What is known about the time constant in first passage percolation?
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The time constant in first passage percolation has been extensively studied in various contexts. In the critical case of first passage percolation on the triangular lattice, it has been shown that the time constant exists and its exact value is given by limn→∞ T(0,∂B(n)) logn=I 2 3πalmost surely, where I=inf{x>0:F(x)>1/2} . In the standard model of first passage percolation on Zd, it has been proven that the time constant converges to a constant με(z) as ε approaches 0, with a specific growth rate depending on the number of nonnull coordinates of z . Additionally, in Gromov-hyperbolic groups, it has been shown that the time constant exists and is almost surely constant in every direction, with the variance of the first passage time growing linearly with word distance . Ashdown-Lecointre studied the maximal edge-traversal time on optimal paths in first passage percolation on Zd and determined its growth rate based on the tail of the edge distribution .
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| Papers (4) | Insight |
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| The paper discusses the regularity of the time constant in first passage percolation, but it does not provide specific information about what is known about the time constant. | |
| The paper discusses the time constant in first-passage percolation and states that it is unbounded when the edge distribution has unbounded support. | |
01 Dec 2022 | The paper investigates the behavior of the time constant in first-passage percolation and provides a formula for the time constant in terms of the parameter ε and the number of nonnull coordinates of z. |
| The paper shows that there exists a "time constant" in first-passage percolation and its exact value is given by limn→∞ T(0,∂B(n)) logn=I 2 3πalmost surely, where I=inf{x>0:F(x)>1/2}. |
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