Fluctuations of the Occupation Density for a Parking Process

Autor(es)

Coletti, Cristian F. ; Gallo, Sandro ; Roldan-Correa, Alejandro ; Valencia, Leon A.

• Tipo de documento: Artigo Científico
• Fonte: Journal of Statistical Physics; v. 191, n. 11, p. 17-pg., 2024-10-30.

Resumo

Consider the following simple parking process on Lambda n:={-n,& mldr;,n}d,d >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _n:= \{-n, \ldots , n\}<^>d,d\ge 1$$\end{document}: at each step, a site i is chosen at random in Lambda n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _n$$\end{document} and if i and all its nearest neighbor sites are empty, i is occupied. Once occupied, a site remains so forever. The process continues until all sites in Lambda n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _n$$\end{document} are either occupied or have at least one of their nearest neighbors occupied. The final configuration (occupancy) of Lambda n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _n$$\end{document} is called the jamming limit and is denoted by X Lambda n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\Lambda _n}$$\end{document}. Ritchie (J Stat Phys 122:381-398, 2006) constructed a stationary random field on Zd\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Z}<^>d$$\end{document} obtained as a (thermodynamic) limit of the X Lambda n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\Lambda _n}$$\end{document}'s as n tends to infinity. As a consequence of his construction, he proved a strong law of large numbers for the proportion of occupied sites in the box Lambda n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda _n$$\end{document} for the random field X. Here we prove the central limit theorem, the law of iterated logarithm, and a gaussian concentration inequality for the same statistics. A particular attention will be given to the case d=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d=1$$\end{document}, in which we also obtain new asymptotic properties for the sequence X Lambda n,n >= 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_{\Lambda _n},n\ge 1$$\end{document}. (AU)

• Processo FAPESP: 22/08948-2 - Uma abordagem probabilística e algébrica para dinâmicas evolutivas
• Beneficiário: Cristian Favio Coletti
• Modalidade de apoio: Auxílio à Pesquisa - Regular