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Critical exponents of correlated percolation of sites not visited by a random walk
- Source :
- Phys. Rev. E 110, 024116 (2024)
- Publication Year :
- 2024
-
Abstract
- 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 $\beta$, which controls the strength of the spanning cluster, and $\gamma$, 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 $\beta$ from the $L$-dependence of $M_2$ and $\gamma$ from the finite size scaling of $S$. The resulting exponent $\beta$ remains close to 1 in all dimensions. The exponent $\gamma$ decreases from $\approx 3.9$ in $d=3$ to $\approx1.9$ in $d=4$ and $\approx 1.3$ in $d=5$ towards $\gamma=1$ expected in $d=6$, which is close to $\gamma=4/(d-2)$.<br />Comment: LaTeX, 8 figures, 9 pages
- Subjects :
- Condensed Matter - Statistical Mechanics
Subjects
Details
- Database :
- arXiv
- Journal :
- Phys. Rev. E 110, 024116 (2024)
- Publication Type :
- Report
- Accession number :
- edsarx.2405.14950
- Document Type :
- Working Paper
- Full Text :
- https://doi.org/10.1103/PhysRevE.110.024116