Joseph S. Alameda, Caroline Bang, Zachary Brennan, David P. Herzog, Jurgen Kritschgau, Elizabeth Sprangel
{"title":"Cutoff in the Bernoulli-Laplace urn model with swaps of order n","authors":"Joseph S. Alameda, Caroline Bang, Zachary Brennan, David P. Herzog, Jurgen Kritschgau, Elizabeth Sprangel","doi":"10.1214/23-ecp569","DOIUrl":"https://doi.org/10.1214/23-ecp569","url":null,"abstract":"","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"75 23","pages":""},"PeriodicalIF":0.5,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139458360","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 investigate the number of holes created by an “earthworm” moving on the two-dimensional integer lattice. The earthworm is modeled by a simple random walk. At the initial time, all vertices are filled with grains of soil except for the position of the earthworm. At each step, the earthworm pushes the soil in the direction of its motion. It leaves a hole (an empty vertex with no grain of soil) behind it. If there are holes in front of the earthworm (in the direction of its step), the closest hole is filled with a grain of soil. Thus the number of holes increases by 1 or remains unchanged at every step. We show that the number of holes is at least O(n3∕4)after n steps.
{"title":"On the size of earthworm’s trail","authors":"Krzysztof Burdzy, Shi Feng, Daisuke Shiraishi","doi":"10.1214/23-ecp556","DOIUrl":"https://doi.org/10.1214/23-ecp556","url":null,"abstract":"We investigate the number of holes created by an “earthworm” moving on the two-dimensional integer lattice. The earthworm is modeled by a simple random walk. At the initial time, all vertices are filled with grains of soil except for the position of the earthworm. At each step, the earthworm pushes the soil in the direction of its motion. It leaves a hole (an empty vertex with no grain of soil) behind it. If there are holes in front of the earthworm (in the direction of its step), the closest hole is filled with a grain of soil. Thus the number of holes increases by 1 or remains unchanged at every step. We show that the number of holes is at least O(n3∕4)after n steps.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704384","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 show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.
{"title":"Small ball probabilities and large deviations for grey Brownian motion","authors":"Stefan Gerhold","doi":"10.1214/23-ecp555","DOIUrl":"https://doi.org/10.1214/23-ecp555","url":null,"abstract":"We show that the uniform norm of generalized grey Brownian motion over the unit interval has an analytic density, excluding the special case of fractional Brownian motion. Our main result is an asymptotic expansion for the small ball probability of generalized grey Brownian motion, which extends to other norms on path space. The decay rate is not exponential but polynomial, of degree two. For the uniform norm and the Hölder norm, we also prove a large deviations estimate.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135704906","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}
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":" ","pages":""},"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":" ","pages":""},"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":"295 1","pages":"0"},"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":"60 1","pages":"0"},"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":"1 1","pages":"0"},"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}
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":"2 1","pages":"0"},"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}
{"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":"74 1","pages":"0"},"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}
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