Aspects of a randomly growing cluster in $\reals^d,d\geq 2
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2025
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915092910571520 |
|---|---|
| author | Frieze, Alan Kannan, Ravi Pegden, Wesley |
| author_facet | Frieze, Alan Kannan, Ravi Pegden, Wesley |
| contents | We consider a simple model of a growing cluster of points in $\Re^d,d\geq 2$. Beginning with a point $X_1$ located at the origin, we generate a random sequence of points $X_1,X_2,\ldots,X_i,\ldots,$. To generate $X_{i},i\geq 2$ we choose a uniform integer $j$ in $[i-1]=\{1,2,\ldots,i-1\}$ and then let $X_{i}=X_j+D_i$ where $D_i=(δ_1,\ldots,δ_d)$. Here the $δ_j$ are independent copies of the Normal distribution $N(0,σ_i)$, where $σ_i=i^{-α}$ for some $α>0$. We prove that for any $α>0$ the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a $β$-dimensional subset of $\Re^d$ from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where $β=\min(d,1/α)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03359 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Aspects of a randomly growing cluster in $\reals^d,d\geq 2 Frieze, Alan Kannan, Ravi Pegden, Wesley Probability We consider a simple model of a growing cluster of points in $\Re^d,d\geq 2$. Beginning with a point $X_1$ located at the origin, we generate a random sequence of points $X_1,X_2,\ldots,X_i,\ldots,$. To generate $X_{i},i\geq 2$ we choose a uniform integer $j$ in $[i-1]=\{1,2,\ldots,i-1\}$ and then let $X_{i}=X_j+D_i$ where $D_i=(δ_1,\ldots,δ_d)$. Here the $δ_j$ are independent copies of the Normal distribution $N(0,σ_i)$, where $σ_i=i^{-α}$ for some $α>0$. We prove that for any $α>0$ the resulting point set is bounded a.s., and moreover, that the points generated look like samples from a $β$-dimensional subset of $\Re^d$ from the standpoint of the minimum lengths of combinatorial structures on the point-sets, where $β=\min(d,1/α)$. |
| title | Aspects of a randomly growing cluster in $\reals^d,d\geq 2 |
| topic | Probability |
| url | https://arxiv.org/abs/2501.03359 |