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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2405.14950 |
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| _version_ | 1866914917419843584 |
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| author | Levi, Raz Halifa Kantor, Yacov |
| author_facet | Levi, Raz Halifa Kantor, Yacov |
| contents | We consider a $d$-dimensional correlated percolation problem of sites {\em not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5. The length of the random walk is ${\cal N}=uL^d$. Close to the critical value $u=u_c$, many geometrical properties of the problem can be described as powers (critical exponents) of $u_c-u$, such as $β$, which controls the strength of the spanning cluster, and $γ$, which characterizes the behavior of the mean finite cluster size $S$. We show that at $u_c$ the ratio between the mean mass of the largest cluster $M_1$ and the mass of the second largest cluster $M_2$ is independent of $L$ and can be used to find $u_c$. We calculate $β$ from the $L$-dependence of $M_2$ and $γ$ from the finite size scaling of $S$. The resulting exponent $β$ remains close to 1 in all dimensions. The exponent $γ$ decreases from $\approx 3.9$ in $d=3$ to $\approx1.9$ in $d=4$ and $\approx 1.3$ in $d=5$ towards $γ=1$ expected in $d=6$, which is close to $γ=4/(d-2)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_14950 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Critical exponents of correlated percolation of sites not visited by a random walk Levi, Raz Halifa Kantor, Yacov Statistical Mechanics We consider a $d$-dimensional correlated percolation problem of sites {\em not} visited by a random walk on a hypercubic lattice $L^d$ for $d=3$, 4 and 5. The length of the random walk is ${\cal N}=uL^d$. Close to the critical value $u=u_c$, many geometrical properties of the problem can be described as powers (critical exponents) of $u_c-u$, such as $β$, which controls the strength of the spanning cluster, and $γ$, which characterizes the behavior of the mean finite cluster size $S$. We show that at $u_c$ the ratio between the mean mass of the largest cluster $M_1$ and the mass of the second largest cluster $M_2$ is independent of $L$ and can be used to find $u_c$. We calculate $β$ from the $L$-dependence of $M_2$ and $γ$ from the finite size scaling of $S$. The resulting exponent $β$ remains close to 1 in all dimensions. The exponent $γ$ decreases from $\approx 3.9$ in $d=3$ to $\approx1.9$ in $d=4$ and $\approx 1.3$ in $d=5$ towards $γ=1$ expected in $d=6$, which is close to $γ=4/(d-2)$. |
| title | Critical exponents of correlated percolation of sites not visited by a random walk |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2405.14950 |