Domain walls, optimal droplets and disorder chaos at zero temperature are studied numerically for the solid-on-solid model on a random substrate. It is shown that the ensemble of random curves represented by the domain walls obeys Schramm’s left passage formula with κ = 4 whereas their fractal dimension is ds = 1.25, and therefore their behavior cannot be described as showing ‘Schramm– (or stochastic) Loewner evolution’ (SLE). Optimal droplets with a lateral size between L and 2L have the same fractal dimension as domain walls but an energy that saturates at a value of order for such that arbitrarily large excitations exist which cost only a small amount of energy. Finally it is demonstrated that the sensitivity of the ground state to small changes of order δ in the disorder is subtle: beyond a crossover length scale Lδ∼δ−1 the correlations of the perturbed ground state with the unperturbed ground state, rescaled using the roughness, are suppressed and approach zero logarithmically.

We consider random search processes alternating stochastically between diffusion and ballistic motion, in which the distribution function of ballistic motion directions varies from point to point in space. The specific space dependence of the directional distribution together with the switching rates between the two modes of motion establishes a spatially inhomogeneous search strategy. We show that the mean first passage times for several standard search problems-narrow escape, reaction partner finding, reaction escape-can be minimized with a directional distribution that is reminiscent of the spatial organization of the cytoskeleton filaments of cells with a centrosome: radial ballistic transport from the center to the periphery and back, and ballistic transport in random directions within a concentric shell of thickness Δ_{opt} along the domain boundary. The results suggest that living cells realize efficient search strategies for various intracellular transport problems economically through a spatial cytoskeleton organization that involves radial microtubules in the central region and only a narrow actin cortex rather than a cell body filled with randomly oriented actin filaments.

Random search processes for targets that are inhomogeneously distributed in a search domain require spatially inhomogeneous search strategies to find the target as fast as possible. Here, we compare systematically the efficiency of homogeneous and inhomogeneous strategies for intermittent search, which alternates stochastically between slow, diffusive motion in which the target can be detected and fast ballistic motion during which targets cannot be detected. We analyze the mean first-passage time of homogeneous and inhomogeneous strategies for three paradigmatic search problems: (1) the narrow escape problem, i.e., the searcher looks for a small area on the boundary of the search domain, (2) reaction kinetics, i.e., the detection of an immobile target in the interior of a search domain, and (3) the reaction-escape problem, i.e., the searcher first needs to find a mobile target before it can escape through a narrow area on the boundary. Using families of inhomogeneous strategies, partially motivated by the organization of the cytoskeleton in cells with a centrosome, we show that they are almost always more efficient than homogeneous strategies.

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We present an efficient Monte Carlo method to simulate reaction-diffusion processes with spatially varying particle annihilation or transformation rates as it occurs for instance in the context of motor-driven intracellular transport. Like Green's function reaction dynamics and first-passage time methods, our algorithm avoids small diffusive hops by propagating sufficiently distant particles in large hops to the boundaries of protective domains. Since for spatially varying annihilation or transformation rates the single particle diffusion propagator is not known analytically, we present an algorithm that generates efficiently either particle displacements or annihilations with the correct statistics, as we prove rigorously. The numerical efficiency of the algorithm is demonstrated with an illustrative example.