks with long-range jumps","authors":"J. Baumler","doi":"10.1214/23-EJP998","DOIUrl":"https://doi.org/10.1214/23-EJP998","url":null,"abstract":"Let $X_1, X_2, ldots$ be i.i.d. random variables with values in $mathbb{Z}^d$ satisfying $mathbb{P} left(X_1=xright) = mathbb{P} left(X_1=-xright) = Theta left(|x|^{-s}right)$ for some $s>d$. We show that the random walk defined by $S_n = sum_{k=1}^{n} X_k$ is recurrent for $din {1,2}$ and $s geq 2d$, and transient otherwise. This also shows that for an electric network in dimension $din {1,2}$ the condition $c_{{x,y}} leq C |x-y|^{-2d}$ implies recurrence, whereas $c_{{x,y}} geq c |x-y|^{-s}$ for some $c>0$ and $s<2d$ implies transience. This fact was already previously known, but we give a new proof of it that uses only electric networks. We also use these results to show the recurrence of random walks on certain long-range percolation clusters. In particular, we show recurrence for several cases of the two-dimensional weight-dependent random connection model, which was previously studied by Gracar et al. [Electron. J. Probab. 27. 1-31 (2022)].","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45417210","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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 use duality techniques to study a combination of the well-known contact process (CP) and the somewhat less-known annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two entries, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameter in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones in the cCP can only be present at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper"Commutative monoid duality"to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.
{"title":"Applying monoid duality to a double contact process","authors":"Jan Niklas Latz, J. Swart","doi":"10.1214/23-ejp961","DOIUrl":"https://doi.org/10.1214/23-ejp961","url":null,"abstract":"In this paper we use duality techniques to study a combination of the well-known contact process (CP) and the somewhat less-known annihilating branching process. As the latter can be seen as a cancellative version of the contact process, we rebrand it as the cancellative contact process (cCP). Our process of interest will consist of two entries, the first being a CP and the second being a cCP. We call this process the double contact process (2CP) and prove that it has (depending on the model parameters) at most one invariant law under which ones are present in both processes. In particular, we can choose the model parameter in such a way that CP and cCP are monotonely coupled. In this case also the above mentioned invariant law will have the property that, under it, ones in the cCP can only be present at sites where there are also ones in the CP. Along the way we extend the dualities for Markov processes discovered in our paper\"Commutative monoid duality\"to processes on infinite state spaces so that they, in particular, can be used for interacting particle systems.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46111330","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that the set of $gamma$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $gamma neq 0$. Our proof relies on the coupling between a GFF and the nested CLE$_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE$_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE$_{kappa}$, $kappa in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.
证明了具有Dirichlet边界条件的平面高斯自由场(GFF)的$gamma$ -厚点集对于所有$gamma neq 0$都是完全不连通的。我们的证明依赖于GFF和嵌套CLE $_4$之间的耦合。特别是，我们证明了GFF的粗点与加权CLE $_4$嵌套域的粗点相同，并建立了嵌套CLE补的几乎肯定的完全不连通$_{kappa}$, $kappa in (8/3,4]$。作为一个推论，我们看到超临界LQG指标的奇点集也是完全不连通的。
{"title":"Thick points of the planar GFF are totally disconnected for all γ≠0","authors":"Juhan Aru, L'eonie Papon, E. Powell","doi":"10.1214/23-ejp975","DOIUrl":"https://doi.org/10.1214/23-ejp975","url":null,"abstract":"We prove that the set of $gamma$-thick points of a planar Gaussian free field (GFF) with Dirichlet boundary conditions is a.s. totally disconnected for all $gamma neq 0$. Our proof relies on the coupling between a GFF and the nested CLE$_4$. In particular, we show that the thick points of the GFF are the same as those of the weighted CLE$_4$ nesting field and establish the almost sure total disconnectedness of the complement of a nested CLE$_{kappa}$, $kappa in (8/3,4]$. As a corollary we see that the set of singular points for supercritical LQG metrics is a.s. totally disconnected.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41263507","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph $G$. We show that the random interlacement point process on any transient transitive graph $G$ is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced by Popov and Teixeira) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if $G$ is non-unimodular.
{"title":"Random interlacement is a factor of i.i.d.","authors":"M'arton Borb'enyi, Bal'azs R'ath, S. Rokob","doi":"10.1214/23-EJP950","DOIUrl":"https://doi.org/10.1214/23-EJP950","url":null,"abstract":"The random interlacement point process (introduced by Sznitman, generalized by Teixeira) is a Poisson point process on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift on a transient graph $G$. We show that the random interlacement point process on any transient transitive graph $G$ is a factor of i.i.d., i.e., it can be constructed from a family of i.i.d. random variables indexed by vertices of the graph via an equivariant measurable map. Our proof uses a variant of the soft local time method (introduced by Popov and Teixeira) to construct the interlacement point process as the almost sure limit of a sequence of finite-length variants of the model with increasing length. We also discuss a more direct method of proving that the interlacement point process is a factor of i.i.d. which works if and only if $G$ is non-unimodular.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45217805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the paper we prove that, for $kappain(0,8)$, the $n$-point boundary Green's function of exponent $frac8kappa -1$ for chordal SLE$_kappa$ exists. We also prove that the convergence is uniform over compact sets and the Green's function is continuous. We also give up-to-constant bounds for the Green's function.
{"title":"Existence of multi-point boundary Green’s function for chordal Schramm-Loewner evolution (SLE)","authors":"Rami Fakhry, Dapeng Zhan","doi":"10.1214/23-ejp936","DOIUrl":"https://doi.org/10.1214/23-ejp936","url":null,"abstract":"In the paper we prove that, for $kappain(0,8)$, the $n$-point boundary Green's function of exponent $frac8kappa -1$ for chordal SLE$_kappa$ exists. We also prove that the convergence is uniform over compact sets and the Green's function is continuous. We also give up-to-constant bounds for the Green's function.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42217503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove higher order concentration bounds for functions on Stiefel and Grassmann manifolds equipped with the uniform distribution. This partially extends previous work for functions on the unit sphere. Technically, our results are based on logarithmic Sobolev techniques for the uniform measures on the manifolds. Applications include Hanson--Wright type inequalities for Stiefel manifolds and concentration bounds for certain distance functions between subspaces of $mathbb{R}^n$.
{"title":"Higher order concentration on Stiefel and Grassmann manifolds","authors":"F. Gotze, H. Sambale","doi":"10.1214/23-ejp966","DOIUrl":"https://doi.org/10.1214/23-ejp966","url":null,"abstract":"We prove higher order concentration bounds for functions on Stiefel and Grassmann manifolds equipped with the uniform distribution. This partially extends previous work for functions on the unit sphere. Technically, our results are based on logarithmic Sobolev techniques for the uniform measures on the manifolds. Applications include Hanson--Wright type inequalities for Stiefel manifolds and concentration bounds for certain distance functions between subspaces of $mathbb{R}^n$.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44016280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $sigma$, satisfying Krylov--R"ockner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,sigma^i)$, both for It^o and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean--Vlasov SDEs, criteria for strong compactness of solutions and Wong--Zakai type theorems.
{"title":"Stability estimates for singular SDEs and applications","authors":"L. Galeati, Chengcheng Ling","doi":"10.1214/23-ejp913","DOIUrl":"https://doi.org/10.1214/23-ejp913","url":null,"abstract":"We consider multidimensional SDEs with singular drift $b$ and Sobolev diffusion coefficients $sigma$, satisfying Krylov--R\"ockner type assumptions. We prove several stability estimates, comparing solutions driven by different $(b^i,sigma^i)$, both for It^o and Stratonovich SDEs, possibly depending on negative Sobolev norms of the difference $b^1-b^2$. We then discuss several applications of these results to McKean--Vlasov SDEs, criteria for strong compactness of solutions and Wong--Zakai type theorems.","PeriodicalId":50538,"journal":{"name":"Electronic Journal of Probability","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2022-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49565674","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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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