Branching random walk in the presence of a hard wall
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2018
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| _version_ | 1866909115912028160 |
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| author | Roy, Rishideep |
| author_facet | Roy, Rishideep |
| contents | We consider a branching random walk on a $d$-ary tree of height $n$ ($n \in \mathbb{N}$), under the presence of a hard wall which restricts each value to be positive, where $d$ is a natural number satisfying $d\geqslant2$. The question of behaviour of Gaussian processes with long range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of {\color{blue}vertices}, and a relation with the expected maximum of the processes has been observed. We find the probability of the event that the branching random {\color{blue}walk} is positive at every vertex in the $n^{th}$ generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1806_02565 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Branching random walk in the presence of a hard wall Roy, Rishideep Probability 60J80, 60G15, 60G70 We consider a branching random walk on a $d$-ary tree of height $n$ ($n \in \mathbb{N}$), under the presence of a hard wall which restricts each value to be positive, where $d$ is a natural number satisfying $d\geqslant2$. The question of behaviour of Gaussian processes with long range interactions, for example the discrete Gaussian free field, under the condition that it is positive on a large subset of {\color{blue}vertices}, and a relation with the expected maximum of the processes has been observed. We find the probability of the event that the branching random {\color{blue}walk} is positive at every vertex in the $n^{th}$ generation, and show that the conditional expectation of the Gaussian variable at a typical vertex, under positivity, is less than the expected maximum by order of $\log n$. |
| title | Branching random walk in the presence of a hard wall |
| topic | Probability 60J80, 60G15, 60G70 |
| url | https://arxiv.org/abs/1806.02565 |