Random parking, Euclidean functionals, and ...
Document type :
Article dans une revue scientifique: Article original
Permalink :
Title :
Random parking, Euclidean functionals, and rubber elasticity
Author(s) :
Gloria, Antoine [Auteur]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Penrose, Mathew [Auteur]
Department of Mathematical Sciences [Bath]
SImulations and Modeling for PArticles and Fluids [SIMPAF]
Penrose, Mathew [Auteur]
Department of Mathematical Sciences [Bath]
Journal title :
Communications in Mathematical Physics
Pages :
31
Publisher :
Springer Verlag
Publication date :
2013
ISSN :
0010-3616
English keyword(s) :
thermodynamic limit
random parking
subadditive ergodic theorem
Euclidean optimization problems
stochastic homogenization
polymer-chain networks
thermodynamic limit.
random parking
subadditive ergodic theorem
Euclidean optimization problems
stochastic homogenization
polymer-chain networks
thermodynamic limit.
HAL domain(s) :
Mathématiques [math]/Equations aux dérivées partielles [math.AP]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Physique mathématique [math-ph]
Mathématiques [math]/Probabilités [math.PR]
English abstract : [en]
We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\R^d$ and of point sets that are (almost) subadditive in ...
Show more >We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\R^d$ and of point sets that are (almost) subadditive in their first variable. Denoting by $\xi$ the random parking measure in $\R^d$, and by $\xi^R$ the random parking measure in the cube $Q_R=(-R,R)^d$, we show, under some natural assumptions on $S$, that there exists a constant $\overline{S}\in \R$ such that % $$ \lim_{R\to +\infty} \frac{S(Q_R,\xi)}{|Q_R|}\,=\,\lim_{R\to +\infty}\frac{S(Q_R,\xi^R)}{|Q_R|}\,=\,\overline{S} $$ % almost surely. If $\zeta \mapsto S(Q_R,\zeta)$ is the counting measure of $\zeta$ in $Q_R$, then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.Show less >
Show more >We study subadditive functions of the random parking model previously analyzed by the second author. In particular, we consider local functions $S$ of subsets of $\R^d$ and of point sets that are (almost) subadditive in their first variable. Denoting by $\xi$ the random parking measure in $\R^d$, and by $\xi^R$ the random parking measure in the cube $Q_R=(-R,R)^d$, we show, under some natural assumptions on $S$, that there exists a constant $\overline{S}\in \R$ such that % $$ \lim_{R\to +\infty} \frac{S(Q_R,\xi)}{|Q_R|}\,=\,\lim_{R\to +\infty}\frac{S(Q_R,\xi^R)}{|Q_R|}\,=\,\overline{S} $$ % almost surely. If $\zeta \mapsto S(Q_R,\zeta)$ is the counting measure of $\zeta$ in $Q_R$, then we retrieve the result by the second author on the existence of the jamming limit. The present work generalizes this result to a wide class of (almost) subadditive functions. In particular, classical Euclidean optimization problems as well as the discrete model for rubber previously studied by Alicandro, Cicalese, and the first author enter this class of functions. In the case of rubber elasticity, this yields an approximation result for the continuous energy density associated with the discrete model at the thermodynamic limit, as well as a generalization to stochastic networks generated on bounded sets.Show less >
Language :
Anglais
Peer reviewed article :
Oui
Audience :
Internationale
Popular science :
Non
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Source :
Submission date :
2025-01-24T10:36:13Z
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