A multivariate extension of the Erdös-Taylor theorem
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866911773962010624 |
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| author | Lygkonis, Dimitris Zygouras, Nikos |
| author_facet | Lygkonis, Dimitris Zygouras, Nikos |
| contents | The Erdös-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfracπ{\log N} \,\mathsf{L}_N$ converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if $\mathsf{L}_N^{(1,2)}=\sum_{n=1}^N 1_{\{S_n^{(1)}= S_n^{(2)}\}}$, then $\tfracπ{\log N}\, \mathsf{L}^{(1,2)}_N$ converges in distribution to an exponential random variable of parameter one. We prove that for every $h \geq 3$, the family $ \big\{ \fracπ{\log N} \,\mathsf{L}_N^{(i,j)} \big\}_{1\leq i<j\leq h}$, of logarithmically rescaled, two-body collision local times between $h$ independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdös-Taylor theorem. We also discuss connections to directed polymers in random environments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_08145 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A multivariate extension of the Erdös-Taylor theorem Lygkonis, Dimitris Zygouras, Nikos Probability Mathematical Physics The Erdös-Taylor theorem [Acta Math. Acad. Sci. Hungar, 1960] states that if $\mathsf{L}_N$ is the local time at zero, up to time $2N$, of a two-dimensional simple, symmetric random walk, then $\tfracπ{\log N} \,\mathsf{L}_N$ converges in distribution to an exponential random variable with parameter one. This can be equivalently stated in terms of the total collision time of two independent simple random walks on the plane. More precisely, if $\mathsf{L}_N^{(1,2)}=\sum_{n=1}^N 1_{\{S_n^{(1)}= S_n^{(2)}\}}$, then $\tfracπ{\log N}\, \mathsf{L}^{(1,2)}_N$ converges in distribution to an exponential random variable of parameter one. We prove that for every $h \geq 3$, the family $ \big\{ \fracπ{\log N} \,\mathsf{L}_N^{(i,j)} \big\}_{1\leq i<j\leq h}$, of logarithmically rescaled, two-body collision local times between $h$ independent simple, symmetric random walks on the plane converges jointly to a vector of independent exponential random variables with parameter one, thus providing a multivariate version of the Erdös-Taylor theorem. We also discuss connections to directed polymers in random environments. |
| title | A multivariate extension of the Erdös-Taylor theorem |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2202.08145 |