Robert M. Ziff

Bio: Robert M. Ziff is an academic researcher from University of Michigan. The author has contributed to research in topics: Percolation threshold & Percolation. The author has an hindex of 54, co-authored 247 publications receiving 10665 citations. Previous affiliations of Robert M. Ziff include State University of New York System & Rockefeller University.

Kinetic phase transitions in an irreversible surface-reaction model.

Abstract: An irreversible kinetic surface-reaction model, based upon the reaction of carbon monoxide and oxygen on a catalyst surface, is presented. It is found by computer simulation that the adsorbed molecules on the surface undergo both first- and second-order kinetic phase transitions. These transitions correspond to the "poisoning" phenomenon seen on catalysts. Interesting transient and periodic behavior is also seen.

Fast Monte Carlo algorithm for site or bond percolation.

Abstract: We describe in detail an efficient algorithm for studying site or bond percolation on any lattice. The algorithm can measure an observable quantity in a percolation system for all values of the site or bond occupation probability from zero to one in an amount of time that scales linearly with the size of the system. We demonstrate our algorithm by using it to investigate a number of issues in percolation theory, including the position of the percolation transition for site percolation on the square lattice, the stretched exponential behavior of spanning probabilities away from the critical point, and the size of the giant component for site percolation on random graphs.

Efficient Monte Carlo algorithm and high-precision results for percolation.

Abstract: We present a new Monte Carlo algorithm for studying site or bond percolation on any lattice. The algorithm allows us to calculate quantities such as the cluster size distribution or spanning probability over the entire range of site or bond occupation probabilities from zero to one in a single run which takes an amount of time scaling linearly with the number of sites on the lattice. We use our algorithm to determine that the percolation transition occurs at p(c) = 0.592 746 21(13) for site percolation on the square lattice and to provide clear numerical confirmation of the conjectured 4/3-power stretched-exponential tails in the spanning probability functions.

Precise determination of the bond percolation thresholds and finite-size scaling corrections for the sc, fcc, and bcc lattices

Abstract: Extensive Monte Carlo simulations were performed to study bond percolation on the simple cubic (sc), face-centered-cubic (fcc), and body-centered-cubic (bcc) lattices, using an epidemic approach. These simulations provide very precise values of the critical thresholds for each of the lattices: ${p}_{c}(\mathrm{sc})=0.2488126\ifmmode\pm\else\textpm\fi{}0.0000005,$ ${p}_{c}(\mathrm{fcc})=0.1201635\ifmmode\pm\else\textpm\fi{}0.0000010,$ and ${p}_{c}(\mathrm{bcc})=0.1802875\ifmmode\pm\else\textpm\fi{}0.0000010$. For $p$ close to ${p}_{c},$ the results follow the expected finite-size and scaling behavior, with values for the Fisher exponent \ensuremath{\tau} $(2.189\ifmmode\pm\else\textpm\fi{}0.002),$ the finite-size correction exponent \ensuremath{\Omega} $(0.64\ifmmode\pm\else\textpm\fi{}0.02),$ and the scaling function exponent \ensuremath{\sigma} $(0.445\ifmmode\pm\else\textpm\fi{}0.01)$ confirmed to be universal.

The kinetics of cluster fragmentation and depolymerisation

Abstract: The equation that describes fragmentation kinetics, such as that which occurs in cluster breakup and polymer chain degradation (depolymerisation), is solved for models where the rate of breakup depends upon the size of the object breaking up. The resulting dynamic scaling behaviour is investigated. Both discrete and continuous models are considered.

Catastrophic cascade of failures in interdependent networks

Abstract: Complex networks have been studied intensively for a decade, but research still focuses on the limited case of a single, non-interacting network. Modern systems are coupled together and therefore should be modelled as interdependent networks. A fundamental property of interdependent networks is that failure of nodes in one network may lead to failure of dependent nodes in other networks. This may happen recursively and can lead to a cascade of failures. In fact, a failure of a very small fraction of nodes in one network may lead to the complete fragmentation of a system of several interdependent networks. A dramatic real-world example of a cascade of failures ('concurrent malfunction') is the electrical blackout that affected much of Italy on 28 September 2003: the shutdown of power stations directly led to the failure of nodes in the Internet communication network, which in turn caused further breakdown of power stations. Here we develop a framework for understanding the robustness of interacting networks subject to such cascading failures. We present exact analytical solutions for the critical fraction of nodes that, on removal, will lead to a failure cascade and to a complete fragmentation of two interdependent networks. Surprisingly, a broader degree distribution increases the vulnerability of interdependent networks to random failure, which is opposite to how a single network behaves. Our findings highlight the need to consider interdependent network properties in designing robust networks.

Stochastic Processes in Physics and Chemistry

Abstract: N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

Sampling-based algorithms for optimal motion planning

Abstract: During the last decade, sampling-based path planning algorithms, such as probabilistic roadmaps (PRM) and rapidly exploring random trees (RRT), have been shown to work well in practice and possess theoretical guarantees such as probabilistic completeness. However, little effort has been devoted to the formal analysis of the quality of the solution returned by such algorithms, e.g. as a function of the number of samples. The purpose of this paper is to fill this gap, by rigorously analyzing the asymptotic behavior of the cost of the solution returned by stochastic sampling-based algorithms as the number of samples increases. A number of negative results are provided, characterizing existing algorithms, e.g. showing that, under mild technical conditions, the cost of the solution returned by broadly used sampling-based algorithms converges almost surely to a non-optimal value. The main contribution of the paper is the introduction of new algorithms, namely, PRM* and RRT*, which are provably asymptotically optimal, i.e. such that the cost of the returned solution converges almost surely to the optimum. Moreover, it is shown that the computational complexity of the new algorithms is within a constant factor of that of their probabilistically complete (but not asymptotically optimal) counterparts. The analysis in this paper hinges on novel connections between stochastic sampling-based path planning algorithms and the theory of random geometric graphs.

Long-Circulating and Target-Specific Nanoparticles: Theory to Practice

Abstract: The rapid recognition of intravenously injected colloidal carriers, such as liposomes and polymeric nanospheres from the blood by Kupffer cells, has initiated a surge of development for "Kupffer cell-evading" or long-circulating particles. Such carriers have applications in vascular drug delivery and release, site-specific targeting (passive as well as active targeting), as well as transfusion medicine. In this article we have critically reviewed and assessed the rational approaches in the design as well as the biological performance of such constructs. For engineering and design of long-circulating carriers, we have taken a lead from nature. Here, we have explored the surface mechanisms, which affords red blood cells long-circulatory lives and the ability of specific microorganisms to evade macrophage recognition. Our analysis is then centered where such strategies have been translated and fabricated to design a wide range of particulate carriers (e.g., nanospheres, liposomes, micelles, oil-in-water emulsions) with prolonged circulation and/or target specificity. With regard to the targeting issues, attention is particularly focused on the importance of physiological barriers and disease states.