Unified scaling for the optimal path length in disordered lattices.

In recent decades, much attention has been focused on the topic of optimal paths in weighted networks due to its broad scientific interest and technological applications. In this work we revisit the problem of the optimal path between two points and focus on the role of the geometry (size and shape) of the embedding lattice, which has received very little attention. This role becomes crucial, for example, in the strong disorder (SD) limit, where the mean length of the optimal path (ℓ[over ¯]_{opt}) for a fixed end-to-end distance r diverges as the lattice size L increases. We propose a unified scaling ansatz for ℓ[over ¯]_{opt} in D-dimensional disordered lattices. Our ansatz introduces two exponents, φ and χ, which respectively characterize the scaling of ℓ[over ¯]_{opt} with r for fixed L, and the scaling of ℓ[over ¯]_{opt} with L for fixed r, both in the SD limit. The ansatz is supported by a comprehensive numerical study of the problem on 2D lattices, yet we also present results in D=3. We show that it unifies well-known results in the strong and weak disorder regimes, including the crossover behavior, but it also reveals novel scaling scenarios not yet addressed. Moreover, it provides relevant insights into the origin of the universal exponents characterizing the scaling of the optimal path in the SD limit. For example, for the fractal dimension of the optimal path in the SD limit, d_{opt}, we find d_{opt}=φ+χ.

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

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 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.

First-passage percolation under extreme disorder: From bond percolation to Kardar-Parisi-Zhang universality.

We consider the statistical properties of arrival times and balls on first-passage percolation (FPP) two-dimensional square lattices with strong disorder in the link times. A previous work showed a crossover in the weak disorder regime, between Gaussian and Kardar-Parisi-Zhang (KPZ) universality, with the crossover length decreasing as the noise amplitude grows. On the other hand, this work presents a very different behavior in the strong-disorder regime. An alternative crossover length appears below which the model is described by bond-percolation universality class. This characteristic length scale grows with the noise amplitude and diverges at the infinite-disorder limit. We provide a thorough characterization of the bond-percolation phase, reproducing its associated critical exponents through a careful scaling analysis of the balls, which is carried out through a continuous mapping of the FPP passage time into the occupation probability of the bond-percolation problem. Moreover, the crossover length can be explained merely in terms of properties of the link-time distribution. The interplay between the characteristic length and the correlation length intrinsic to bond percolation determines the crossover between the initial percolation-like growth and the asymptotic KPZ scaling.