Using the technique of evolving sets, we explore the connection between entropy growth and transience for simple random walks on connected infinite graphs with bounded degree. In particular we show that for a simple random walk starting at a vertex $x_0$, if the entropy after $n$ steps, $E_n$ is at least $Cn$ where the $C$ is independent of $x_0$, then the random walk is transient. We also give an example which demonstrates that the condition of $C$ being independent of $x_0$ is necessary.
{"title":"Transience of simple random walks with linear entropy growth","authors":"B. Morris, Hamilton Samraj Santhakumar","doi":"10.1214/23-ecp532","DOIUrl":"https://doi.org/10.1214/23-ecp532","url":null,"abstract":"Using the technique of evolving sets, we explore the connection between entropy growth and transience for simple random walks on connected infinite graphs with bounded degree. In particular we show that for a simple random walk starting at a vertex $x_0$, if the entropy after $n$ steps, $E_n$ is at least $Cn$ where the $C$ is independent of $x_0$, then the random walk is transient. We also give an example which demonstrates that the condition of $C$ being independent of $x_0$ is necessary.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48022816","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}
{"title":"A central limit theorem for some generalized martingale arrays","authors":"L. Pratelli, P. Rigo","doi":"10.1214/23-ecp534","DOIUrl":"https://doi.org/10.1214/23-ecp534","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44536096","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 study the distribution of the maximum gap size in one-dimensional hard-core models. First, we sequentially pack rods of length 2 into an interval of length L at random, subject to the hard-core constraint that rods do not overlap. We find that in a saturated packing, with high probability there is no gap of size 2−o(L−1) between adjacent rods, but there are gaps of size at least 2−Lε−1 for all ε>0. We subsequently study a dependent thinning-based variant of the hard-core process, the one-dimensional “ghost” hard-core model. In this model, we sequentially pack rods of length 2 into an interval of length L at random, such that placed rods neither overlap with previously placed rods nor previously considered candidate rods. We find that in the infinite time limit, with high probability the maximum gap between adjacent rods is smaller than logL but at least (logL)1−ε for all ε>0.
{"title":"Maximum gaps in one-dimensional hard-core models","authors":"Dingding Dong, Nitya Mani","doi":"10.1214/23-ecp552","DOIUrl":"https://doi.org/10.1214/23-ecp552","url":null,"abstract":"We study the distribution of the maximum gap size in one-dimensional hard-core models. First, we sequentially pack rods of length 2 into an interval of length L at random, subject to the hard-core constraint that rods do not overlap. We find that in a saturated packing, with high probability there is no gap of size 2−o(L−1) between adjacent rods, but there are gaps of size at least 2−Lε−1 for all ε>0. We subsequently study a dependent thinning-based variant of the hard-core process, the one-dimensional “ghost” hard-core model. In this model, we sequentially pack rods of length 2 into an interval of length L at random, such that placed rods neither overlap with previously placed rods nor previously considered candidate rods. We find that in the infinite time limit, with high probability the maximum gap between adjacent rods is smaller than logL but at least (logL)1−ε for all ε>0.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135448459","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 an example of a subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate. This answers a question of Amir and Yang (2022) in the negative.
{"title":"A subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate","authors":"Pengfei Tang","doi":"10.1214/23-ecp544","DOIUrl":"https://doi.org/10.1214/23-ecp544","url":null,"abstract":"We construct an example of a subperiodic tree whose intermediate branching number is strictly less than the lower intermediate growth rate. This answers a question of Amir and Yang (2022) in the negative.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136003494","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}
Jorge González Cázares, David Kramer-Bang, Aleksandar Mijatović
We characterise the Hölder continuity of the convex minorant of most Lévy processes. The proof is based on a novel connection between the path properties of the Lévy process at zero and the boundedness of the set of r-slopes of the convex minorant.
{"title":"Hölder continuity of the convex minorant of a Lévy process","authors":"Jorge González Cázares, David Kramer-Bang, Aleksandar Mijatović","doi":"10.1214/23-ecp549","DOIUrl":"https://doi.org/10.1214/23-ecp549","url":null,"abstract":"We characterise the Hölder continuity of the convex minorant of most Lévy processes. The proof is based on a novel connection between the path properties of the Lévy process at zero and the boundedness of the set of r-slopes of the convex minorant.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135213384","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}
{"title":"Intrinsic ultracontractivity and uniform convergence to the Q-process for symmetric Markov processes","authors":"Hanjun Zhang, Huasheng Li, Saixia Liao","doi":"10.1214/23-ecp550","DOIUrl":"https://doi.org/10.1214/23-ecp550","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135310698","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 present a criterion for positive definiteness of the matrix-valued function f(t):=exp(−|t|α[B++B−sign(t)]), where α∈(0,2] and B± are real symmetric and antisymmetric d×d matrices. We also find a criterion for positive definiteness of its multidimensional generalization f(t):=exp(−∫Sd−1|t⊤s|α[B++B−sign(t⊤s)]dΛ(s)) where Λ is a finite measure on the unit sphere Sd−1⊂Rd under a more restrictive assumption that B± commute and are normal. The associated stationary Gaussian random field may be viewed as as a generalization of the univariate fractional Ornstein-Uhlenbeck process. This generalization turns out to be particularly useful for the asymptotic analysis of Rd-valued Gaussian random fields. Another possible application of these findings may concern variogram modelling and general stationary time series analysis.
{"title":"A matrix-valued Schoenberg’s problem and its applications","authors":"Pavel Ievlev, Svyatoslav Novikov","doi":"10.1214/23-ecp562","DOIUrl":"https://doi.org/10.1214/23-ecp562","url":null,"abstract":"In this paper we present a criterion for positive definiteness of the matrix-valued function f(t):=exp(−|t|α[B++B−sign(t)]), where α∈(0,2] and B± are real symmetric and antisymmetric d×d matrices. We also find a criterion for positive definiteness of its multidimensional generalization f(t):=exp(−∫Sd−1|t⊤s|α[B++B−sign(t⊤s)]dΛ(s)) where Λ is a finite measure on the unit sphere Sd−1⊂Rd under a more restrictive assumption that B± commute and are normal. The associated stationary Gaussian random field may be viewed as as a generalization of the univariate fractional Ornstein-Uhlenbeck process. This generalization turns out to be particularly useful for the asymptotic analysis of Rd-valued Gaussian random fields. Another possible application of these findings may concern variogram modelling and general stationary time series analysis.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704903","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 prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.
{"title":"On convergence of volume of level sets of stationary smooth Gaussian fields","authors":"Dmitry Beliaev, Akshay Hegde","doi":"10.1214/23-ecp543","DOIUrl":"https://doi.org/10.1214/23-ecp543","url":null,"abstract":"We prove convergence of Hausdorff measure of level sets of smooth Gaussian fields when the levels converge. Given two coupled stationary fields $f_1, f_2$ , we estimate the difference of Hausdorff measure of level sets in expectation, in terms of $C^2$-fluctuations of the field $F=f_1-f_2$. The main idea in the proof is to represent difference in volume as an integral of mean curvature using the divergence theorem. This approach is different from using Kac-Rice type formula as main tool in the analysis.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135213966","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 modified two-neighbour bootstrap percolation in two dimensions each site of $mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $tau$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $pto0$ it holds that [taule expleft(frac{pi^2}{6p}-frac{clog(1/p)}{sqrt p}right)] for some positive constant $c$, while the classical model is known to lack the logarithmic factor.
{"title":"Sensitive bootstrap percolation second term","authors":"Ivailo Hartarsky","doi":"10.1214/23-ecp535","DOIUrl":"https://doi.org/10.1214/23-ecp535","url":null,"abstract":"In modified two-neighbour bootstrap percolation in two dimensions each site of $mathbb Z^2$ is initially independently infected with probability $p$ and on each discrete time step one additionally infects sites with at least two non-opposite infected neighbours. In this note we establish that for this model the second term in the asymptotics of the infection time $tau$ unexpectedly scales differently from the classical two-neighbour model, in which arbitrary two infected neighbours are required. More precisely, we show that for modified bootstrap percolation with high probability as $pto0$ it holds that [taule expleft(frac{pi^2}{6p}-frac{clog(1/p)}{sqrt p}right)] for some positive constant $c$, while the classical model is known to lack the logarithmic factor.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48395239","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 prove a large deviation principle for the greedy exploration of configuration models, building on a time-discretized version of the method proposed by [2] and [4] for jointly constructing a random graph from a given degree sequence and its exploration. The proof of this result follows the general strategy to study large deviations of processes proposed by [9], based on the convergence of non-linear semigroups. We provide an intuitive interpretation of the LD cost function using Crámer’s theorem for the average of random variables with appropriate distribution, depending on the degree distribution of explored nodes. The rate function can be expressed in a closed-form formula, and the large deviations trajectories can be obtained through explicit associated optimization problems. We then deduce large deviations results for the size of the independent set constructed by the algorithm. As a particular case, we analyze these results for d-regular graphs.
{"title":"Large deviations for the greedy exploration process on configuration models","authors":"Bermolen Paola, Goicoechea Valeria, Jonckheere Matthieu","doi":"10.1214/23-ecp541","DOIUrl":"https://doi.org/10.1214/23-ecp541","url":null,"abstract":"We prove a large deviation principle for the greedy exploration of configuration models, building on a time-discretized version of the method proposed by [2] and [4] for jointly constructing a random graph from a given degree sequence and its exploration. The proof of this result follows the general strategy to study large deviations of processes proposed by [9], based on the convergence of non-linear semigroups. We provide an intuitive interpretation of the LD cost function using Crámer’s theorem for the average of random variables with appropriate distribution, depending on the degree distribution of explored nodes. The rate function can be expressed in a closed-form formula, and the large deviations trajectories can be obtained through explicit associated optimization problems. We then deduce large deviations results for the size of the independent set constructed by the algorithm. As a particular case, we analyze these results for d-regular graphs.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136003485","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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