{"title":"On the Radial Growth of Ballistic Aggregation and Other Aggregation Models","authors":"Tillmann Bosch, Steffen Winter","doi":"10.1007/s10955-024-03256-1","DOIUrl":null,"url":null,"abstract":"<p>For a class of aggregation models on the integer lattice <span>\\({{\\mathbb {Z}}}^d\\)</span>, <span>\\(d\\ge 2\\)</span>, in which clusters are formed by particles arriving one after the other and sticking irreversibly where they first hit the cluster, including the classical model of diffusion-limited aggregation (DLA), we study the growth of the clusters. We observe that a method of Kesten used to obtain an almost sure upper bound on the radial growth in the DLA model generalizes to a large class of such models. We use it in particular to prove such a bound for the so-called ballistic model, in which the arriving particles travel along straight lines. Our bound implies that the fractal dimension of ballistic aggregation clusters in <span>\\({{\\mathbb {Z}}}^2\\)</span> is 2, which proves a long standing conjecture in the physics literature.</p>","PeriodicalId":667,"journal":{"name":"Journal of Statistical Physics","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Statistical Physics","FirstCategoryId":"101","ListUrlMain":"https://doi.org/10.1007/s10955-024-03256-1","RegionNum":3,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"PHYSICS, MATHEMATICAL","Score":null,"Total":0}
引用次数: 0
Abstract
For a class of aggregation models on the integer lattice \({{\mathbb {Z}}}^d\), \(d\ge 2\), in which clusters are formed by particles arriving one after the other and sticking irreversibly where they first hit the cluster, including the classical model of diffusion-limited aggregation (DLA), we study the growth of the clusters. We observe that a method of Kesten used to obtain an almost sure upper bound on the radial growth in the DLA model generalizes to a large class of such models. We use it in particular to prove such a bound for the so-called ballistic model, in which the arriving particles travel along straight lines. Our bound implies that the fractal dimension of ballistic aggregation clusters in \({{\mathbb {Z}}}^2\) is 2, which proves a long standing conjecture in the physics literature.
期刊介绍:
The Journal of Statistical Physics publishes original and invited review papers in all areas of statistical physics as well as in related fields concerned with collective phenomena in physical systems.