Shape effects in the fluctuations of random isochrones on a square lattice

Abstract

We consider the isochrone curves in first-passage percolation on a 2D square lattice, i.e. the boundary of the set of points which can be reached in less than a given time from a certain origin. The occurrence of an instantaneous average shape is described in terms of its Fourier components, highlighting a crossover between a diamond and a circular geometry as the noise level is increased. Generally, these isochrones can be understood as fluctuating interfaces with an inhomogeneous local width which reveals the underlying lattice structure. We show that once these inhomogeneities have been taken into account, the fluctuations fall into the Kardar-Parisi-Zhang (KPZ) universality class with very good accuracy, where they reproduce the Family-Vicsek Ansatz with the expected exponents and the Tracy-Widom histogram for the local radial fluctuations.

Figures (10)

• Figure 1: Random balls $B(t)$ for an FPP system on a square lattice, centered on the origin of coordinates for different times. (a) Balls obtained for a uniform link-time distribution using $\mu=5$ and $\text{CV}=0.57$. (b) Balls obtained for a uniform link-time distribution using $\mu=5$ and $\text{CV}=0.11$. Colors are changed every $\Delta t=100$, and the total lattice size is $500\times 500$ for panel (a) and $250\times 250$ for panel (b).
• Figure 2: (a) Time evolution of the scaled IAS, $\rho(\theta,t)$, for two uniform link-time distributions, using $\text{CV}=0.57$ and $\text{CV}=0.11$. (b) Limit shapes for different link-time distributions, all of them with expected value $\mu=5$. The dashed line represents the circumference, and the perfect diamond shape, given by Eq. \ref{['eq:diamond']}, is shown with a red continuous line.
• Figure 3: Fourier decomposition of the limit shape, $a_n$ for the uniform and Weibull link-time distributions, with $\mu=5$, employing $\text{CV}=0.57$ in panel (a) and $\text{CV}=0.11$ in panel (b). Insets: Time-evolution of some selected Fourier coefficients of the scaled IAS, $a_n(t)$, using both $\text{CV}=0.57$ and $\text{CV}=0.11$ for the uniform and Weibull link-time distributions.
• Figure 4: Deviation of the interfacial radii of the isochrones, $\sigma_r$ (symbols) and of the IAS radii, $\sigma_R$ (solid lines) for several uniform distributions. The dashed line shows a power law behavior with exponent $\beta=1$.
• Figure 5: Time evolution of the roughness $W(t)$, defined with respect to the IAS, as shown in Eq. \ref{['eq:true_W']}, for uniform distributions. Dashed lines indicate power-law behavior with exponent $1/3$.